Schaums outline electric circuits 4th ed (2003)

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THE SOLUTION MANUAL

Contents

1 Introduction

1.1 ELECTRICAL QUANTITIES AND SI UNITS
1.2 FORCE, WORK, AND POWER
1.3 ELECTRIC CHARGE AND CURRENT
1.4 ELECTRIC POTENTIAL
1.5 ENERGY AND ELECTRICAL POWER
1.6 CONSTANT AND VARIABLE FUNCTIONS

2 Circuit Concepts

2.1 PASSIVE AND ACTIVE ELEMENTS
2.2 SIGN CONVENTIONS
2.3 VOLTAGE- CURRENT RELATIONS
2.4 RESISTANCE
2.5 INDUCTANCE
2.6 CAPACITANCE
2.7 CIRCUIT DIAGRAMS
2.8 NONLINEAR RESISTORS

3 Circuit Laws

3.1 INTRODUCTION
3.2 KIRCHHOFF¡¯S VOLTAGE LAW
3.3 KIRCHHOFF¡¯S CURRENT LAW
3.4 CIRCUIT ELEMENTS IN SERIES
3.5 CIRCUIT ELEMENTS IN PARALLEL
3.6 VOLTAGE DIVISION
3.7 CURRENT DIVISION
3.1 Find V3 and its polarity if the current I in the circuit of Fig. 3- 7 is 0.40 A.
3.2 Obtain the currents I1 and I2 for the network shown in Fig. 3- 8.
3.3 Find the current I for the circuit shown in Fig. 3- 9.
3.4 Find the equivalent resistance for the circuit shown in Fig. 3- 10.
3.5 Determine the equivalent inductance of the three parallel inductances shown in Fig. 3- 11.
3.6 Express the total capacitance of the three capacitors in Fig. 3- 12.
3.7 The circuit shown in Fig. 3- 13 is a voltage divider, also called an attenuator. When it is a single
3.8 Find all branch currents in the network shown in Fig. 3- 14( a).

4 Analysis Methods

4.1 THE BRANCH CURRENT METHOD
4.2 THE MESH CURRENT METHOD
4.3 MATRICES AND DETERMINANTS
4.4 THE NODE VOLTAGE METHOD
4.5 INPUT AND OUTPUT RESISTANCES
4.6 TRANSFER RESISTANCE
4.7 NETWORK REDUCTION
4.8 SUPERPOSITION
4.9 THE VENIN¡¯S AND NORTON¡¯S THEOREMS
4.10 MAXIMUM POWER TRANSFER THEOREM
4.1 Use branch currents in the network shown in Fig. 4- 17 to . nd the current supplied by the 60- V
4.2 Solve Problem 4.1 by the mesh current method.
4.3 Solve the network of Problems 4.1 and 4.2 by the node voltage method. See Fig. 4- 19.
4.4 In Problem 4.2, obtain Rinput; and use it to calculate I1.
4.5 Obtain Rtransfer; and Rtransfer; for the network of Problem 4.2 and use them to calculate I2 and
4.6 Solve Problem 4.1 by use of the loop currents indicated in Fig. 4- 20.
4.7 Write the mesh current matrix equation for the network of Fig. 4- 21 by inspection, and solve for
4.8 Solve Problem 4.7 by the node voltage method.
4.9 For the network shown in Fig. 4- 23, . nd Vs which makes I0 ¼ 7: 5 mA.
4.10 In the network shown in Fig. 4- 24, . nd the current in the 10- resistor.
4.11 Find the voltage Vab in the network shown in Fig. 4- 25.
4.12 For the ladder network of Fig. 4- 26, obtain the transfer resistance as expressed by the ratio of Vin
4.13 Obtain a The ´ venin equivalent for the circuit of Fig. 4- 26 to the left of terminals ab.
4.14 Use superposition to . nd the current I from each voltage source in the circuit shown in Fig. 4- 30.
4.15 Obtain the current in each resistor in Fig. 4- 31( a), using network reduction methods.
4.16 Find the value of the adjustable resistance R which results in maximum power transfer across the

