am 5 202 ac circuit analysis

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AM 5-202

BASIC ELECTRONICS

AC CIRCUIT ANALYSIS

December 2011

DISTRIBUTION RESTRICTION: Approved for Pubic Release. Distribution is unlimited.

DEPARTMENT OF THE ARMYMILITARY AUXILIARY RADIO SYSTEM

FORT HUACHUCA ARIZONA 85613-7070


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AM 5 201 Basic Electronics DC Circuits Analysis

ii Ver. 1.0

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AM 5 201 Basic Electronics - DC Circuit Analysis

Ver. 1.0 iii

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iv Ver. 1.0

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AM 5 201 Basic Electronics - DC Circuit Analysis

Ver. 1.0 v

CONTENTS

1 AC CIRCUIT ANALYSIS................................................................................................. 1-11.1 REFERENCE ..........................................................................................................................1-11.1.1 Introduction..................................................................................................................................... 1-11.1.2 Alternating Current ......................................................................................................................... 1-11.1.3 Frequency and Cycle .....................................................................................................................1-21.1.4 Resistance in AC Circuits ............................................................................................................... 1-21.1.5 Inductance in an AC Circuit ............................................................................................................ 1-22 INDUCTIVE REACTANCE ............................................................................................. 2-12.1 POWER .................................................................................................................................2-12.1.1 Power Factor .................................................................................................................................. 2-22.1.2 More Cosine o ................................................................................................................................ 2-22.2 INDUCTIVE REACTANCE IN SERIES AND PARALLEL ....................................................................2-23 CAPACITIVE REACTANCE ........................................................................................... 3-13.1 PARALLEL RESONANCE ..........................................................................................................3-23.1.1 Impedance of a Parallel Resonant Circuit ...................................................................................... 3-33.1.2 Resonant Frequency and Bandwidth ............................................................................................. 3-3


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vi Ver. 1.0

IMPROVEMENTS

(Suggested corrections, or changes to this document, should be submitted through your StateDirector to the Regional Director. Any Changes will be made by the National documentation team.

DISTRIBUTIONDistribution is unlimited.

VERSIONS

The Versions are designated in the footer of each page if no version number is designated theversion is considered to be 1.0 or the original issue. Documents may have pages with differentversions designated; if so verify the versions on the Change Page at the beginning of eachdocument.

REFERENCES

The following references apply to this manual:

Allied Communications Publications (ACP):

ACP - 167 - Glossary of Communications Electronics Terms

US Army FM/TM Manuals

1. TM 5-811-3 - Electrical Design, Lightning and Static Electricity Protection2. TM 5-682 - Facilities Engineering Electrical Facilities Safety3. TM 5-690 - Grounding and Bonding in Command, Control, Communications, Computer,

Intelligence, Surveillance, and Reconnaissance (C4ISR) Facilities4. TM 11-661 Electrical Fundamentals, Direct Current

5. TM-664 Basic Theory and Use of Electronic Test Equipment

US Army Handbooks

1. MIL-HDBK 1857 - Grounding, Bonding and Shielding Design Practices

Commercial References

1. Basic Electronics, Components, Devices and Circuits; ISBN 0-02-81860-X, By William P Handand Gerald Williams Glencoe/McGraw Hill Publishing Co.

2. Standard Handbook for Electrical Engineers - McGraw Hill Publishing Co.

CONTRIBUTORSThis document has been produced by the Army MARS Technical Writing Team under the authority of ArmyMARS HQ, Ft Huachuca, AZ. The following individuals are subject matter experts who made significantcontributions to this document.

William P Hand


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AM 5-202 Basic Electronics - AC Circuit Analysis

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1 AC CIRCUIT ANALYSIS

1.1 REFERENCE

Basic Electronics, Components, Devices and Circuits; ISBN 0-02-81860-X By William P Hand andGerald Williams.

1.1.1 Introduction

Alternating current (ac) is probably the most common, and most important, available form ofelectricity. Alternating current is a current that begins at zero, rises to some set value, and thenfalls to zero again. It then reverses its direction of current flow and rises to the same set value inthe reverse direction, and then falls to zero again. This reversal of current flow direction is incontrast to direct current (dc), which always maintains the same direction of flow. Standardalternating current can be plotted on a graph as shown in Figure 1-1. The graph shows how thewaveform is produced by an alternating current generator as the armature (rotating part) rotatesthrough 360 circular degrees for each cycle.

Figure 1-1A Sine Wave Voltage

1.1.2 Alternating Current

As the generator armature moves through one 360 degree rotation (full circle), the generatorvoltage goes through one complete cycle, as shown in Figure 1-1. The curve displayed in the figurecan also be described by the mathematical equation

e = Em sin t ( = 2Nf)

where e equals the voltage, Em the maximum value of generated voltage, and wt the angularvelocity multiplied by the time.

When a generator produces an ac voltage, the current arising from it varies in step with thevoltage. Like the voltage, the current can be represented graphically by a sine wave and by thefollowing equation:

i = Im sin t


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1-2 Ver. 1.0

where i equals the current, Im the maximum value of generated current, and t the angularvelocity multiplied by the time.

