ENT 171

Week 1  Ohm's Law: voltage, current, resistance

Ohm's Law

The most important principle in electricity/electronics

Purchase the three required books for the course (see Books). Read Chapter one in the textbook (several times), and complete the homework set for chapter one. I will collect these problems next week. Read through the first laboratory experiment (for next week). Consult the course SYLLABUS- Course Documents.

"Must Remember" Summary: I can't stress enough how important this topic is. As an electronics engineering and/or a quality assurance associate, you will use Ohm's Law many times each day to test and troubleshoot components, circuits, and systems. You must fully understand the CONCEPTS of Ohm's Law- not just crunching numbers. The relationship between the BIG THREE- Voltage, Current and Resistance is paramount to your success in your career! Consider the following everyday examples:

• Why does a circuit continually trip the circuit breaker?  (low resistance- due to overloading- is causing excessive current to flow)
• How do I divide a given voltage or current? (add resistance in series, parallel)
• How can I prevent electrical shock hazards and improve safety? (reduce the electrical resistance from the equipment chassis to ground)
• Why is an important test point voltage on a PCB reading lower than it should ? (measuring equipment loading is occurring during measurement)

Hopefully you are fully motivated. Lets get started.

As presented in your textbook, Ohm's Law is explained by using the simple mechanical analogy of a sealed-water system. The electrical concept of Voltage is like the mechanical force of a pump that imparts energy to water (representing atomic electrons) flowing (Current) in the sealed leak-less pipes (circuit devices and wiring). Energy is lost in our system (in the form of heat loss) due to pipe restrictions/bends, and internal corrosion/scaling (Resistance). The energy supplied by the pump forces the water to flow through the elements of the sealed pipe system (Circuit), where energy supplied by the pump = energy delivered and consumed by the pipe elements (Voltage Rise =  Voltage Dropped). This is conservation of energy. As you increase the pump's output power, the water flow rate increases in direct proportion. If the pipe's bends and restrictions are caused to increase, this will cause additional resistance to the flow of water and the flow rate will diminish. Ohm's Law deals with the relationship between voltage, current, and resistance in every electrical device and system/sub-system/circuit of connected devices. The potential difference (or voltage, V) in energy across an element is proportional to the current (I) through the element's resistance (R). Batteries and power supplies ADD energy to the electrical system, and electrical devices (resistors) consume that energy. A power supply (voltage source) will only furnish the power demanded by the circuit/system that it is part of. Ohm's Law is given by: V = I R, I = V/R, and  R = V/I  where V is the potential difference between two points (Voltage) which includes a resistance R. I is the resulting current flowing through the circuit resistance. You must have V to cause I to occur. Ohm's Law can be applied to individual devices (one at a time), and to the entire circuit (all devices at once). Total Voltage, Total Current, and Total Resistance refer to the entire circuit. We measure: voltage (V)  in UNITS of volts (v), current (I) in UNITS of amps (a), and resistance in UNITS of ohms ( ).

Matter and Atoms- mini solar systems

The term "atom" comes from the Greek "atomos" which is defined as the smallest, indivisible piece of matter. Current theory describes the atom as an electron cloud model. This model is depicted with a nucleus containing both positively charged particles called "protons" and neutrally charged particles called "neutrons." Orbiting the nucleus at a high rate of speed are negatively charged particles called "electrons."   Protons and neutrons are heavier than electrons and reside in the nucleus, which is the center of the atom. Protons have a positive electrical charge, and neutrons have no electrical charge. Electrons are extremely lightweight and are negatively charged. Like charges repel and opposites attract. Electrons orbit the atom and are electrically attracted to the proton (+) charge. The electron orbit at a radius 10,000 times greater than that of the nucleus.

HISTORY: The first subatomic particle to be identified was the electron, in 1898. Ten years later, Ernest Rutherford discovered that atoms have a very dense nucleus, which contains protons. In 1932, James Chadwick discovered the neutron, another particle located within the nucleus. Rutherford performed early experiments of shooting alpha particles (helium nuclei) at sheets of gold to show that atoms were, in fact, mostly empty space. Some of the alpha particles passed through the foil as expected, but some particles bounced back. Alpha particles carry positive charge. This meant that there was a small concentration of positive charges in the atom. This model is similar to a solar system where the nucleus is the sun and the electrons orbit the nucleus similar to the planetary orbits. The solid behavior of atoms is due to the electromagnetic repulsion of the electrons in the outer orbits. When you press your hand on a table, the solidness you feel is due to the electrons from the atoms of your hand pushing away the electrons of the atoms of the table (electrostatic repulsion).

The Atomic Nucleus

The nucleus of an atom is made up of  protons, and a similar number of neutrons. It is held together by the tight cohesive force known as the strong nuclear force. This force between the protons and neutrons overcomes the repulsive electrical force that would, according to the rules of electricity, otherwise push the protons apart. Protons and neutrons are 1,860 times heavier than electrons. Virtually all the mass of the atom resides in the nucleus. The nucleus is only 1/100,000th the diameter of the atom (like the size of a baseball compared to that of a ball park) and yet nearly all the mass of the atom is in that tiny nucleus. There are just over one-hundred elements known to man. Each element may also have several isotopes (different numbers of neutrons), but generally only a few will be stable (not radioactive). Heavy atoms tend to be radioactive. Normally, the number of electrons and protons is the same, and since the electrical charges on electrons and protons is equal but opposite, the atom has no net electrical charge. The number of protons determines which chemical element the atom belongs to.

Electrons

Electrons are negatively charged subatomic particles, and they create what we call "electricity" when they flow, or static electricity when many of them build up in one place, or are taken away. Visit the following weblink for an interesting illustration of static electricity   External energy (electric field) is required to push electrons from one point to another. The electrons have negative electrical charge, and their movement between atoms is responsible for electrical current. They can also be removed from atoms by rubbing different materials together, e.g. by combing your hair. This is static electricity. The electrical charge of protons and electrons in a neutral atom are exactly equal but opposite. Usually there are the same number of protons and electrons in an atom, and their electrical charges cancel each other. When an atom has a surplus of electrons the atom is termed a negative ion, when at a deficit electron(s) it is known as a positive ion.The electron is the lightweight particle that "orbits" outside of the atomic nucleus. Electrons surround the atom in pathways called orbitals. The inner orbitals surrounding the atom tightly bind electrons, the outermost orbitals bind the electrons much more lightly. Conductors (metals- especially copper) have many electrons in the outer orbitals that are almost free to migrate away from the atom (conduction band). Insulators have very few electrons in the conduction band- all the electrons are tightly bound to the nucleus.

In AC and DC circuits containing purely resistive loads ( lights and heaters) Ohm's Law can be used to compute current, voltage and resistance in the circuit. In a resistive DC circuit, both current and voltage are fixed, steady values. In an AC resistive circuit, the current alternates exactly in step with the voltage. In either case, Ohms Law can be applied. In a resistive circuit Ohm's Law states that: voltage is equal to current times resistance. Ohm's Law V = I x R where:
V = voltage (volts)
I = current (amps)
R = resistance (ohms)

Ohm's Law is named after the German physicist George Simon Ohm.

Example:
A wire with a resistance of 10 ohm's is connected to a 9-volt battery. To determine the current flow in the wire, use ohm's law and divide 9 volts by 10 ohms. The current flow in the wire equals 0.9 amperes. Replace the 9-volt battery with a 1.5 volt battery. Using the same wire the calculated current flow is 1.5 volts divided by 10 ohms, which produces a current flow of 0.15 amps. The larger voltage results in a higher potential difference (electrical "pressure") to force more current through the given resistance of 10 ohms.

Key terms:

• Electron (e): A small negative charge of electricity.
• Voltage (V): The electrical potential that causes a current to flow in a closed circuit. Voltage is measured in volts (or millivolts or microvolts) using a voltmeter.
• Current (I): The flow of electrons that results when a voltage is applied to a circuit. This unit is measured in amps (or milliamps, or microamps) with an ammeter.
• Resistance (R): Resistance is the impeding or impedance to current flow; all electrical equipment has an internal resistance—measured in ohms (or kilohms or megohms) with an ohmmeter.
• Direct current (DC): Direct current is an electrical current that flows through a circuit in the same direction at all times with constant amplitude.
• Alternating current (AC): Alternating current is a periodic current that reverses at regularly occurring intervals of time and has alternately positive and negative values.

The lower the circuit resistance, the higher the current flow. Lower resistance also occurs when several electrical devices are connected in parallel. This increase in circuit loading increases the current. Also, when a circuit fault (such as an internal short circuit) occurs, unsafe high values of current can occur. Protective devices such as fuses or circuit breakers prevent this unsafe condition by opening the circuit and preventing current flow. Higher voltages can cause increased current flow. Line voltages in American homes are 120 volts (RMS). When devices are in series, the current is identical at every point in the circuit, and the applied voltage is DIVIDED amongst the circuit resistances. Voltage applied (input) must equal voltage dropped but the resistances.

Consider this basic mechanical analogy: a single pipe carrying water that has connectors in line. The flow of water cannot divide and must be the same around the series loop. The total resistance in a series circuit therefore is the sum of the individual device resistances.  If any of the individual devices fail and the circuit is interrupted, current flow to the other devices is also interrupted.

In parallel circuits, the current divides between the various branches and the total resistance for two devices is found by dividing the product of the individual resistances by their sum. Electrical equipment in commercial or industrial facilities is connected in parallel. Failure of any individual device will not affect the other devices in other branches.

NOTE: In this course voltages will range in value from about 0.5 v to a maximum of 30 v; currents will be small- from about 0.001a (1 milli amp = 1ma) to about 0.1a (100ma); and resistance will range from 10 ohms to 10,000,000 ohms (10 M ohm). If you measure or calculate values outside of these ranges its a good chance that you made an error in arithmetic (excluding some of the lab software simulations). You should learn the three basic multiplication factors immediately: Kilo = K = 1000; Mega = M = 1,000,000; and milli = m = 0.001 (one thousandth). Examples: 1000m = (1000)(0.001) = 1, (10K)(1m) = 10, m x K = 1, 1/m = 1K = K, 1/K = 1m = m

Click on the following web links to view illustrations of current, voltage , and resistance. Use your browser's "Back"  button to return from each link.