5 Amplifiers and Operational Amplifier Circuits

5.1 AMPLIFIER MODEL
5.2 FEEDBACK IN AMPLIFIER CIRCUITS
5.3 OPERATIONAL AMPLIFIERS
5.4 ANALYSIS OF CIRCUITS CONTAINING IDEAL OP AMPS
5.5 INVERTING CIRCUIT
5.6 SUMMING CIRCUIT
5.7 NONINVERTING CIRCUIT
5.8 VOLTAGE FOLLOWER
5.9 DIFFERENTIAL AND DIFFERENCE AMPLIFIERS
5.10 CIRCUITS CONTAINING SEVERAL OP AMPS
5.11 INTEGRATOR AND DIFFERENTIATOR CIRCUITS
5.12 ANALOG COMPUTERS
5.13 LOW- PASS FILTER
5.14 COMPARATOR
5.1 In Fig. 5- 3, let vs ¼ 20 V, Rs ¼ 10 , Ri ¼ 990 , k ¼ 5, and Ro ¼ 3 . Find ( a) the The ´ venin
5.2 In the circuits of Figs. 5- 4 and 5- 5 let R1 ¼ 1k and R2 ¼ 5k . Find the gains ¼ v2= vs in
5.3 Let R1 ¼ 1k , R2 ¼ 5k , and Ri ¼ 50 k in the circuit of Fig. 5- 33. Find v2= vs for k ¼ 1, 10,
5.4 Let again R1 ¼ 1k and R2 ¼ 5k in the circuit of Fig. 5- 33.
5.5 Let again R1 ¼ 1 k and R2 ¼ 5 k in the circuit of Fig. 5- 33. Replace the circuit to the left of
5.6 Find the output voltage of an op amp with A 10 and Vcc 10V for v 0 and sin t ( V). ¼ ¼ ¼ ¼
5.7 Repeat Problem 5.6 for ¼ sin 2 t ( V) and v ¼ 0: 5V.
5.8 In the circuit of Fig. 5- 35 vs ¼ sin 100t. Find v1 and v2.
5.9 Saturation levels for the op amps in Fig. 5- 31 are þVcc ¼ 5V and Vcc ¼ 5 V. The reference
5.10 Find v in the circuit of Fig. 5- 36.
5.11 In the circuit of Fig. 5- 37 . nd vC ( the voltage at node C), i1, Rin ( the input resistance seen by the
5.12 Find v2 in Problem 5.11 by replacing the circuit to the left of nodes A- B in Fig. 5- 37 by its
5.13 Find vC, i1, v2, and Rin, the input resistance seen by the 21- V source in Fig. 5- 38.
5.14 In the circuit of Fig. 5- 38 change the 21- V source by a factor of k. Show that vC, i1, v2 in
5.15 Find v2 and vC in Problem 5.13 by replacing the circuit to the left of node C in Fig. 5- 38
5.16 ( a) Find the The ´ venin equivalent of the circuit to the left of nodes A- B in Fig. 5- 39( a) and then
5.17 Find v2 as a function of i1 in the circuit of Fig. 5- 40( a).
5.18 A transducer generates a weak current i1 which feeds a load Rl and produces a voltage v1 across
5.19 Determine the resistor values which would produce a current- to- voltage conversion gain of
5.20 Find i2 as a function of v1 in the circuit of Fig. 5- 43.
5.21 A practical current source ( is in parallel with internal resistance Rs) directly feeds a load Rl as in
5.22 Find vo in the circuit of Fig. 5- 45.
5.25 In Fig. 5- 48 choose resistors for a di . erential gain of 10 so that vo 10 ¼ ðv2 v1Þ.
5.26 Resistors having high magnitude and accuracy are expensive. Show that in the circuit of Fig. 5-
5.27 Show that in the circuit of Fig. 5- 50 i1 ¼ i2, regardless of the circuits of N1 and N2.
5.28 Let N1 be the voltage source v1 and N2 be the resistor R2 in the circuit of Fig. 5- 50. Find the
5.29 A voltage follower is constructed using an op amp with a . nite open- loop gain A and Rin ¼ 1