1.1.3 Frequency and Cycle

While the coil in a generator rotates 360 (one complete revolution), the output voltage goesthrough one complete cycle. During one cycle, the voltage increases from zero to positive Em in

one direction, decreases to zero, increases in the opposite direction to negative Em, and thendecreases again to zero, The first 180 (one-half of the voltage cycle) is called the positivealternation and the last 180, from 180 to 360, is called the negative alternation. The value ofthe Em voltage at 90 is called the amplitude or peak voltage. The time required for a positive anda negative alternation is called the period. The number of complete cycles per second is the

frequency of the sine wave. When the angular velocity, , at which the coil rotates, is expressed in

radians per second, the mathematical relation between and f is given by the equation

= 2Nf

1.1.4 Resistance in AC Circuits

Resistance is the property by which a conductor opposes the flow of current. The resistance of aconductor opposes alternating current in the same way that it opposes direct current.

1.1.5 Inductance in an AC Circuit

The discussion of induction you learned that a coil opposes a change in the current through it bybuilding up a counter voltage. This counter voltage is an induced voltage that is equal to

where ei is the counter voltage, L the inductance in henrys, Li the change in current, and Lt the

change in time. The term Li / Lt is the rate of change in current with respect to time (how fast thecurrent changes). Reference Figure 1-2.

Figure 1-2

Phase Shift

In alternating current, the instantaneous value of i is

ei = Llm cos t

This is the equation for the instantaneous value of the alternating voltage. It is also the equation ofa cosine curve, a curve that has the same shape as a sine wave curve but differs in phase from it by900 (I j 4 cycle). This phase difference exists because the counter voltage reaches its maximum notat the time of maximum current, but at the time the current is changing most rapidly; that is, at


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the time when i is zero. The counter voltage is in such a direction as to oppose the change incurrent. Hence, if i is increasing, the counter voltage will be in the opposite direction to thecurrent. Figure 202-2 illustrates this condition. When i is decreasing, the direction of the voltage isthe same as that of the current. The counter voltage (ec) lags the current (i) by 90 degrees.

ec

Figure 1-3Voltage and Current Relationships in an Inductor

An Analogy in Figure 1-4 what is in the black box? By Ohm's law 10 volts will drive amp of current

through 20 ohms of resistance:

Figure 1-4Black Box Example

Because there are 10 ohms of resistance visible in the drawing, we must assume that the black boxcontains a 10 ohm resistor.

There is another alternative, however. A 5 volt battery connected in opposition to the battery Blmakes the total potential applied across the 10 ohm resistor only 5 volts (reference Figure 1-5)again by Ohm's law,


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Figure 1-5The Secret of the Black Box


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2 INDUCTIVE REACTANCEThe counter voltage produced in a coil with an alternating current passing through it opposes theapplied voltage. As in the previous analogy, the opposing voltage reduces the current. This apparentopposition to current flow in an inductor is called inductive reactance. The unit of measurement isthe ohm. The higher the inductance value of the coil, the greater will be the counter voltage, and

larger counter voltages mean higher reactances. The counter voltage is also dependent upon howfast the field is changing. The rate of change for alternating current is determined by the frequency(frequency implies a cyclical change). Inductive reactance is found by using the formula,

XL = 2NfL

where XL = the inductive reactance in ohms, 2N = 6.28, L = the inductance in henrys, and f = thefrequency in hertz.

Example 6-1Problem:

Find the inductive reactance of a 10 Henry inductor at a frequency of 60 hertz (Hz):

Solution:

XL = 2NfL

XL = 6.28 x 60 x10 = 3768

2.1 POWER

In a dc circuit, power is equal to E x I (voltage times current). In an ac circuit, the actual power isless than the voltage-current product, whenever there is any phase shift in the circuit. This is truebecause maximum voltage and maximum current do not occur at the same time. The maximumvoltage-current product is never realized and thus the maximum power is not produced.

The voltage-current product (E x I) is called apparent power. The true power depends upon thephase angle and is expressed by the formula:

true power = apparent power X cosine of the phase angle,

or true power = E x I cosine o

The cosine is simply the ratio of resistance to Impedance.

The cosine is a trigonometric relationship defined as:


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2.1.1 Power Factor

The cosine of o (theta) is also known as the power factor. It is often multiplied by 100 so that it canbe expressed as a percentage. In the case of the previous example the cosine of o was found to be0.6. Multiplying by 100, the power factor is 60%. This is interpreted to mean that the true power isequal to 60% of the apparent power.

2.1.2 More Cosine o

1. Series Circuits Only:

The cosine of o can also be expressed as:

2. Parallel Circuits Only:

The cosine of o can also be expressed as:

where ERis the voltage across the resistor, IR is the current through it, EZ is the voltage acrossthe total impedance, and IZ is the circuit current. These quantities can also be plotted on a vectordiagram.

If cosine values are plotted against time, the result will be a curve identical in shape to the sinecurve, but displaced in time by 90.