A simple complete circuit consists of a closed loop of devices (that offer resistance) connected by low-resistance conductors to each other and to a voltage source (Electro Motive Force = EMF). The sum of the voltages around a complete circuit is zero (energy supplied = energy consumed). An increase of potential energy in a circuit (increased voltage at the source) causes (electron) charge to move from a lower to a higher potential energy level. Resistance is energy consumption in the circuit which drives charge from a higher to a lower potential, and energy is released (in the form of heat).  The electrons that orbit every atom in the universe behave in this manner- add energy and the outer electrons change to a higher orbit radius around the nucleus (positively charged Protons bonded to neutrally charged Neutrons) - remove energy and the electrons drop down to a lower orbit. Click on the following web links to view atomic structure. Use your browser's "Back"  button to return.   The means of adding energy to electrons is through an Electric Force Field (E field). EMF  (generated within a circuit's voltage source) is conducted through a circuit of devices (resistors) that are connected by conductors (metal wires). Insulators are high resistance elements that greatly reduce current flow. Click on the

following web links to view properties of conductors and insulators. Use your browser's "Back"  button to return.

NOTE: The conductors act as a low resistance "conduit" to propagate the flow of electric field energy throughout a circuit (or system of circuits). If there is a break (discontinuity) in any part of the simple single-loop circuit, the E field is prevented from adding energy to the circuit, and NO CURRENT FLOWS. This can be viewed as infinite resistance (electrical OPEN). The opposite is an circuit element of zero resistance (electrical SHORT). A light switch represents both states: acting as an OPEN when the switch is in the off position, and a SHORT when it is in the on position (allowing the energy to pass through the switch to the lamp it controls). Computers use digital switches to create and manipulate data (0, 1 binary states) electronically.

Click on the following web link to take a short fun quiz on atomic structure (4 questions). Then use your browser's "Back"  button (4 times) to return.

Because energy is conserved, the potential difference across the terminals of an EMF (voltage source) must be equal to the potential difference across the rest of the circuit. That is, Ohm's Law will be satisfied: The symbols V and are often used interchangeably. Technically,  represents a voltage rise, and V a voltage drop (however, this distinction has been lost over the years)

Here is a simple simulated experiment  to re-enforce understanding of Ohm's Law. Note that the red box represents the circuit voltage source (EMF), and the gray boxes are meters to measure voltage (voltmeter) and current (ammeter). You are to click on the buttons to increase/decrease the applied voltage and resistance, and observe the resulting current. Does Ohm's Law accurately predict the result? (YES it does!). Use your browser's "Back"  button to return. Press here    to begin the experiment.

Example 1: The Simple Loop Circuit and application of Ohm's Law

Question
An EMF source of 6.0V is connected to a purely resistive lamp and a current of 2.0 amperes flows. All the wires are resistance-free. What is the resistance of the lamp?

Schematic diagram of the circuit in this problem.

1. Where in the circuit does the gain in potential energy (voltage) occur?

2. Where in the circuit does the loss of potential energy (voltage) occur?

3. Calculate the unknown lamp resistance. R =?

Solution:

The gain of potential energy occurs within the battery =6.0V (EMF adds energy)
No energy is lost in the wires, since they are assumed to be resistance-free (V = I x R = 2A X 0
= 0v). The potential that was gained (6.0V) must be consumed by the resistor. So, according to Ohm's Law:

V = I R

R=V / I = 6v/ 2A = 3ohm

R =  3.0

Example  2: Series resistive circuit.

A 10v supply powers a simple closed circuit loop consisting of five 100 ohm resistors connected one-following-the-other (series). What is the total circuit resistance, and the resulting circuit current that flows in the loop?

Solution: Total  R = 100 ohm + 100 ohm + 100 ohm + 100 ohm + 100 ohm = 500 The total circuit current  I = V/R = 10v/500ohms = 0.2 a = 200ma.

• Now calculate the voltage developed (dropped) across each resistor:
Resistor voltage = (resistor Current) x (resistor's Resistance) = (200ma)  x (100
) = 2000 mv = 2v. Each resistor is 100 ohms, so each resistor will drop the same voltage. Why is 200ma the current that flows through each resistor? Because there is only one current path in a simple closed loop circuit, all the resistors (and the power supply) share the same current of 200 ma.

• Show how the total voltage supplied is equal to the sum of the individual voltage drops in the loop:
Each resistor drops 2 volts, so the total resistor voltage drop is 5 x 2v = 10v. This is the same as the voltage supplied. Therefore Voltage supplied = Voltage dropped in the closed loop. As we will see latter, this is a statement of the second most important Law in electricity/electronics (Kirchoff's Voltage Law = KVL).

Example  3:  Ohm's Law Concepts. You will be tested extensively on these concepts! You don't truly understand Ohm's Law without knowledge of these concepts (used a lot in troubleshooting). Review how fractions and proportions work.

• If voltage increases but current remains constant, what can be said about the resistance?  (Increasing, decreasing, constant, equal to zero, equal to infinity)

Solution
:  R = V / I, therefore as voltage increases (numerator of the fraction), and the current is unchanging (denominator) the Resistance (fraction itself) must be INCREASING at the same rate as the voltage. (1/1 < 2/1 < 3/1 < 4/1, etc)

• If current is constant while the resistance decreases, what can be said about the voltage?

Solution:  V = I X R, (constant value of current)( decreasing value of resistance) = Voltage level that must be DECREASING at the same rate as the resistance.

• If resistance is decreasing while the voltage is increasing, what can be said about the current?
I = V/R
, therefore the current must be increasing (at a high rate).

Click on the following web link to take a short fun quiz on Ohm's Law (4 questions). Then use your browser's "Back"  button (four times) to return.

Week 2

Lab Experiments: 1
Practice ENT 171 Quiz #1

Voltage - The most important circuit parameter. Voltage symbol is "V" or "E".

Read chapters 2 , 3, 4 and complete the homework for these chapters. Be prepared to complete Lab Experiment 1, 2  during the posted Lab hours (L210C). Study the handout for the Oscilloscope and the Function Generator. You must master the concept of a potential difference, polarity, RMS, Peak, Peak-to-Peak, voltmeter equivalent circuit and internal resistance.

REVIEW: Now what makes current flow and how is the rate of flow affected?

– For the water system analogy of pipes, it would be the water pressure (due to a pump) forcing water to flow.

– The electrical analogy is the voltage (V) source forcing electron flow through a complete electrical circuit

– In the water model, an increase in water pressure leads to increased flow rate. The actual flow rate is determined by the pressure and the resistance of the pipes to the flow due to friction etc. Large pipes have low resistance, whereas small pipes have high resistance.

– In electric circuits, the current I increases linearly with applied voltage V as expressed by Ohm’s law. V = I X R

Voltage is the potential difference between two points. This potential difference is expressed by a POLARITY marker. Polarity ( + and - or red and black) indicates this difference in energy. Consider the positive point to be at A HIGHER electrical energy level relative to the negative point. It is obvious that you must have TWO points to express a difference in potential.

• Voltage does not exist at a single point. (V = 0 because there is no difference between a point and itself)

• If you track through any loop of an electrical circuit or system, you will discover that the sum of all voltage RISES exactly equals the sum of all the voltage DROPS in the loop. This is true for ALL loops in all circuits and systems. As you track through the loop, if you travel through a component from the positive point (terminal) through the component to the negative terminal- you have encountered a VOLTAGE DROP. Traveling from negative to positive constitutes a voltage RISE for that component. The polarity marker is not arbitrarily set- it is determined by current.

• It is vital that you remember that the DIRECTION of current flow determines the polarity of every component in every loop of a circuit or system.

Example: Assume that you have four resistances and a 10v power supply connected in a single loop (series connection). If the first three resistance drop a collective 7v, what must the voltage drop be across the fourth resistance? (Ans. 3v).

A voltage diagram of a circuit loop (conducting current) shows how and where each voltage rise and drop occurs. Pay close attention to when a potential difference is positive (+) or negative (-) and the difference in a voltage from one arbitrary circuit point to another and in the reverse ( second point back to the first point). Also note that the direction of current flow (clockwise vs counterclockwise) determines the polarity of the voltage developed across all circuit components (except for the voltage source).

Current Flow is the flow rate of electrons. One ampere of current is a flow rate of 6,250,000,000,000,000,000 electrons per second. As you can see, this is a very large number and the ampere (amp) is a large unit of measure. We normally use the "milli" or even "micro" multipliers to scale down this large unit.

There are two ways of referring to the flow of current: Electron versus Conventional. Don't get this confused!

• Electron flow (technically correct) specifies that current flows OUT of the NEGATIVE terminal of a power supply- through the circuit- back to the POSITIVE power supply terminal. This is really what happens. The negative terminal of a power supply has a SURPLUS of electrons, and the POSITIVE terminal has an electron deficiency. When a power supply is connected to a circuit, lost of "free" electrons flow from NEGATIVE power supply terminal - through the circuit- back to the POSITIVE terminal (electrons are attracted to the positively-charged atoms lacking electrons - opposite charges attract).

• CONVENTIONAL current flow is the opposite to that described above. POSITIVE - to - NEGATIVE flow. We will use CONVENTIONAL current flow because it is the most popular model, and the solid-state electronic symbols you will learn in ENT 178, 180, 281 all use arrows to denote conventional flow direction.

EXAMPLE:

The following is a completely worked series resistive circuit example. Pay special attention to the subscripts (Vab, Vbc, etc) as they relate how the potential difference (voltage) is observed- voltage rise or a drop. For example, Vab = the voltage from point "a" to point "b". Polarity is established by the direction of current flow through a component (in the + terminal and out the - terminal).

From the example above, determine Vad, Vce, Vda
Ans. Vad = Vae = - 12v
Vce = Vcd =  - 5v
Vda = Vdc + Vcb + Vba = (+5v) + (+3v) + (+4v) = + 12v

Make sure you understand potential difference and how voltage symbol subscripts indicate the direction you are traveling through a component (to get from one location to another). You can travel from a given point in a loop to another point in two different ways (clockwise or counter-clockwise). Most important is to remember that the direction that current flows through circuit resistances (determined by the polarity of the voltage source) sets polarity of each resistor. Practice the exercise above until you have mastered these concepts. This is what voltage is all about- a potential difference between any two points in a circuit or system of connected circuits.

SCHEMATIC DIAGRAM Symbols You Must Know

Basic Components: DC voltage source-battery & resistors

battery- dc                                 Resistor Types

simple resistor           surface-mount resistor

potentiometer (variable resistor) and schematic symbol

The following photos illustrate other types of resistors- both fixed and variable.

 1/4 Watt "Fixed" . Single Turn Trimmer "Pot" Multi-turn Trimmer . Fixed Symbol . Potentiometer (Pot) Symbol Potentiometer
 Potentiometer Sliding Potentiometer

Two main components are: the battery or power supply and the resistor.  Both are shown above.

• The power supply (applied voltage source) can be a chemical battery which stores an amount of potential energy or a variable voltage electronic power supply (see below) that plugs into a wall receptacle.
• The resistor can also be of different forms as shown above, Axial lead component resistor which has the color bands- the  most common.

Voltage is the potential difference between two points (in a circuit, across a voltage source's terminals, etc). If you travel through from + to - you have passed through a voltage drop, and from - to + designates a voltage rise. The voltage rises must equal the voltage drops around every loop in a circuit/system. Your textbook explains this important property of voltage. Make sure that you understand voltage concepts and applications.

DC Signals- Voltage and Current

DC (Direct Current) represents a voltage that does not vary over time (constant unless manually changed). The schematic symbol for a DC voltage source is shown above. A plot of a dc voltage or current would be a straight horizontal line over time. Examples of dc voltages sources are: Batteries, and the Lab DC Power Supply (photo below). These sources generate an electric field proportional to the voltage amplitude setting. A battery (we use a 6v lantern type) has a + and a - terminal. The battery's voltage is fixed (until it dies), and the DC Supply's output voltage can be adjusted up to 30v. We use special banana cable lug plugs to attach to the battery terminals 9see photo).

As the photo shows, there are two separate dc supplies built into the chassis (one on the left side and the other on the right side) that can be used independently. The voltage knob (black) adjusts the output dc voltage (indicated by the small inaccurate panel meter) and the output connection is at the bottom of the supply (red banana jack is + positive, and black banana jack is - negative). The center ground (black jack) connection is not used in this course. Connecting the ground terminal (instead of the negative terminal) into a circuit will not work.

Week 3

Voltage-
Peak-to-peak, Peak, RMS values

AC Signals- Voltage and Current: SINUSOIDAL Waveforms

Information is often expressed as voltage variation over time. Voltage signals are important to understand. Analog signals vary continuously, digital signals vary in discreetly (0, 1 states- low, higher voltage states). This course will deal exclusively with analog signals, and with a subset known as SINUSOIDAL waveforms (see illustration below). Sinusoidal waveforms are of a single frequency f (measured in cycles per second = Hz). The frequency indicates who many complete cycles (360 degrees) are completed in one second of time. The ac Function Generator (FG) in the lab generates ac voltage waveforms that can be adjusted (amplitude, and frequency). The Period of a waveform T is the time it takes to complete one cycle (360 degrees). T is the inverse of f. T = 1/ f

## AC waveforms

AC voltage switches polarity over time, but does so in a very particular manner. When graphed over time, the "wave" traced by this voltage of alternating polarity from an alternator takes on a distinct shape, known as a sine wave:

In the voltage plot above, the change from one polarity to the other is a smooth one, the voltage level changing most rapidly at the zero ("crossover") point and most slowly at its peak. If we were to graph the trigonometric function of "sine" over a horizontal range of 0 to 360 degrees, we would find the exact same pattern:

Angle             Sine (angle)
in degrees
0  ...............0.0000  -- zero crossing -- (start marker)
15 ............... 0.2588
30 ............... 0.5000
45 ............... 0.7071
60 ............... 0.8660
75 ............... 0.9659
90 ............... 1.0000  -- positive peak
105 ..............  0.9659
120 ..............  0.8660
135 ..............  0.7071
150 ..............  0.5000
165 ..............  0.2588
180 ..............  0.0000  -- zero
195 .............. -0.2588
210 .............. -0.5000
225 .............. -0.7071
240 .............. -0.8660
255 .............. -0.9659
270 .............. -1.0000  -- negative peak
285 .............. -0.9659
300 .............. -0.8660
315 .............. -0.7071
330 .............. -0.5000
345 .............. -0.2588
360 ..............  0.0000  -- zero -- (finish marker)

If we were to follow the changing voltage from any point on the sine wave graph to that point when the wave shape begins to repeat itself, we would have marked exactly one cycle of that wave. The time that it takes to complete the single cycle is called the PERIOD (T measured in units of seconds), and the number of cycles in one SECOND of time is called the FREQUENCY (f in units of hertz or Hz). This is most easily shown by spanning identical peaks, but may be measured between any corresponding points on the graph. The degree marks on the horizontal axis of the graph represent the phase domain of the trigonometric sine function.

Sinusoidal Waveform:

One way of expressing the amplitude of different waveforms is to mathematically average the values of all the points on a waveform's graph to a single value. This amplitude measure is known simply as the average value of the waveform. If we average all the points on the waveform algebraically (that is, to consider their sign, either positive or negative), the average value for most waveforms is technically zero, because all the positive points cancel out all the negative points over a full cycle:

This, of course, will be true for any waveform having equal-area portions above and below the "zero" line of a plot. However, as a practical measure of a waveform's aggregate value, "average" is usually defined as the mathematical mean of all the points' absolute values over a cycle. In other words, we calculate the practical average value of the waveform by considering all points on the wave as positive quantities, as if the waveform looked like this:

Polarity-insensitive mechanical meter movements (meters designed to respond equally to the positive and negative half-cycles of an alternating voltage or current) register in proportion to the waveform's (practical) average value, because the inertia of the pointer against the tension of the spring naturally averages the force produced by the varying voltage/current values over time. Conversely, polarity-sensitive meter movements vibrate uselessly if exposed to AC voltage or current, their needles oscillating rapidly about the zero mark, indicating the true (algebraic) average value of zero for a symmetrical waveform.

RMS voltage and current values of an ac waveform will be used most of the time throughout this course. RMS is the effective value which means that a 1v rms signal will "heat a resistor" the same as a 1v DC signal.

Click on the following web links to take a short fun quiz on ac waveforms. Then use your browser's "Back"  button (twice) to return.

Week 4

ENT 171 Practice Quiz #1
Voltage- Instrumentation, Model, Rv

Once we have a source of accurate frequency, how do we compare that against an unknown frequency to obtain a measurement? One way is to use a CRT as a frequency-comparison device. Cathode Ray Tubes typically have means of deflecting the electron beam in the horizontal as well as the vertical axis. Metal plates are used to electrostatically deflect the electrons- with a pair of plates to the left and right of the beam as well as a pair of plates above and below the beam.

• ## The Horizontal Sensitivity control (left-middle of the photo) adjust how fast the electron gun "sweeps" across the display (from left to right). Sweep speed. Knob setting (seconds per horizontal division) X (number of divisions per cycle) = Period T. The waveform frequency f = 1 / T

• The Trigger control (full left side of photo) adjusts when the sweep begins (voltage level can be manually set, or left in "auto" mode). Trigger synchronizes the waveform to the oscope sweep so that the voltage waveform appears stationary and can be measured. We will use the "auto" setting so that the oscope will trigger on any signal that is inputted (vertical mode"). Your first Lab (this week) is experimenting with the controls of an oscope and becoming familiar with how the controls adjust the display.

You will use the Function Generator (FG) to generate the ac voltage signal that you will connect to a channel of the oscope for observation and measurement. The output connector (with banana adapter connected) is shown in the photo (lower left corner). The FG has two main controls: Amplitude, and Frequency.

The amplitude knob (gray knob in the left center of the photo) adjusts the voltage level of the generated waveform, but does not indicate what the level is (a meter or the oscope must be used to measure the voltage output of the FG).

The frequency of the voltage waveform is adjusted with two separate FG knobs. The large gray wheel knob (photo's right) is the fine adjustment (frequency magnitude coefficient), and the push buttons (photo bottom center row) provide the frequency multiplier (1, 10, 100, 1000, 10K, 100K, 1M). Although you receive an approximate indication of the FG output voltage's frequency- it show be more accurately measured using the oscope's timebase (to find T, then calculate f from T). To set, for instance, a frequency of 3K Hz, the frequency wheel knob is set to 0.3 and the frequency pushbutton is set to 10K. The frequency is therefore 0.3 X 10KHz = 3KHz

Period and frequency are mathematical reciprocals of one another. That is to say, if a wave has a period of 10 seconds, its frequency will be 0.1 Hz, or 1/10 of a cycle per second:

F = 1/Period = 1/T

An instrument called an oscilloscope is used to display a changing voltage over time on a graphical screen. General-purpose oscilloscopes have the ability to display voltage from virtually any voltage source, plotted as a graph with time as the independent variable. The relationship between period and frequency is very useful to know when displaying an AC voltage or current waveform on an oscilloscope screen. By measuring the period of the wave on the horizontal axis of the oscilloscope screen and reciprocating that time value (in seconds), you can determine the frequency in Hertz.

Voltage and current are by no means the only physical variables subject to variation over time. Much more common to our everyday experience is sound, which is nothing more than the alternating compression and decompression (pressure waves) of air molecules, interpreted by our ears as a physical sensation. Because alternating current is a wave phenomenon, it shares many of the properties of other wave phenomena, like sound. For this reason, sound (especially structured music) provides an excellent analogy for relating AC concepts. In musical terms, frequency is equivalent to pitch. Low-pitch notes such as those produced by a tuba or bassoon consist of air molecule vibrations that are relatively slow (low frequency). High-pitch notes such as those produced by a flute or whistle consist of the same type of vibrations in the air, only vibrating at a much faster rate (higher frequency). Here is a table showing the actual frequencies for a range of common musical notes:

## Voltmeter usage

A multimeter is an electrical instrument capable of measuring voltage, current, and resistance (one at a time). Digital multimeters have numerical displays, like digital clocks, for indicating the quantity of voltage, current, or resistance. Analog multimeters indicate these quantities by means of a moving pointer over a printed scale.

Digital multimeter can be used to measure voltage (dc or ac), current (dc or ac) and resistance.

LEARNING OBJECTIVES

• How to measure voltage
• Characteristics of voltage: existing between two points
• Selection of proper meter range

EQUIPMENT ILLUSTRATION
Note where the test leads attach to the multimeters.

Your most fundamental "eyes" in the world of electricity and electronics will be a device called a multimeter (MM). Multimeters measure voltage, current, and resistance (one at a time)

Voltage is the measure of electrical "push" ready to motivate electrons to move through a conductor. Voltage is the potential difference in energy level from one point to another. It is analogous to pressure in a fluid system: the force that moves fluid through a pipe. Your multimeter comes with some basic instructions (setup procedures). Read them well. Digital multimeters use autoranging. Autoranging finds the best measurement range to display the particular quantity being measured.

The analog lab meter's measurement range setting is manually set by the operator. Unless you know ABSOLUTELY the maximum value of the voltage to be measure, set the analog multimeter's selector switch to the highest-value "DC volt" position available. Autoranging multimeters will have a single position for DC voltage, in which case you need to set the switch to that one position. Touch the red test probe to the positive (+) terminal of a voltage source, and the black test probe to the negative (-) terminal of the same voltage source. The meter provides you with indication of the measured amplitude (voltage, current, or resistance level). Reverse the test probe connections to the voltage source if the meter's indication is negative (on an analog meter, a negative value is indicated by the pointer deflecting left instead of to the right).

If the selector switch on an analog MM has been set to an very high-range position, the indication will be too small for an accurate measurement. Move the selector switch to the next lower DC voltage range setting and reconnect to the voltage source. The indication should be higher now, as indicated by a greater deflection of the analog meter pointer (needle).  For the best results, move the selector switch to the lowest-range setting that does not "over-range" the meter. An over-ranged analog meter is said to be "pegged," as the needle will be forced all the way to the right-hand side of the scale, past the full-range scale value. An over-ranged digital meter sometimes displays the letters "OL", or a series of dashed lines (if used in manual range mode).

Connect the MM (as a voltmeter) ACROSS THE COMPONENT = PARALLEL CONNECTION to measure the component's voltage. Since we do not want circuit current to flow through the voltmeter (current flowing through the voltmeter can't simultaneously flow through the resistor we are connected to. For this reason we want the voltmeter to have a very large (ideally infinite) internal resistance. The voltmeters large resistance prevents circuit current from bypassing around circuit element(s).

The equivalent circuit (model) of an "ideal" voltmeter is an infinite resistance (open) where no current flows through the voltmeter. A practical voltmeter's equivalent circuit is that of the "ideal" voltmeter shunted (in parallel- both terminals shared) by the non-infinite internal resistance of the particular voltmeter- Rv. Rv is then the internal resistance of a voltmeter where the larger Rv is, the better the voltmeter. Typical Rv ranges from 10Kohms (analog multimeter) , 3M ohms (digital MM) , up to 10M ohms for an oscilloscope (oscope). The Rv "rule of thumb" is that you want the voltmeter internal resistance to be at LEAST 10 TIMES the resistance of the component across which you are measuring the voltage (using the VM).

Week 5

Ammeters, Ra (ammeter internal resistance)

## AMMETER Configuration

LEARNING OBJECTIVES

• How to measure current with a multimeter
• Selection of proper meter range

INSTRUCTIONS

Current is the measure of the rate of electron "flow" in a circuit. It is measured in the unit of the Ampere, simply called "Amp," (A).

The most common way to measure current in a circuit is to break the circuit open and insert an "ammeter" in series (in-line) with the circuit so that all electrons flowing through the circuit also have to go through the meter. Because measuring current in this manner requires the meter be made part of the circuit, it is a more difficult type of measurement to make than either voltage or resistance. The internal resistance of an ammeter (Ra) should ideally be zero ohms, and can be found from the equipment specifications. Our Lab digital ammeter has an Ra of approximately 10ohms. The analog Simpson 260 (ammeter operation) exhibits an internal resistance that ranges from about 10ohms (100mA current range setting) to a high of about 250ohms (1mA range setting). The Simpson analog ammeter can be used to measure only DC currents- not AC currents.

Some digital meters, like the unit shown below, have a separate jack to insert the red test lead plug when measuring current. Other meters, like most analog meters (including our Simpson 260), use the same jacks for measuring voltage, resistance, and current. Consult the equipment manual for details on measuring current.

When an ammeter is placed in series with a circuit, it ideally drops no voltage as current goes through it. In other words, it acts very much like a piece of wire, with very little resistance from one test probe to the other. Consequently, an ammeter will act as a short circuit if placed in parallel (across the terminals of) a substantial source of voltage. If this is done, a surge in current will result, potentially damaging the meter:

Be very careful to avoid this scenario!

Ammeters are generally protected from excessive current by means of a small fuse located inside the meter housing. If the ammeter is accidently connected across a substantial voltage source, the resultant surge in current will "blow" the fuse and render the meter incapable of measuring current until the fuse is replaced. Be very careful to avoid this scenario. You may test the condition of a multimeter's fuse by switching it to the resistance mode and measuring continuity through the test leads (and through the fuse). On a meter where the same test lead jacks are used for both resistance and current measurement, simply leave the test lead plugs where they are and touch the two probes together. On a meter where different jacks are used, this is how you insert the test lead plugs to check the fuse.

You properly configure the ammeter by breaking the circuit under test open at any point and connecting the meter's test probes to the two points of the break to measure current. As usual, if your meter is manually-ranged, begin by selecting the highest range for current, then move the selector switch to lower range positions until the strongest indication is obtained on the meter display without over-ranging it. If the meter indication is "backwards," (left motion on analog needle, or negative reading on a digital display), then reverse the test probe connections and try again. When the ammeter indicates a normal reading (not "backwards"), the meter is properly configured. Here are some configuration tips:

• Identify what wire or component terminal you wish to measure current through.
• Pull that wire or terminal free- let it hang in mid-air.
• Connect the ammeter between the two unconnected wire ends (the two that were hanging in mid-air). You are now assured of measuring current through the wire or terminal initially identified.

Week 6

Circuit connections: Basic Series & Parallel

Lab Experiment: 3

Circuit configurations- Series and parallel connections

This instructional module is one of the most important of the course! Please review this information several times carefully, including: assigned homework, class notes, and the hands-on Lab experiment. Seek assistance from your instructor if needed. Understanding series and parallel connection is VITAL to your success in electronics and quality assurance.

Series is the type of connection whereby the components are wired such that the output terminal of one component connects to the input terminal of the second component and so on in serial fashion. When a voltage source is applied (input terminal component one to the output terminal of the last component), a single current flows through all of the components. Current is the same through series connected components. The voltage dropped across series connected components divides: the applied voltage equals the sum of the serial component voltages.

Challenge questions (from the textbook reading): Which of the following diagrams correctly indicates how you could connect two components in SERIES?

Which diagram shows the proper connection of an ammeter?

Ammeter is connected in series with the current to be measured- (answer is c)

Basics of Series Resistive Circuits:
"Current is the same, voltage divides"

I -->        - - -      I -->

R1        R2        R3       R4       R5                    Rn

Several resistors attached "in series" one after another. The current I flowing through the first resistor R1 flows also through resistors R2, R3, R4, R5 and through resistor Rn where n represents the total number of resistors in the circuit. The current I is the same throughout the circuit, I is constant throughout the circuit.

If we apply Ohm's Law for each individual resistor in the circuit, we will conclude that the voltage through the first resistor is V1 = I X R1, the voltage through the second resistor is V2 = I X R2 and so on all the way through the n-th resistor for which Vn = I X Rn. For any resistor Rx in the circuit the voltage drop across that resistor is Vx = I X Rx.

The total amount of resistance in the circuit is the sum of all the individual resistances and by Ohm's Law :V = I  X Rt  or in other words the total voltage drop is the current times the total resistance in the circuit. The total resistance of the circuit is the sum of the individual resistances, or: Rtotal = R1 + R2 + R3 + R4 + R5 + ... + Rn The total voltage in the circuit is given by Ohm's Law as : Vtotal = I X Rtotal.

Finally, we note that the total voltage drop in the circuit can also be expressed as the sum of all the individual voltages: The total voltage in the circuit is the sum of the individual voltages:  Vtotal = V1 + V2 + V3 + V4 + V5 +  ... + Vn or voltage divides.

Example:

The resistors R1, R2 and R3 are 100Ω, 50Ω and 20Ω respectively. The current I is 2A.  Find: the voltages V, V1, V2, V3.