6 Waveforms and Signals

6.1 INTRODUCTION
6.2 PERIODIC FUNCTIONS
6.3 SINUSOIDAL FUNCTIONS
6.4 TIME SHIFT AND PHASE SHIFT
6.5 COMBINATIONS OF PERIODIC FUNCTIONS
6.6 THE AVERAGE AND EFFECTIVE ( RMS) VALUES
6.7 NONPERIODIC FUNCTIONS
6.8 THE UNIT STEP FUNCTION
6.9 THE UNIT IMPULSE FUNCTION
6.10 THE EXPONENTIAL FUNCTION
6.11 DAMPED SINUSOIDS
6.12 RANDOM SIGNALS
6.1 Find the maximum and minimum values of v ¼ 1 þ 2 sinð! t þ Þ, given ! ¼ 1000 rad/ s and ¼ 3
6.2 In a microwave range measurement system the electromagnetic signal v1 ¼ Asin 2 ft, with
6.3 Show that if periods T1 and T2 of two periodic functions v1ðtÞ and v2ðtÞ have a common multiple,
6.4 Show that the average of cos t is 1/ 2. ð! þ Þ
6.5 Let Vdc Vac cos t Show that V V V ac. vðtÞ ¼ þ ð! þ Þ. ¼ þ
6.6 Let f1 and f2 be two di . erent harmonics of f0. Show that the e . ective value of
6.7 The signal vðtÞ in Fig. 6- 16 is sinusoidal. Find its period and frequency. Express it in the form
6.8 Let v1 ¼ cos 200 t and v2 ¼ cos 202 t. Show that v ¼ v1 þ v2 is periodic. Find its period, Vmax,
6.9 Convert vðtÞ ¼ 3 cos 100t þ 4 sin 100t to Asinð100t þ Þ.
6.10 Find the average and e . ective value of v2ðtÞ in Fig. 6- 1( b) for V1 ¼ 2, V2 ¼ 1, T ¼ 4T1.
6.11 Find V3; and V3; in Fig. 6- 1( c) for T ¼ 100T1.
6.12 Referring to Fig. 6- 1( d), let T ¼ 6 and let the areas under the positive and negative sections of
6.13 Find the average and e . ective value of the half- recti . ed cosine wave v1ðtÞ shown in Fig. 6- 17( a).
6.14 Find the average and e . ective value of the full- recti . ed cosine wave v2ðtÞ ¼ Vmj cos 2 t= Tj shown
6.15 A 100- mH inductor in series with 20- resistor [ Fig. 6- 18( a)] carries a current i as shown in
6.16 A radar signal sðtÞ, with amplitude Vm ¼ 100 V, consists of repeated tone bursts. Each tone
6.17 An appliance uses Veff ¼ 120V at 60 Hz and draws Ieff ¼ 10A with a phase lag of 608. Express
6.18 A narrow pulse is of 1- A amplitude and 1- ms duration enters a 1- mF capacitor at t ¼ 0, as shown
6.19 The narrow pulse is of Problem 6.18 enters a parallel combination of a 1- mF capacitor and a
6.20 Plot the function vðtÞ which varies exponentially from 5V at t ¼ 0 to 12V at t ¼ 1 with a time
6.21 The voltage v ¼ V0e for a > 0 is connected across a parallel combination of a resistor and a