2.2 INDUCTIVE REACTANCE IN SERIES AND PARALLEL

When inductances are connected in series and are not close enough to be in the magnetic field ofeach other, the inductances and their inductive reactances add like resistances connected in series.Thus, in a series circuit the sum of the inductive reactances can be expressed by the equation,

and the sum. of the inductances by the equation,

When inductances are connected in parallel, their inductances and the inductive reactances add bythe sum of the reciprocals method, like resistances connected in parallel. In a parallel circuit, thesum of the inductive reactances is expressed by the equation,


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and the sum of the inductances, by the equation,


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3 CAPACITIVE REACTANCEA capacitor also exhibits an opposition to current in an ac circuit. The mechanism is similar to thatof inductive reactance in the sense that the opposition is due to an opposing voltage instead ofheat-producing resistance.

Capacitive reactance (Xc) also produces a 90 phase shift, but in the opposite direction from thephase shift in an inductor. In a capacitor, the current leads the voltage by 90 where current lags by

90 in an inductor. Figure 3-1shows a vector diagram of resistance, capacitive reactance, andinductive reactance.

The reactance of a capacitor is also dependent upon the frequency of the ac sine wave current.However, capacitive reactance decreases as the frequency increases as opposed to inductivereactance which increases as the frequency increases.

Figure 3-1

Resistance, Capacitive Reactance, and Inductive Reactance

The formula for capacitive reactance is

Where Xc = capacitive reactance

2N= 6.28

f = the frequency in hertz (cycles per second)C = capacitance in farads

Example 6-7

Problem: Find the capacitive reactance of a 1 fd capacitor at 60 Hz.

Solution:


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Therefore

Xc = 2650 Ohms

If there is only capacitance in the circuit, the special forms of ac Ohms law apply.

where I = current, E = voltage, and Xc = capacitive reactance.

3.1 PARALLEL RESONANCE

The parallel resonant circuit shown in Figure 3-2 is often called a tank circuit. The unique resonantcondition provides energy storage in the capacitor that is exactly equal to the energy storage in themagnetic field of the inductor. Assuming the capacitor to be fully charged to start, the capacitorwill discharge through the inductor storing the capacitor's stored energy in the inductor's magnetic

field. When the capacitor is discharged, the inductor's field begins to collapse, driving its storedenergy back into the capacitor. Thus, current will continue to circulate from inductor to capacitorand back again.

Figure 3-2Resonant tank circuit

If there were no losses in the circuit, the current would circulate forever. In real circuits there isalways some resistance and this resistance gradually dissipates the energy in the form of heat. The


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smaller the resistance (in dotted lines) the faster the circulating energy is dissipated.

"Q" is measured by the relationship Q= XL/ R. It can also be written as Q = Xc/R because, atresonance, XL = Xc. A high value of the quality Q means the energy of a tank circuit will circulatelonger than it will with a lower value Q.

3.1.1 Impedance of a Parallel Resonant Circuit

Because XL cancels Xc at resonance, the impedance is simply the resistance if the resistance is inparallel as shown in Figure 1-8. If the resistance is in series, the impedance approaches infinity. Thereason for this is the circulating current. The only current demanded by the parallel tank is thatwhich is lost in heat by the series resistance. With a small series resistance and a large value for XL(and Xc), the current required to maintain the circulating current is very small. A small currentmeans high impedance. A parallel tank has high impedance at resonance.

3.1.2 Resonant Frequency and Bandwidth

Every parallel inductor-capacitor circuit will be resonant at some frequency. When you examineFigure 3-3, you will see that as the frequency increases, X L increases and Xc decreases. The XC curvein the figure is going downward while the XL curve is going upward. The two curves must inevitablycross somewhere. The point at which they cross (point 0) is the resonant frequency, because at thispoint XL = Xc. The resonant frequency is designated fo.

The resonant frequency can be determined for any inductor / capacitor combination by using thefollowing formula:

Where

fa is the frequency of resonance 2N is the constant; 2 X 3.14159 L is the inductance in henrys C isthe capacitance in farads

Curve A in Figure 3-3 is called the resonant frequency curve, or bandwidth curve. Resonance doesnot occur at a single frequency because all real inductors have'

some resistance. The more resistance there is in the circuit, the flatter and wider the curve will be.A narrow, tall curve results when the Q is high (Q = XLI R) and will be squat and:

X and Z

FREQUENCY

Figure 3-3XL, Xc, and the Resonant Frequency.


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broad when the Q is low. The bandwidth defined as those frequencies within the cu.1 where thecurve is above 70.7% of the ic'_ curve height.

Figure 3-4 shows a high Q and a low I resonant frequency curve. Note the band of frequenciescovered by the low Q curve is wider than that covered by the high Q tank circuit. In manyapplications, resistance is deliberately added to the circuit to make it respond to a wider band offrequencies. In other applications the resistance is kept small to respond to only a narrow band offrequencies.

The bandwidth of a circuit can be found by the equation,

Bandwidth

Where bandwidth is measured at the 70.7% point on the resonance curve


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fa = the resonant frequencyQ = the figure of merit of the tankQ = XL/R, where XL is the inductive reactance at the resonant frequency, and R is the seriesresistance in the tank.

Figure 3-4 Q and bandwidth.


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NOTES: