BASIC ELECTRONICS AC CIRCUIT ANALYSIS. December 2011


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1 AM BASIC ELECTRONICS AC CIRCUIT ANALYSIS December 2011 DISTRIBUTION RESTRICTION: Approved for Pubic Release. Distribution is unlimited. DEPARTMENT OF THE ARMY MILITARY AUXILIARY RADIO SYSTEM FORT HUACHUCA ARIZONA
2 AM Basic Electronics DC Circuits Analysis This Page Intentionally Left Blank ii Ver. 1.0
3 AM Basic Electronics  DC Circuit Analysis CHANGE PAGE LIST OF EFFECTIVE PAGES INSERT LATEST CHANGED PAGES. DISTROY SUPERSEDED PAGES NOTE The portion of this text effected by the changes is indicated by a vertical line in the outer margins of the page. Changes to illustrations are indicated by shaded or screened areas or by miniature pointing hands. Changes of issue for original and changed pages are: ORIGIONAL..0. Page Change NO. No. Title Page NO. Change No. Page No. Change No. *Zero in this column indicates an original page A Change 0 US Army 2. RETAIN THIS NOTICE AND INSERT BEFORE TABLE OF CONTENTS. 3. Holders of AM will verify that page changes and additions indicated above have been entered. This notice page will be retained as a check sheet. This issuance, together with appended pages, is a separate publication. Each notice is to be retained by the stocking points until the standard is completely revised of canceled. Ver. 1.0 iii
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5 AM Basic Electronics  DC Circuit Analysis CONTENTS 1 AC CIRCUIT ANALYSIS REFERENCE Introduction Alternating Current Frequency and Cycle Resistance in AC Circuits Inductance in an AC Circuit INDUCTIVE REACTANCE POWER Power Factor More Cosine o INDUCTIVE REACTANCE IN SERIES AND PARALLEL CAPACITIVE REACTANCE PARALLEL RESONANCE Impedance of a Parallel Resonant Circuit Resonant Frequency and Bandwidth Ver. 1.0 v
6 AM Basic Electronics DC Circuits Analysis IMPROVEMENTS (Suggested corrections, or changes to this document, should be submitted through your State Director to the Regional Director. Any Changes will be made by the National documentation team. DISTRIBUTION Distribution is unlimited. VERSIONS The Versions are designated in the footer of each page if no version number is designated the version is considered to be 1.0 or the original issue. Documents may have pages with different versions designated; if so verify the versions on the Change Page at the beginning of each document. REFERENCES The following references apply to this manual: Allied Communications Publications (ACP): ACP Glossary of Communications Electronics Terms US Army FM/TM Manuals 1. TM Electrical Design, Lightning and Static Electricity Protection 2. TM Facilities Engineering Electrical Facilities Safety 3. TM Grounding and Bonding in Command, Control, Communications, Computer, Intelligence, Surveillance, and Reconnaissance (C4ISR) Facilities 4. TM Electrical Fundamentals, Direct Current 5. TM664 Basic Theory and Use of Electronic Test Equipment US Army Handbooks 1. MILHDBK Grounding, Bonding and Shielding Design Practices Commercial References 1. Basic Electronics, Components, Devices and Circuits; ISBN X, By William P Hand and Gerald Williams Glencoe/McGraw Hill Publishing Co. 2. Standard Handbook for Electrical Engineers  McGraw Hill Publishing Co. CONTRIBUTORS This document has been produced by the Army MARS Technical Writing Team under the authority of Army MARS HQ, Ft Huachuca, AZ. The following individuals are subject matter experts who made significant contributions to this document. William P Hand vi Ver. 1.0
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9 1 AC CIRCUIT ANALYSIS 1.1 REFERENCE Basic Electronics, Components, Devices and Circuits; ISBN X By William P Hand and Gerald Williams Introduction Alternating current (ac) is probably the most common, and most important, available form of electricity. Alternating current is a current that begins at zero, rises to some set value, and then falls to zero again. It then reverses its direction of current flow and rises to the same set value in the reverse direction, and then falls to zero again. This reversal of current flow direction is in contrast to direct current (dc), which always maintains the same direction of flow. Standard alternating current can be plotted on a graph as shown in Figure 11. The graph shows how the waveform is produced by an alternating current generator as the armature (rotating part) rotates through 360 circular degrees for each cycle Alternating Current Figure 11 A Sine Wave Voltage As the generator armature moves through one 360 degree rotation (full circle), the generator voltage goes through one complete cycle, as shown in Figure 11. The curve displayed in the figure can also be described by the mathematical equation e = E m sin ωt (ω = 2Nf) where e equals the voltage, Em the maximum value of generated voltage, and wt the angular velocity multiplied by the time. When a generator produces an ac voltage, the current arising from it varies in step with the voltage. Like the voltage, the current can be represented graphically by a sine wave and by the following equation: i = Im sin ωt 11 Ver. 1.0