7 First- Order Circuits

7.1 INTRODUCTION
7.2 CAPACITOR DISCHARGE IN A RESISTOR
7.3 ESTABLISHING A DC VOLTAGE ACROSS A CAPACITOR
7.4 THE SOURCE- FREE RL CIRCUIT
7.5 ESTABLISHING A DC CURRENT IN AN INDUCTOR
7.6 THE EXPONENTIAL FUNCTION REVISITED
7.7 COMPLEX FIRST- ORDER RL AND RC CIRCUITS
7.8 DC STEADY STATE IN INDUCTORS AND CAPACITORS
7.9 TRANSITIONS AT SWITCHING TIME
7.10 RESPONSE OF FIRST- ORDER CIRCUITS TO A PULSE
7.11 IMPULSE RESPONSE OF RC AND RL CIRCUITS
7.12 SUMMARY OF STEP AND IMPULSE RESPONSES IN RC AND RL CIRCUITS
7.13 RESPONSE OF RC AND RL CIRCUITS TO SUDDEN EXPONENTIAL EXCITATIONS
7.14 RESPONSE OF RC AND RL CIRCUITS TO SUDDEN SINUSOIDAL EXCITATIONS
7.15 SUMMARY OF FORCED RESPONSE IN FIRST- ORDER CIRCUITS
7.16 FIRST- ORDER ACTIVE CIRCUITS
7.1 At t ¼ 0 just before the switch is closed in Fig. 7- 20, vC ¼ 100 V. Obtain the current and
7.2 In Problem 7.1, obtain the power and energy in the resistor, and compare the latter with the initial
7.3 An RC transient identical to that in Problems 7.1 and 7.2 has a power transient
7.4 The switch in the RL circuit shown in Fig. 7- 21 is moved from position 1 to position 2 at t ¼ 0.
7.5 For the transient of Problem 7.4 obtain pR and pL.
7.6 A series RC circuit with R ¼ 5k and C ¼ 20 mF has a constant- voltage source of 100V applied
7.7 The switch in the circuit shown in Fig. 7- 22( a) is closed at t ¼ 0, at which moment the capacitor
7.8 Obtain the current i, for all values of t, in the circuit of Fig. 7- 23.
7.9 In Fig. 7- 24( a), the switch is closed at t ¼ 0. The capacitor has no charge for t < 0. Find iR, iC,
7.10 In Fig. 7- 25, the switch is opened at t ¼ 0. Find iR, iC, vC, and vs.
7.11 The switch in the circuit of Fig. 7- 26 is closed on position 1 at t ¼ 0 and then moved to 2 after one
7.12 A series RL circuit has a constant voltage V applied at t ¼ 0. At what time does vR ¼ vL?
7.13 A constant voltage is applied to a series RL circuit at t ¼ 0. The voltage across the inductance is
7.14 In Fig. 7- 28, switch S1 is closed at t ¼ 0. Switch S2 is opened at t ¼ 4 ms. Obtain i for t > 0.
7.15 In the circuit of Fig. 7- 29, the switch is closed at t ¼ 0, when the 6- mF capacitor has charge
7.16 In the circuit shown in Fig. 7- 30, the switch is moved to position 2 at t ¼ 0. Obtain the current i2
7.17 In Fig. 7- 31, the switch is closed at t ¼ 0. Obtain the current i and capacitor voltage vC, for
7.18 The switch in the two- mesh circuit shown in Fig. 7- 32 is closed at t ¼ 0. Obtain the currents i1
7.19 A series RL circuit, with R ¼ 50 and L ¼ 0: 2 H, has a sinusoidal voltage
7.20 For the circuit of Fig. 7- 33, obtain the current iL, for all values of t.
7.21 The switch in Fig. 7- 34 has been in position 1 for a long time; it is moved to 2 at t ¼ 0. Obtain
7.22 The switch in the circuit shown in Fig. 7- 35 is moved from 1 to 2 at t ¼ 0. Find vC and vR, for
7.23 Obtain the energy functions for the circuit of Problem 7.22.
7.24 A series RC circuit, with R ¼ 5k and C ¼ 20 mF, has two voltage sources in series,

8 Higher-Order Circuits and Complex Frequency

8.1 INTRODUCTION
8.2 SERIES RLC CIRCUIT
Overdamped Case ð > ! 0Þ
Critically Damped Case ð ¼ ! 0Þ
Underdamped or Oscillatory Case ð < ! 0Þ
8.3 PARALLEL RLC CIRCUIT
Overdamped Case > ! ð 0Þ
Underdamped ( Oscillatory) Case > ð! Þ
8.4 TWO- MESH CIRCUIT
8.5 COMPLEX FREQUENCY
8.6 GENERALIZED IMPEDANCE ( R; L; C) IN s- DOMAIN
8.7 NETWORK FUNCTION AND POLE- ZERO PLOTS
8.8 THE FORCED RESPONSE
8.9 THE NATURAL RESPONSE
8.10 MAGNITUDE AND FREQUENCY SCALING
8.11 HIGHER- ORDER ACTIVE CIRCUITS
8.1 A series RLC circuit, with R ¼ 3k , L ¼ 10 H, and C ¼ 200 mF, has a constant- voltage source,
8.2 A series RLC circuit, with R ¼ 50 ; L ¼ 0: 1H; and C ¼ 50 mF, has a constant voltage V ¼ 100V
8.3 Rework Problem 8.2, if the capacitor has an initial charge Q0 ¼ 2500 mC.
8.4 A parallel RLC network, with R ¼ 50: 0 , C ¼ 200 mF, and L ¼ 55: 6 mH, has an initial charge
8.5 In Fig. 8- 19, the switch is closed at t ¼ 0. Obtain the current i and capacitor voltage vC, for
8.6 For the time functions listed in the . rst column of Table 8- 2, write the corresponding amplitude
8.7 For each amplitude and phase angle in the . rst column and complex frequency s in the second
8.8 An amplitude and phase angle of 10 2 458V has an associated complex frequency .
8.9 A passive network contains resistors, a 70- mH inductor, and a 25- mF capacitor. Obtain the
8.10 For the circuit shown in Fig. 8- 20, obtain v at t 0: 1 s for source current i 10 cos 2t ( A), ¼ ðaÞ ¼
8.11 Obtain the impedance ZinðsÞ for the circuit shown in Fig. 8- 21 at ( a) s ¼ 0, ( b) s ¼ j4 rad/ s,
8.12 Express the impedance ZðsÞ of the parallel combination of L ¼ 4H and C ¼ 1 F. At what
8.13 The circuit shown in Fig. 8- 22 has a voltage source connected at terminals ab. The response to
8.14 Obtain HðsÞ for the network shown in Fig. 8- 23, where the excitation is the driving current IðsÞ
8.15 For the two- port network shown in Fig. 8- 24 . nd the values of R1, R2, and C, given that the
8.16 Construct the pole- zero plot for the transfer admittance function
8.17 Obtain the natural frequencies of the network shown in Fig. 8- 26 by driving it with a conveniently
8.18 Repeat Problem 8.17, now driving the network with a conveniently located voltage source.
8.19 A 5000- rad/ s sinusoidal source, V ¼ 100 08V in phasor form, is applied to the circuit of
8.20 Refer to Fig. 8- 28. Obtain Vo= Vi for s j4 10 rad/ s. Scale the network with HðsÞ ¼ ¼
8.21 A three- element series circuit contains R ¼ 5 , L ¼ 4 H, and C ¼ 3: 91 mF. Obtain the series

9 Sinusoidal Steady-State Circuit Analysis

9.1 INTRODUCTION
9.2 ELEMENT RESPONSES
9.3 PHASORS
9.4 IMPEDANCE AND ADMITTANCE
9.5 VOLTAGE AND CURRENT DIVISION IN THE FREQUENCY DOMAIN
9.6 THE MESH CURRENT METHOD
9.7 THE NODE VOLTAGE METHOD
9.8 THE VENIN¡¯S AND NORTON¡¯S THEOREMS
9.9 SUPERPOSITION OF AC SOURCES
9.1 A 10- mH inductor has current i ¼ 5: 0 cos 2000t ( A). Obtain the voltage vL.
9.2 A series circuit, with R ¼ 10 and L ¼ 20 mH, has current i ¼ 2: 0 sin 500t ( A). Obtain total
9.3 Find the two elements in a series circuit, given that the current and total voltage are
9.4 A series circuit, with R ¼ 2: 0 and C ¼ 200 pF, has a sinusoidal applied voltage with a frequency
9.5 The current in a series circuit of R ¼ 5 and L ¼ 30mH lags the applied voltage by 808.
9.6 At what frequency will the current lead the voltage by 308 in a series circuit with R ¼ 8 and
9.7 A series RC circuit, with R ¼ 10 , has an impedance with an angle of 458 at f1 ¼ 500 Hz. Find
9.8 A two- element series circuit has voltage V ¼ 240 08V and current I ¼ 50 608 A. Determine
9.9 For the circuit shown in Fig. 9- 18, obtain Zeq and compute I.
9.10 Evaluate the impedance Z1 in the circuit of Fig. 9- 19.
9.11 Compute the equivalent impedance Zeq and admittance Yeq for the four- branch circuit shown in
9.12 The total current I entering the circuit shown in Fig. 9- 20 is 33: 0 13: 08 A. Obtain the branch
9.13 Find Z1 in the three- branch network of Fig. 9- 21, if I ¼ 31: 5 24: 08 A for an applied voltage
9.14 The constants R and L of a coil can be obtained by connecting the coil in series with a known
9.15 In the parallel circuit shown in Fig. 9- 24, the e . ective values of the currents are: Ix ¼ 18: 0A,
9.16 Obtain the phasor voltage VAB in the two- branch parallel circuit of Fig. 9- 26.
9.17 In the parallel circuit shown in Fig. 9- 27, VAB ¼ 48: 3 308 V. Find the applied voltage V.
9.18 Obtain the voltage Vx in the network of Fig. 9- 28, using the mesh current method.
9.19 In the netwrok of Fig. 9- 29, determine the voltage V which results in a zero current through the
9.20 Solve Problem 9.19 by the node voltage method.
9.21 Use the node voltage method to obtain the current I in the network of Fig. 9- 31.
9.22 Find the input impedance at terminals ab for the network of Fig. 9- 32.
9.23 For the network in Fig. 9- 32, obtain the current in the inductor, Ix, by . rst obtaining the transfer
9.24 For the network in Fig. 9- 32, . nd the value of the source voltage V which results in
9.25 For the network shown in Fig. 9- 33, obtain the input admittance and use it to compute node
9.26 For the network of Problem 9.25, compute the transfer admittance Ytransfer; and use it to obtain
9.27 Replace the active network in Fig. 9- 34( a) at terminals ab with a The ´ venin equivalent.
9.28 For the network of Problem 9.27, obtain a Norton equivalent circuit ( Fig. 9- 35).
9.29 Obtain the The ´ venin equivalent for the bridge circuit of Fig. 9- 36. Make the voltage of a with
9.30 Replace the network of Fig. 9- 37 at terminals ab with a Norton equivalent and with a The ´ venin