The voltage V1 is the voltage through resistor R1.  By Ohm's Law: V1 =  I X R1 = 2A X 100Ω = 200V.Similarly, V2 = I X R2 = 2A X  50Ω = 100V  and  V3 = I X  R3 = 2A X  20Ω = 40V. Now V = V1 + V2 + V3 = 200V + 100V + 40V = 340V. Alternatively, another way of obtaining V is by first getting the total Resistance Rt = 100Ω + 50Ω + 20Ω = 170Ω and V = I X Rt = 2A X 170Ω = 340V

Voltage Dividers: Series connections

V1                           V2                           V3                      V4

Since the current throughout a Series circuit is the same 'throughout the entire circuit, Ohm's Law  predicts I = V1/R1 = V2/R2 =V3/R3 =V4/R4

and also I = Vtotal/Rt  or  I = V/Rt.  Also V/Rt = V1/R1 = V2/R2 = V3/R3 = V4/R4 or  generally, Vx/Rx = V/Rt   and therefore:   Vx = Rx * V/Rt

The voltage across any resistance Rx is given by the ratio of that resistance to the total resistance Rt times the total voltage V across the circuit.  This is called The Voltage Divider Rule.

Example:

We are given the values of the resistors and the total voltage as V = 10V. We want to find out the voltages V1, V2 and V3 without knowing the current

By the Voltage Divider Rule we can find V1, V2 and V3 without knowing the value of the current I.Thus:  V1 = V X R1/Rt  ( Rt = 1KΩ + 5KΩ + 2KΩ = 8KΩ)  and V1 = 10 X 1/8 =1.25V.

Similarly: V2 = V X R2/Rt = 10 X 5/8  = 6.25V  and  V3 = V  X R3/Rt = 10 X 2/8 = 2.5V. Note that the highest the resistance value the higher the voltage drop.  Also, Notice that V1 + V2 + V3 = 1.25V + 6.25V + 2.5V  = 10V = V

Parallel connection of components occurs when two or more components share both terminals (all input terminals connected, all output terminals connected). In this type of connection, the voltage across all parallel connected components is the same, and an applied voltage to a parallel circuit causes different currents to flow through each parallel component- voltage is the same and current divides (the opposite of series connections).

Question: Which diagram shows the proper connection of a voltmeter?

Since voltage is measured by a voltmeter, the voltmeter is connected in parallel (across) with the component- (answer is a)

Basics of Parallel Resistive Circuits
"Voltage is the same, branch currents divide"

I1   -->          R1

I2   -->         R2

I3   -->         R3

I4   -->         R4

In  -->         Rn

+ V -

The potential V is the same for all resistors. The current coming into the left node (that is the left contact where all resistors are attached to) will

split such that the total current is the sum of the currents through all of the resistors- as a tree and its branches.

The total current I is the sum of the individual currents through the resistors: I total = I1 + I2 + I3 + I4 + ... + In.

V = V1 = I1X R1 = V2 = I2 X R2 = V3 =I3 X R3 = V4 = I4 X R4 = ... Vn = In X Rn. V = I X Req, where Req is the equivalent resistance of the circuit holds true by Ohm's Law.  I = I1 + I2 + I3 + I4 + ... + In        and V/Req = V1/R1 + V2/R2 + V3/R3 + V4/R4 + ... +Vn/Rn  So, V/Req =  V/R1  +  V/R2  +  V/R3  + V/R4  +  ... + V/Rn   both sides by the voltage V:  1/Req = 1/R1 + 1/R2 + 1/R3 + 1/R4 + ... 1/Rn    or:

Req  = 1/ (1/R1 + 1/R2 + 1/R3 + 1/R4 + ... 1/Rn) general formula for 'N' number of parallel connected resistances

Example of a Parallel Resistive Circuit

R1 = 100 Ohms and R2 = 20 Ohms are the two parallel resistors in the circuit.  The current will split into the two currents I1 and I2. The supply voltage V= 10 volts is shared by R1 and R2. The equivalent resistance Req can be solved by using 1/Req = 1/R1 + 1/R2  = ( R2 + R1)/(R1X R2).  Req is the inverse of the above or Req = (R1X R2)/(R1 + R2), or in general for two resistors we can say that the equivalent resistance Req is the product over the sum of the resistances. Here Req = 100*20/120 = 16.67 Ohms. Since V1 = V2 = V,  I1 = V/R1 = 10/100 =0.1A and I2 = V/R2 = 10/20 = 0.5A  The total current is I = I1 + I2  = 0.1A + 0.5A = 0.6A.  For two resistances in parallel:

Req = (R1X R2)/(R1 + R2).

Observations:

• It should be noted that if "n" resistances in parallel are identical, then Req = R / n. For example, two 100 Ohm resistors in parallel combine to 100Ohms/2 = 50 Ohms.

• Also important is to realize that the total equivalent resistance of a parallel circuit is smaller than the smallest parallel resistance value. A parallel circuit of 10, 20, and 30 Ohms must have an equivalent combined resistance of less than 10 Ohms.

Weeks 7, 8

Resistance and the use of the ohmmeter
Carbon resistor 3 band color code

Lab Experiment 4
ENT 171 Practice Quiz 2

Multimeter used as an ohmmeter to measure component resistance

Before you attempt to measure resistance using an analog MM, you must determine the resistance range and set the MM (now an ohmmeter), in that range. Now comes "zeroing" the ohmmeter to eliminate the ohmmeter's internal resistance and the lead resistance. With the test lead tips touching each other, adjust the "zero" knob on the front panel so that the needle indicates a resistance of zero ohms (top scale).

To check a fuse, or circuit breaker for proper operation- touch the leads together (see illustration below).

To measure resistance with an analog multimeter (after "zeroing"), first set  the appropriate resistance range and take the reading (see below). Be sure not to physically touch the resistor terminals when measuring resistance with your fingers, or your body resistance will influence the measurement.

Another method to measure resistance is to use two multimeters simultaneously- one to measure current (ammeter) and the other to measure voltage (voltmeter). This is a lab experiment in ENT 171. Set your multimeter to the appropriate voltage range and measure voltage across the resistor as it is being powered by the voltage source. Set your second multimeter to the highest current range available. Break the circuit and connect the ammeter within that break, so it becomes a part of the circuit, in series with the battery and resistor. Select the best current range: whichever one gives the strongest meter indication without over-ranging the meter. If your multimeter uses "auto-ranging" (most digital MMs do) you need not bother with manually setting ranges. Use the Ohm's Law  to calculate circuit current. Compare this calculated figure with the measured figure for circuit current. Taking the measured figures for voltage and current, use the Ohm's Law equation to calculate circuit resistance. Compare this calculated figure with the measured figure for circuit resistance. Finally, taking the measured figures for resistance and current, use the Ohm's Law equation to calculate circuit voltage. Compare this calculated figure with the measured figure for circuit voltage. There should be close agreement between all measured and all calculated figures (within two decimal places). Any differences in respective quantities of voltage, current, or resistance are most likely due to meter inaccuracies or operator error. These differences should be rather small, no more than several percent.

## RESISTANCE and Ohmmeter usage

This section describes how to measure electrical resistance Be sure to never measure the resistance of any electrically "live" object or circuit. In other words, do not attempt to measure the resistance of a battery or any other source of substantial voltage using a multimeter set to the resistance ("ohms") function. Failing to heed this warning will likely result in meter damage and even personal injury.

LEARNING OBJECTIVES

• Determination and comprehension of "electrical continuity"
• Determination and comprehension of "electrically common points"
• How to measure resistance
• Characteristics of resistance: existing between two points
• Selection of proper meter range
• Relative conductivity of various components and materials

Resistance is the measure of electrical "friction" as electrons move through a conductor. It is measured in the unit of the "Ohm," that unit symbolized by the capital Greek letter omega (Ω).  The resistance function is usually denoted by the unit symbol for resistance: the Greek letter omega (Ω), or sometimes by the word "ohms." You must first "Zero" the analog ohmmeter. To zero, touch the two test probe leads of your meter together. When you do, the meter should register 0 ohms of resistance. If you are using an analog meter, you will notice the needle deflect full-scale when the probes are touched together, and return to its resting position when the probes are pulled apart. The resistance scale on an analog multimeter is reverse-printed from the other scales: zero resistance in indicated at the far right-hand side of the scale, and infinite resistance is indicated at the far left-hand side. There should also be a small adjustment knob or "wheel" on the analog multimeter to calibrate it for "zero" ohms of resistance. Touch the test probes together and move this adjustment until the needle exactly points to zero at the right-hand end of the scale. Although your multimeter is capable of providing quantitative values of measured resistance, it is also useful for qualitative tests of continuity: whether or not there is a continuous electrical connection from one point to another. You can, for instance, test the continuity of a piece of wire by connecting the meter probes to opposite ends of the wire and checking to see the the needle moves full-scale. What would we say about a piece of wire if the ohmmeter needle didn't move at all when the probes were connected to opposite ends? Its an OPEN! Digital multimeters set to the "resistance" mode indicate non-continuity by displaying some non-numerical indication on the display. Some models say "OL" (Open-Loop), while others display dashed lines.

An important concept in electricity, closely related to electrical continuity, is that of points being electrically common to each other. Electrically common points are points of contact on a device or in a circuit that have negligible (extremely small) resistance between them. We could say, then, that points within a breadboard column (vertical in the illustrations) are electrically common to each other, because there is electrical continuity between them. Conversely, breadboard points within a row (horizontal in the illustrations) are not electrically common, because there is no continuity between them. Continuity describes what is between points of contact, while commonality describes how the points themselves relate to each other.

Consider a 10,000 ohm (10 kΩ) resistor. This resistance value is indicated by a series of color bands (see below): Brown, Black, Orange, and then another color representing the precision of the resistor, Gold (+/- 5%) or Silver (+/- 10%). Some resistors have no color for precision, which marks them as +/- 20%. Other resistors use five color bands to denote their value and precision, in which case the colors for a 10 kΩ resistor will be Brown, Black, Black, Red, and a fifth color for precision.

OHMMETER Configuration

Connect the meter's test probes across the resistor as such, and note its indication on the resistance scale:

If the needle points very close to infinity ohms, you need to select a higher resistance range on the meter. If you are using a digital multimeter, you should see a numerical figure on the display, with a  "K" symbol on the right-hand side denoting the metric prefix for "kilo" (thousand). While most digital meters are "auto-ranged", some can be manually-ranged (over ride), and require appropriate range selection just as the analog meter. Experiment with different range switch positions and see which one gives you the best indication. The best analog ohmmeter range gives a needle indication in the upper-third of the scale (topmost OHMS scale).

Try reversing the test probe connections on the resistor. Does this change the meter's indication at all? (ans. NO). What does this tell us about the resistance of a resistor? What happens when you only touch one probe to the resistor? What does this tell us about the nature of resistance, and how it is measured?

When you touch the meter probes to the resistor terminals, try not to touch both probe tips to your fingers. If you do, you will be measuring the parallel combination of the resistor and your own body, which will tend to make the meter indication lower than it should be. Make sure that you are measuring ONLY the resistance of the component(s) desired- not other connecting circuitry. Disconnect (isolate) the resistance to be measured. NEVER USE AN OHMMETER IN A CIRCUIT CONTAINING A VOLTAGE SOURCE!!! Don't forget to ZERO the meter before taking a measurement.

Resistance is the measure of friction to electron flow through an object. The more resistance there is between two points, the harder it is for electrons to move (flow) between those two points. Given that electric shock is caused by a large flow of electrons through a person's body, and increased body resistance acts as a safeguard by making it more difficult for electrons to flow through us, what can we ascertain about electrical safety from the resistance readings? Does conductive moisture increase or decrease shock hazard to people?

3 Band Carbon Resistor Color Code:

You will learn the three band carbon resistor color code. Refer to Chapter 7 in your textbook.

Also click on the following link for a useful illustration

Practice various resistor color codes at the following website

Week 9

Series Circuits - Resistive Circuits

A simplified series circuit is made up of three elements:

• a source,
• resistors, in which the electrical energy is utilized and is commonly referred to as loads,
• and ideal conductors, with no assumed resistance, to connect the elements in series.

Consider a typical household electrical outlet as a 120-volt source for the circuit and the light bulbs as resistors in a closed circuit. The wires are assumed to have no resistance themselves and connect each of the light bulbs in a single, closed circuit.

Since there is only one path for current to flow in this series circuit, the current or electron flow must be the same in each segment of the circuit. This means that the current leaving the source is equal to the amount of current through each resistance.

There are three rules governing the simple series circuits of resistive elements. They are:

1. The current flow is the same through each element of the series circuit.
2. The combined resistance of the various loads in series is the sum of the separate resistances.
3. The voltage across the source or power supply is equal to the sum of the voltage drops across the separate loads connected in series.

The following three facts based on the rules (above), allow for calculation of current and voltage drops in a series circuit:
1. Ohm's Law can be applied to the whole circuit or any part of the circuit.
2.. We can determine the current flow in the circuit by dividing the total circuit resistance into the total applied voltage.
3.. We determine each resistor's voltage by multiplying the current (step 5) by the resistor's resistance (Ohm's Law).

RESISTORS IN SERIES- Calculations

Resistors can be connected in series; that is, the current flows through them one after another. The circuit in Figure 1 shows three resistors connected in series, and the direction of current is indicated by the arrow.

 Resistors connected in series.

Note that since there is only one path for the current to travel, the current through each of the resistors is the same:

I = I1 = I2 = I3 = IR1 = IR2 = IR3 (both notations used here)

Also, the voltage drops across the resistors must add up to the total voltage supplied by the battery (energy in = energy out):

Since V = I R, then

But Ohm's Law must also be satisfied for the complete circuit:

Therefore:

equivalent resistance for resistors connected in series.

Here is an interesting interactive exercise to help you with this concept and the use of Ohm's Law
Use the "BACK" button to return to this place when you are finished.
Press here

Kirchhoff's Voltage Law

 Kirchhoff's Voltage Law is a result of the conservative electrostatic field (conservation of energy). It states that the total voltage around a closed loop must be zero. If this were not the case, then when we travel around a closed loop, the voltages would be indeterminate.

The total voltage around loop 1 should sum to zero, as does the total voltage in loop2. Furthermore, the loop which consists of the outer part of the circuit (the path ABCD) should also sum to zero.

We can adopt the convention that potential gains (going from lower to higher potential: going in the  "-" terminal and going out the "+" terminal) is taken to be positive. Potential losses (in the "+" terminal and out the "-" terminal) will then be negative.

Here are a number of simulated experiments based on Kirchoff's Laws. They are in order of increasing difficulty. Use the "back" button to return to this place.

Experiment 1. Press here.
Experiment 2. Press here.
Experiment 3. Press here.
Experiment 4. Press here.
Experiment 5. Press here.

DC CIRCUITS SELF TEST

The following are some sample problems you must try.

Remember "m" is milli = 1/1000 = 0.001 factor, and "K" is kilo = 1000

All measurements of AC voltage and current must be expressed in the same terms (peak, peak-to-peak, average, or RMS). Normally we use RMS values, oscilloscope voltage measurements are first measured in peak (p) or peak-to-peak (pp) and then converted (via calculator) to the  RMS value. Multimeter voltage readings are in RMS (multimeters are calibrated in RMS). If the source voltage is given in peak volts, then all currents and voltages subsequently calculated are cast in terms of peak units. If the source voltage is given in RMS volts, then all calculated currents and voltages are cast in  RMS units as well. This holds true for any calculation based on Ohm's Laws, Kirchhoff's Laws, etc. Unless otherwise stated, all values of voltage and current in AC circuits are generally assumed to be RMS rather than peak, average, or peak-to-peak.

Connecting components in series means to connect them in-line with each other, so that there is but a single path for electrons to flow through them all. If you connect batteries so that the positive of one connects to the negative of the other, you will find that their respective voltages add. This is called "Series Aiding".

Measure the voltage across each battery individually as they are connected, then measure the total voltage across them both, as shown by the following:

Try connecting batteries of different sizes in series with each other, for instance a 6-volt battery with a 9-volt battery. What is the total voltage in this case? Try reversing the terminal connections of just one of these batteries, so that they are opposing each other like this:

How does the total voltage compare in this situation to the previous one  Note the polarity of the total voltage as indicated by the voltmeter indication and test probe connection. Remember, if the meter's digital indication is a positive number, the red probe is positive (+) and the black probe negative (-); if the indication is a negative number, the polarity is "backward". Analog meters simply will not read properly if reverse-connected, because the needle tries to move the wrong direction (left instead of right). THIS CAN DAMAGE THE ANALOG METER!!!

Be careful in connecting analog multimeters into DC circuits. Observe the polarity correctly- double-check before applying power.

Week 10

Chapter 9
Lab Experiment 5

Shorts and Opens:

A short is a zero ohm resistive path. If the short is located across (parallel with) a component in a circuit- it electrically eliminates (bypasses) that component's resistance (the combined resistance of the component and the short = 0 ohms). This electrical elimination of component resistance will cause the total circuit resistance to decrease and the current to increase. This increase in current can cause problems in the remaining series components if the current increase stresses (damages) the components. The voltage across the shorted component is zero. This zero voltage is the symptom of a shorted component and can assist you in troubleshooting the short.

An open in a series circuit disables the entire circuit- no current flows anywhere in the simple series circuit. All component voltages are zero (except at the open) since there is no complete current path and V =I X R = 0 X R = 0. The open's voltage equals the applied voltage- this is how the open can be troubleshot using a voltmeter.

## Total input voltage "divides" into smaller component voltage drops (series circuit)

LEARNING OBJECTIVES

• Voltmeter use
• Ammeter use
• Ohmmeter use
• Use of Ohm's Law
• Use of Kirchhoff's Voltage Law ("KVL")
• Voltage divider design

SCHEMATIC DIAGRAM

Connection of three resistors in series, and to a 6-volt battery, is shown in the illustrations. We will use Ohm's Law (I=E/R) to calculate circuit current, then verify this calculated value by measuring current with an ammeter. Note the input ports used on the digital ammeter.

If your resistor values are between 1 kΩ and 100 kΩ, and the battery voltage approximately 6 volts, the current should be a very small value, in the milliamp (mA) to microamp (1µA = 0.000001A) range. When you measure current with a digital meter, the meter will show the appropriate metric prefix symbol (m or µ) in some portion of the display. These metric prefixes are easy to overlook when reading the display of a digital meter, so pay attention!

The measured value of current should agree closely with your Ohm's Law calculation. Now, take that calculated value for current and multiply it by the respective resistances of each resistor to predict each resistor's voltage drop (E=IR). Switch you multimeter to the "voltage" mode and measure the voltage dropped across each resistor, verifying the accuracy of your predictions. Again, there should be close agreement between the calculated and measured voltage figures (resistor tolerance causes error).

Each resistor voltage drop will be some fraction or percentage of the total voltage, hence the name voltage divider given to this circuit. This fractional value is determined by the proportion of the resistance of the particular resistor to the total resistance. If a resistor drops 50% of the total battery voltage in a voltage divider circuit, that proportion of 50% will remain the same. So, if the total voltage is 6 volts, the voltage across that resistor will be 50% of 6, or 3 volts. If the total voltage is 20 volts, that resistor will drop 10 volts, or 50% of 20 volts.

Kirchhoff's Voltage Law. Identify each unique point in the circuit with a number. Points that are electrically common (directly connected to each other ) have the same number. An example using the numbers 0 through 3 is shown here in both wiring diagram and schematic form.

Using a digital voltmeter (this is important!), measure voltage drops around the loop formed by the points 0-1-2-3-0.

Using the voltmeter to "step" around the circuit in this manner yields three positive voltage figures and one negative:

These figures, algebraically added ("algebraically" = respecting the polarities), should equal zero. This is the fundamental principle of Kirchhoff's Voltage Law: that the algebraic sum of all voltages in a "loop" add to zero. Voltages rises = Voltage Drops.