10 where i equals the current, Im the maximum value of generated current, and ωt the angular velocity multiplied by the time Frequency and Cycle While the coil in a generator rotates 360 (one complete revolution), the output voltage goes through 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 then decreases again to zero, The first 180 (onehalf of the voltage cycle) is called the positive alternation and the last 180, from 180 to 360, is called the negative alternation. The value of the Em voltage at 90 is called the amplitude or peak voltage. The time required for a positive and a negative alternation is called the period. The number of complete cycles per second is the frequency of the sine wave. When the angular velocity, ω, at which the coil rotates, is expressed in radians per second, the mathematical relation between ω and f is given by the equation ω = 2Nf Resistance in AC Circuits Resistance is the property by which a conductor opposes the flow of current. The resistance of a conductor opposes alternating current in the same way that it opposes direct current Inductance in an AC Circuit The discussion of induction you learned that a coil opposes a change in the current through it by building 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 the current changes). Reference Figure 12. Figure 12 Phase Shift In alternating current, the instantaneous value of i is e i = Ll m cos ωt This is the equation for the instantaneous value of the alternating voltage. It is also the equation of a cosine curve, a curve that has the same shape as a sine wave curve but differs in phase from it by 900 (I j 4 cycle). This phase difference exists because the counter voltage reaches its maximum not at the time of maximum current, but at the time the current is changing most rapidly; that is, at 12 Ver. 1.0
11 the time when i is zero. The counter voltage is in such a direction as to oppose the change in current. Hence, if i is increasing, the counter voltage will be in the opposite direction to the current. Figure illustrates this condition. When i is decreasing, the direction of the voltage is the same as that of the current. The counter voltage (ec) lags the current (i) by 90 degrees. e c Figure 13 Voltage and Current Relationships in an Inductor An Analogy in Figure 14 what is in the black box? By Ohm's law 10 volts will drive ½ amp of current through 20 ohms of resistance: Figure 14 Black Box Example Because there are 10 ohms of resistance visible in the drawing, we must assume that the black box contains a 10 ohm resistor. There is another alternative, however. A 5 volt battery connected in opposition to the battery Bl makes the total potential applied across the 10 ohm resistor only 5 volts (reference Figure 15) again by Ohm's law, 13 Ver. 1.0
12 Figure 15 The Secret of the Black Box 14 Ver. 1.0
13 2 INDUCTIVE REACTANCE The counter voltage produced in a coil with an alternating current passing through it opposes the applied voltage. As in the previous analogy, the opposing voltage reduces the current. This apparent opposition to current flow in an inductor is called inductive reactance. The unit of measurement is the 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 how fast 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, X L = 2NfL where X L = the inductive reactance in ohms, 2N = 6.28, L = the inductance in henrys, and f = the frequency in hertz. Example 61 Problem: Find the inductive reactance of a 10 Henry inductor at a frequency of 60 hertz (Hz): Solution: 2.1 POWER X L = 2NfL X L = 6.28 x 60 x10 = 3768 In a dc circuit, power is equal to E x I (voltage times current). In an ac circuit, the actual power is less than the voltagecurrent product, whenever there is any phase shift in the circuit. This is true because maximum voltage and maximum current do not occur at the same time. The maximum voltagecurrent product is never realized and thus the maximum power is not produced. The voltagecurrent product (E x I) is called apparent power. The true power depends upon the phase 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: 21 Ver. 1.0
14 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 can be expressed as a percentage. In the case of the previous example the cosine of o was found to be 0.6. Multiplying by 100, the power factor is 60%. This is interpreted to mean that the true power is equal to 60% of the apparent power 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 E R is the voltage across the resistor, I R is the current through it, E Z is the voltage across the total impedance, and I Z is the circuit current. These quantities can also be plotted on a vector diagram. If cosine values are plotted against time, the result will be a curve identical in shape to the sine curve, but displaced in time by INDUCTIVE REACTANCE IN SERIES AND PARALLEL When inductances are connected in series and are not close enough to be in the magnetic field of each 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 by the sum of the reciprocals method, like resistances connected in parallel. In a parallel circuit, the sum of the inductive reactances is expressed by the equation, 22 Ver. 1.0
15 and the sum of the inductances, by the equation, 23 Ver. 1.0
16
17 3 CAPACITIVE REACTANCE A capacitor also exhibits an opposition to current in an ac circuit. The mechanism is similar to that of inductive reactance in the sense that the opposition is due to an opposing voltage instead of heatproducing resistance. Capacitive reactance (X c ) also produces a 90 phase shift, but in the opposite direction from the phase shift in an inductor. In a capacitor, the current leads the voltage by 90 where current lags by 90 in an inductor. Figure 31shows a vector diagram of resistance, capacitive reactance, and inductive 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 inductive reactance which increases as the frequency increases. Figure 31 Resistance, Capacitive Reactance, and Inductive Reactance The formula for capacitive reactance is Where X c = capacitive reactance 2N= 6.28 f = the frequency in hertz (cycles per second) C = capacitance in farads Example 67 Problem: Find the capacitive reactance of a 1 µfd capacitor at 60 Hz. Solution: 31 Ver. 1.0
18 Therefore X c = 2650 Ohms If there is only capacitance in the circuit, the special forms of ac Ohms law apply. where I = current, E = voltage, and X c = capacitive reactance. 3.1 PARALLEL RESONANCE The parallel resonant circuit shown in Figure 32 is often called a tank circuit. The unique resonant condition provides energy storage in the capacitor that is exactly equal to the energy storage in the magnetic field of the inductor. Assuming the capacitor to be fully charged to start, the capacitor will 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 stored energy back into the capacitor. Thus, current will continue to circulate from inductor to capacitor and back again. Figure 32 Resonant tank circuit If there were no losses in the circuit, the current would circulate forever. In real circuits there is always some resistance and this resistance gradually dissipates the energy in the form of heat. The 32 Ver. 1.0
19 smaller the resistance (in dotted lines) the faster the circulating energy is dissipated. "Q" is measured by the relationship Q= X L / R. It can also be written as Q = X c /R because, at resonance, X L = X c. A high value of the quality Q means the energy of a tank circuit will circulate longer than it will with a lower value Q Impedance of a Parallel Resonant Circuit Because X L cancels X c at resonance, the impedance is simply the resistance if the resistance is in parallel as shown in Figure 18. If the resistance is in series, the impedance approaches infinity. The reason for this is the circulating current. The only current demanded by the parallel tank is that which is lost in heat by the series resistance. With a small series resistance and a large value for XL (and X c ), the current required to maintain the circulating current is very small. A small current means high impedance. A parallel tank has high impedance at resonance Resonant Frequency and Bandwidth Every parallel inductorcapacitor circuit will be resonant at some frequency. When you examine Figure 33, you will see that as the frequency increases, X L increases and X c decreases. The X C curve in the figure is going downward while the X L curve is going upward. The two curves must inevitably cross somewhere. The point at which they cross (point 0) is the resonant frequency, because at this point X L = X c. The resonant frequency is designated f o. The resonant frequency can be determined for any inductor / capacitor combination by using the following formula: Where fa is the frequency of resonance 2N is the constant; 2 X L is the inductance in henrys C is the capacitance in farads Curve A in Figure 33 is called the resonant frequency curve, or bandwidth curve. Resonance does not 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 = X L I R) and will be squat and: X and Z FREQUENCY Figure 33 X L, X c, and the Resonant Frequency. 33 Ver. 1.0
20 broad when the Q is low. The bandwidth defined as those frequencies within the cu.1 where the curve is above 70.7% of the ic'_ curve height. Figure 34 shows a high Q and a low I resonant frequency curve. Note the band of frequencies covered by the low Q curve is wider than that covered by the high Q tank circuit. In many applications, resistance is deliberately added to the circuit to make it respond to a wider band of frequencies. In other applications the resistance is kept small to respond to only a narrow band of frequencies. 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 34 Ver. 1.0
21 fa = the resonant frequency Q = the figure of merit of the tank Q = X L /R, where X L is the inductive reactance at the resonant frequency, and R is the series resistance in the tank. Figure 34 Q and bandwidth. 35 Ver. 1.0
22 NOTES: 36 Ver. 1.0
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