POLARITIES- See Diagram (directly above):

• V(pt "0" to pt "3") = V03 = + polarity (voltage rise), and V30 = - polarity (voltage drop)
• V33 = 0
• V32 = - polarity, and V23 = + polarity
• V21 = - , and V12 = +
• V10 = - , and V01 = +
• V00 = 0
• V02 = + , and V20 = -
• V01 = + , and V10 = -
• V31 = - , and V13 = +

It is important to realize that the "loop" stepped around does not have to be the same path that current takes in the circuit, or even a legitimate current path at all. The loop in which we tally voltage drops can be any collection of points, so long as it begins and ends with the same point. For example, we may measure and add the voltages in the loop 1-2-3-1, and they will form a sum of zero as well:

Try stepping between any set of points, in any order, around your circuit and see for yourself that the algebraic sum always equals zero. This Law holds true no matter what the configuration of the circuit: series, parallel, series-parallel, or even an irreducible network.

Kirchhoff's Voltage Law is a powerful concept, allowing us to predict the magnitude and polarity of voltages in a circuit by developing mathematical equations for analysis based on the truth of all voltages in a loop adding up to zero.

Resistance Measurement Error- Zero that ohmmeter!

Most ohmmeters operate on the principle of applying a small voltage across an unknown resistance and inferring resistance from the amount of current drawn by the resulting circuit -both the voltage and current quantities employed by the meter are quite small. This presents a problem for measurement of low resistances, as a low resistance specimen may be of much smaller resistance value than the meter circuitry itself. One of the many sources of error in measuring small resistances with an ordinary ohmmeter is the resistance of the ohmmeter's own test leads. Being part of the measurement circuit, the test leads may contain more resistance than the resistance of the test specimen, incurring significant measurement error by their presence.

ZEROING the Ohmmeter:
You should first short the lead tips together and record the displayed resistance and then subtract this lead resistance from your final resistance measurement (resistor connected across the ohmmeter leads) to get an accurate reading. The analog Simpson ohmmeter has a zero level control for this purpose.

Component Isolation
Another even more important form of ohmmeter measurement error is to not isolate the resistance under test from the rest of the circuitry that it is attached to in a circuit. You must NEVER use an ohmmeter in a live circuit- this can damage the ohmmeter and/or the component being measured. You also must electrically isolate the component to be measured so that any additional circuitry doesn't cause a combined total resistance.

Now answer the following review questions for practice.

Question : Ohm's Law
Question : Series Circuit

## Parallel Connections

LEARNING OBJECTIVES

• Voltage source regulation
• Boosting current capacity through parallel connections

SCHEMATIC DIAGRAM

Wiring Diagram

Connecting batteries in parallel with each other- each battery only has to supply a fraction of the total current demanded by the lamp. Voltage remains constant. Parallel connections involve making all the positive (+) battery terminals electrically common to each other by connection through jumper wires, and all negative (-) terminals common to each other as well.

Consider the current of one battery and compare it to the total current (light bulb current). Shown here is how to measure single-battery current (leftmost battery):

By breaking the circuit for just one battery, and inserting our ammeter within that break, we tap into the current of that one battery and are therefore able to measure it. The current path for a single battery BRANCH is established. Measuring total current involves a similar procedure: make a break somewhere in the path that total current must take (bulb), then insert the ammeter within than break:

Speculate on the difference in current between the single-battery and total measurements. How do the branch currents relate to the total circuit current (bulb)?

To obtain maximum brightness from the light bulb, a series-parallel connection is required. Two 6-volt batteries connected series-aiding will provide 12 volts. Connecting two of these series-connected battery pairs in parallel improves their current-sourcing ability:

Placing all four 6v batteries in series would generate 4 X 6v = 24v, but would have limited current producing ability.

Resistors in parallel develop the same voltage, but the individual branch currents are different (parallel component voltage is the same, and the parallel branch currents divide). The next section describes the currents at each junction (node) of parallel connections.

Kirchhoff's Current Law (KCL)

 This fundamental law results from the conservation of charge. It applies to a junction or node in a circuit -- a point in the circuit where charge has several possible paths to travel. In the diagram above we see that IA is the only current flowing into the node. However, there are three paths for current to leave the node, and these current are represented by IB, IC, and ID. Once charge has entered into the node, it has no place to go except to leave (this is known as conservation of charge). The total charge flowing into a node must be the same as the the total charge flowing out of the node. Charge does not vanish nor build-up. Therefore: IB + IC + ID = IA  Sum of currents entering each circuit node = sum of all currents leaving that node

Kirchoff's laws (KVL, KCL)

There are two laws necessary for solving circuit problems. Kirchoff's Voltage (KVL) and Current (KCL) laws.

1. The voltages around a closed path in a circuit must sum to zero (KVL). The voltage drops being negative, while the gains are positive.

2. The sum of the currents entering a node must equal the sum of the currents exiting a node. (KCL)

The first law is a simple statement of the meaning of potential energy. Since every point on a circuit has a unique potential energy value, traveling around the circuit, through any path must bring you back to the starting reference potential. Using a gravitational analogy: If one hikes taking several paths, then finishes at the same point, the sum of the elevation changes of each path will balance out.

 KCL is a statement of the conservation of current. For the node on the right, i1= i2+i3. If all currents had been defined as entering the node, then the sum of the currents would be zero.

Parallel Shorts
Shorts in a parallel circuit are catastrophic
. A short across any component in a parallel circuit forces all component voltages to zero and can cause excessive current to flow from the voltage source through the short. You have effectively bypassed all circuit resistance causing the total circuit current to become unboundedly large V/R = V/0 = large current! You should safeguard at all times against a parallel short circuit. The symptom of a short in a parallel will be excessively large current flow (causing fuse to blow or circuit breaker to trip). Its a good idea to design in a fuse or circuit breaker to protect a parallel circuit.

Parallel opens
An open of a component in parallel (switched off or burned out component) will have minimal effect on circuit operation and can be hard to troubleshoot. All other parallel components will have the same voltage and will draw current from the voltage source- only the opened component will not draw current (but will have the same parallel voltage!). The total circuit current will be lower than normal because the resulting total resistance will be larger than normal. As resistance is eliminated in a parallel circuit the total combined resistance increases!

Now answer the following review questions for practice.

Question : Parallel Circuit

Weeks 12- 14  Series-Parallel Resistive Circuits

Lab Experiments: 6, 7

ENT 171 Practice Quizzes: 3, 4

Basics of Series-Parallel Resistive Circuits

. . .     . . . . . .

Combination resistive circuits, otherwise known as Series-Parallel resistive circuits combine resistors in series with resistors in parallel as shown in the Figure above.  This kind of circuit can be separated into two different cases: For the part(s) of the circuit that has resistors in series we use our knowledge of series circuits and apply the corresponding principles, for the part of the circuit that has resistors in parallel we use our knowledge of parallel circuits and apply the corresponding principles.

Kirchoff's Voltage Law (KVL) is extended to include any loop on the circuit such that:  "The voltage drops around any closed loop in the circuit add up to zero."   Also, Kirchhoff's Current Law (KCL) is  extended to include any node on the circuit such that:  "The  currents coming into any node in the circuit add up to zero."

The equivalent resistance of the circuit Req can be obtained by making the appropriate combinations (either parallel or series)  until we get one unique resistance.  The rule that Vtotal = IX Req still applies as well as Ohm's Law and both should be kept in mind when solving combination circuits.

The photo below is taken in our Lab. Can you determine the type of circuit shown? (Ans: Series-parallel). Is this an AC or DC circuit (look closely at the Fluke multimeter)?

Some examples of finding total circuit resistance: Rt = Req

In the circuit above there are four resistors and a battery V=10 volts.  To obtain Req for this circuit notice that R2 and R3 are in parallel, combining these two resistors R2||R3 = 1/(1/2 + 1/10) .  Obtaining the common denominator and combining gives R2||R3 = 1/(6/10) = 10/6 Kohms  or R2||R3 = 1.67 Kohms.   The value of 1.67 Kohms will then be in series with R4, yielding 6.67 Kohms.  The (R4- R2- R3) combination is in PARALLEL with R1which yields a total circuit resistance (Rt) of: (6.67K)(1K)/(6.67K + 1K) = 870 ohms.

NOTE: The total current I can be calculated by: I = V/Rt = 10v/870ohms = 11.5mA. Since there is 10mA flowing through R1 (10v/1Kohm), the current flowing through R4 must be 1.5 mA (KCL: 11.5mA - 10MA = 1.5mA).

Applying Kirchhoff's laws and Ohm's law in series-parallel circuits (KVL, KCL)

For this example, the battery voltage in the circuit is 5 V and all the resistors have a value of 2KOhms, except for R2 which is  1Kohm.  Find all the voltages and currents in the circuit as well as the equivalent resistance Req.

First solve for Rt ( Req).
R4 and R5 in series give 4Kohms, which if combined with R3 (parallel) results in 1.33Kohms.  The 1.33Kohm can be combined with R2 (parallel) to give a value of   570ohms. Finally, Req = 2K + 570ohms =
2.57Kohms.

Applying KCL to the circuit the total current coming from the DC source  I1 goes through resistor R1 and splits into I2 which goes through resistor R2, I3 which goes through R3 and I4 goes through R4 and R5.  KCL : I1 = I2 + I3 + I4.  Since the current I1 is the total current in the circuit (It), applying Ohm's law  I1 = Vtotal/Req which gives  I1 = 5V/2.57K =  1.95mA.   The voltage across R1, V1 = I1 * R1 = 1.95mA* 2K= 3.9V.

Applying KVL on the first loop, V2 (the voltage across R2) and the voltage V1 have to add up to be 5V. Thus, V2 = 5V - V1 = 5V - 3.9V = 1.1V.    Since R2 and R3 are in parallel this same voltage is seen through resistor R3, making V3 = V2 = 1.1V.

Using Ohm's Law:  I2 and I3 can be easily found since we already have the voltage across both resistors and know the resistor values.  I2 = V2/R2 = 1.1V/1K = 1.1mA , and similarly, I3 = V3/R3 = 1.1V/2K = 0.55mA

Finally, the current I4 can be found by applying KCL and noting that I4 = I1 - I2 - I3.  This equation gives I4 = 1.95mA - 1.1mA - 0.55mA = 0.3mA.   Since R4 and R5 have the same 2Kohm value, both V4 and V5 are equal, yielding:
V4 = V5 = I4* 2Kohms = 0.3mA * 2K = 0.6V

Another example (compound series circuit)

Two 1KOhm resistors in series with three parallel 3KOhm resistors.  Combining the 3K resistors gives a resistance of 1Kohm.  This 1Kohm resistance is in series with R1 and R2 resulting in  Req = 3KOhms

I1 = I2 = 5V/3K = 1.67mA.   The voltages V1 and V2 can be calculated by Ohms Law, since both resistors are the same value and carry the same current  V1 = V2. This gives V1 = V2 = 1K * 1.67mA = 1.67V.

By KVL the voltage across the 3Kohm resistor combination plus V1 & V2 must equal  5V. It follows that the voltage across the 3KOhm resistors is V(3K) = 5V - 1.67V - 1.67V = 1.66V. Also, V3 = V4 = V5 = 1.66V

KCL: I3 = I4 = I5 = 1.66V/3KOhms = 0.553mA. KCL can be used to check the results:I3 + I4 + I5 = 1.67mA = I2 = 1.67mA = I1.

Application of Kirchoff's laws: LOOP Equations

Kirchoff's laws can be used to solve the following circuits:

The left-side figure can be treated by breaking up the circuit into pieces and applying the rules for adding resistances in parallel and series. The resistors R3, R4 and R5 can be treated as a single resistor  R345, which can then be combined in parallel with R2, to give an effective resistance R2345. The DC source sees a net resistance Rt = R1 + R2345.

The middle and right-side figures above can't be solved as simply. However Ohms Law, KVL, and KCL can be used to solve each circuit. Give it a try!.

EXAMPLE

Find the currents through all the resistors in the circuit below:

Solution:

Summing the voltages around the left and right loops (KVL) gives the following two equations (polarity counts!):

1.

One can then reduce the problem to "2 equations with 2 unknowns" by substituting for i1 and obtaining 2 new equations.

Knowing the values for: R1, R2, R3, Va, and Vb will enable the solution of each current.

Now answer the following review questions for practice.

Question : Combination Circuit 1
Question : Combination Circuit 2

Now you are ready for the circuit Puzzles (26 each). Click on the weblink icon to enter Solve as many as possible to fortify your understanding of circuits.

Special Topics: Week 15

Chapter 12

Current divider rule, Thevenin's theorem, Phase, Lab addendum

## Current divider:

LEARNING OBJECTIVES

• Voltmeter use
• Ammeter use
• Ohmmeter use
• Use of Ohm's Law
• Use of Kirchhoff's Current Law (KCL)
• Current divider design

SCHEMATIC DIAGRAM

The three resistors are connected in parallel to each other, and with the 6-volt battery, as shown in the illustrations above.  Consider the voltage across each of the three resistors. In a series circuit, current is equal through all components at any given time. In a parallel circuit, voltage is the common to all components. The total current divides into branch currents.

Ohm's Law (I=E/R) is used to calculate current through each resistor.

With the digital ammeter connected as shown, all three indications should be positive, not negative.

Now, measure total circuit current, keeping the ammeter's red probe on the same point of the circuit, but disconnecting the wire leading to the positive (+) side of the battery and touching the black probe to it:

Note both the sign (direction) of the current as indicated by the ammeter shown above (dependant upon ammeter terminal placement).  Kirchhoff's Current Law concerns branch currents adding at a node of a circuit, and Kirchhoff's Voltage Law describes voltages adding around a series loop. Both of Kirchhoff's Laws allow for the generation of equations describing several variables in a circuit, which may then be solved for unknown variables.

Now consider the current measurements as all positive numbers: the first three representing the current through each resistor, and the fourth representing total circuit current as a positive sum of the three "branch" currents. Each resistor (branch) current is a fraction of the total current. A parallel resistive circuit is called a current divider. The total resistance of a parallel combination will be a value less than the smallest branch resistance The same (total) resistance can be measured when connecting an ohmmeter across any one of a set of the parallel-connected branch resistances. All parallel components are connected between two sets of electrically common points (bus or a "rail"). Since the meter cannot distinguish between points common to each other by way of direct connection- the resistance across one set of resistor terminals, is to read the resistance of the entire combination. The same is true for voltage, which is why battery voltage could be read across any one of the resistors (or across the battery terminals).

The ratio of resistor current to total current is the same as the ratio of total resistance to individual resistance. For example, if a 1 kΩ resistor is part of a current divider circuit with a total resistance of 100Ω, that resistor will conduct 1/10 of the total current

THEVENIN'S Theorem

Thevenin's equivalent circuit model

Any electrical system of components can be "modeled" by a Thevenin equivalent circuit consisting of a Thevenin voltage source (open-output circuit voltage) in series with an equivalent Thevenin resistance (internal resistance). What must be defined:

• where is the output located in the circuit
• which element constitutes the output load resistance?

The circuit (minus the load) constitutes a "blackbox" that is being modeled by Thevenin's Theorem. The load is distinct from the "blackbox" circuitry. The Thevenin Voltage and Resistance describe the "blackbox". Both Thevenin Voltage and Resistance and calculated (or measured) with the Load REMOVED, since the Load is not part of the "blackbox" circuitry.

The Norton equivalent simply replaces a series Thevenin voltage source & resistance with an equivalent current source in PARALLEL with a Norton resistance (see below).

# Thevenin's Theorem

Any combination of voltage sources and resistances (impedances) with two terminals can be replaced by a single voltage source "e" and a single series resistance "r". This circuitry is called a "blackbox".

The value of e is the open circuit voltage at the terminals (output load removed). The value of r is e divided by the current with the output terminals short circuited. The Thevenin resistance r represents the internal resistance of the "blackbox".

# Thevenin Voltage

The Thevenin voltage e is an ideal voltage source equal to the open circuit voltage at the terminals (load removed).

# Thevenin/Norton Resistance

The Thevenin resistance r  is the resistance measured at terminals AB with all voltage sources replaced by their internal resistances (zero ohms ideally) and all current sources replaced by open circuits (ideal internal resistance). It can be determined that:

The same resistance is used in the Norton equivalent.

# Thevenin Example- DO THIS!!

Replacing a network by its Thevenin equivalent can simplify the analysis of a complex circuit.  The Thevenin resistance is the resistance looking back from AB with replaced by a short circuit.

 For = , = , = ,

 and voltage = V

 the Thevenin voltage is V
 since and form a simple voltage divider.

 The Thevenin resistance is .

## AC phase- Basis of modern data communication

Things start to get complicated when we need to relate two or more AC voltages or currents that are out of step with each other. By "out of step," I mean that the two waveforms are not synchronized: that their peaks and zero points do not match up at the same points in time. The following graph illustrates an example of this:

The two waves shown above (A versus B) are of the same amplitude and frequency, but they are out of phase with each other- called a phase shift. The starting point of a sine wave is zero amplitude at zero degrees, progressing to full positive amplitude at 90 degrees, zero at 180 degrees, full negative at 270 degrees, and back to the starting point of zero at 360 degrees. We can use this angle scale along the horizontal axis of our waveform plot to express just how far out of step one wave is with another:

The shift between these two waveforms is about 45 degrees, the "A" wave being ahead of the "B" wave. A sampling of different phase shifts is given in the following graphs to better illustrate this concept:

Because the waveforms in the above examples are at the same frequency, they will be out of step by the same angular amount at every point in time. For this reason, we can express phase shift for two or more waveforms of the same frequency as a constant quantity for the entire wave, and not just an expression of shift between any two particular points along the waves. That is, it is safe to say something like, "voltage 'A' is 45 degrees out of phase with voltage 'B'." Whichever waveform is ahead in its evolution is said to be leading and the one behind is said to be lagging.

Phase shift, like voltage, is always a measurement relative between two things. There's really no such thing as a waveform with an absolute phase measurement because there's no known universal reference for phase. Typically in the analysis of AC circuits, the voltage waveform of the power supply is used as a reference for phase, that voltage stated as "xxx volts at 0 degrees." Any other AC voltage or current in that circuit will have its phase shift expressed in terms relative to that source voltage.

Recommended hand tool purchases:

When working with wire, you need a tool to "strip" the plastic insulation off the ends so that bare copper metal is exposed. This tool is called a wire stripper, and it is a special form of plier with several knife-edged holes in the jaw area sized just right for cutting through the plastic insulation and not the copper, for a multitude of wire sizes, or gauges. Shown here are two different sizes of wire stripping pliers:

Wire stripping pliers

Needle-nose pliers are designed to grasp small objects, and are especially useful for pushing wires into stubborn breadboard holes.

Needle-nose pliers

For projects involving printed-circuit board assembly or repair, a small soldering iron and a spool of "rosin-core" solder are essential tools. I recommend a 25 watt soldering iron, no larger for printed circuit board work, and the thinnest solder you can find. Do not use "acid-core" solder! Acid-core solder is intended for the soldering of copper tubes (plumbing), where a small amount of acid helps to clean the copper of surface impurities and provide a stronger bond. If used for electrical work, the residual acid will cause wires to corrode. Also, you should avoid solder containing the metal lead, opting instead for silver-alloy solder. If you do not already wear glasses, a pair of safety glasses is highly recommended while soldering, to prevent bits of molten solder from accidently landing in your eye should a wire release from the joint during the soldering process and fling bits of solder toward you.

Soldering iron and solder ("rosin core")