10 AC Power

10.1 POWER IN THE TIME DOMAIN
10.2 POWER IN SINUSOIDAL STEADY STATE
10.3 AVERAGE OR REAL POWER
10.4 REACTIVE POWER
10.5 SUMMARY OF AC POWER IN R, L, AND C
10.6 EXCHANGE OF ENERGY BETWEEN AN INDUCTOR AND A CAPACITOR
10.7 COMPLEX POWER, APPARENT POWER, AND POWER TRIANGLE
10.8 PARALLEL- CONNECTED NETWORKS
10.9 POWER FACTOR IMPROVEMENT
10.10 MAXIMUM POWER TRANSFER
10.11 SUPERPOSITION OF AVERAGE POWERS
10.1 The current plotted in Fig. 10- 2( a) enters a 0.5- mF capacitor in series with a 1- k resistor. Find
10.2 A 1- V ac voltage feeds ( a) a1- resistor, ( b) a load Z ¼ 1 þ j, and ( c) a load Z ¼ 1 j. Find P
10.3 Obtain the complete power information for a passive circuit with an applied voltage v ¼
10.4 A two- element series circuit has average power 940W and power factor 0.707 leading. Determine
10.5 Find the two elements of a series circuit having current i ¼ 4: 24 cos ð5000t þ 458Þ A, power 180
10.6 Obtain the power information for each element in Fig. 10- 14 and construct the power triangle.
10.7 A series circuit of R ¼ 10 and XC ¼ 5 has an e . ective applied voltage of 120 V. Determine
10.8 Impedances Zi ¼ 5: 83 59: 08 and Z2 ¼ 8: 94 63: 438 are in series and carry an e . ective
10.9 Obtain the total power information for the parallel circuit shown in Fig. 10- 16.
10.10 Find the power factor for the circuit shown in Fig. 10- 17.
10.11 If the total power in the circuit Fig. 10- 17 is 1100 W, what are the powers in the two resistors?
10.12 Obtain the power factor of a two- branch parallel circuit where the . rst branch has Z1 ¼ 2 þ j4 ,
10.13 A voltage, 28.28 608 V, is applied to a two- branch parallel circuit in which Z1 ¼ 4 308 and
10.14 Determine the total power information for three parallel- connected loads: load # 1, 250 VA,
10.15 Obtain the complete power triangle and the total current for the parallel circuit shown in Fig. 10-
10.16 Obtain the complete power triangle for the circuit shown in Fig. 10- 20, if the total reactive power
10.17 A load of 300 kW, with power factor 0.65 lagging, has the power factor improved to 0.90 lagging
10.18 Find the capacitance C necessary to improve the power factor to 0.95 lagging in the circuit shown
10.19 A circuit with impedance Z ¼ 10: 0 608 has the power factor improved by a parallel capacitive
10.20 A transformer rated at a maximum of 25kVA supplies a 12- kW load at power factor 0.60
10.21 Referring to Problem 10.20, if the additional load has power factor 0.866 leading, how many kVA
10.22 An induction motor with a shaft power output of 1.56 kW has an e . ciency of 85 percent. At

11 Polyphase Circuits

11.1 INTRODUCTION
11.2 TWO- PHASE SYSTEMS
11.3 THREE- PHASE SYSTEMS
11.4 WYE AND DELTA SYSTEMS
11.5 PHASOR VOLTAGES
11.6 BALANCED DELTA- CONNECTED LOAD
11.7 BALANCED FOUR- WIRE, WYE- CONNECTED LOAD
11.8 EQUIVALENT Y- AND - CONNECTIONS
11.9 SINGLE- LINE EQUIVALENT CIRCUIT FOR BALANCED THREE- PHASE LOADS
11.10 UNBALANCED DELTA- CONNECTED LOAD
11.11 UNBALANCED WYE- CONNECTED LOAD
11.12 THREE- PHASE POWER
11.13 POWER MEASUREMENT AND THE TWO- WATTMETER METHOD
11.1 The two- phase balanced ac generator of Fig. 11- 22 feeds two identical loads. The two voltage
11.2 Solve Problem 11.1 given Vp ¼ 110 Vrms and Z ¼ 4 þ j3 .
11.3 Repeat Problem 11.2 but with the two voltage sources of Problem 11.1 908 out of phase.
11.4 Show that the line- to- line voltage VL in a three- phase system is 3 times the line- to- neutral .
11.5 A three- phase, ABC system, with an e . ective voltage 70.7 V, has a balanced - connected load
11.6 A three- phase, three- wire CBA system, with an e . ective line voltage 106.1 V, has a balanced Yconnected
11.7 A three- phase, three- wire CBA system, with an e . ective line voltage 106.1 V, has a balanced -
11.8 A three- phase, three- wire system, with an e . ective line voltage 176.8 V, supplies two balanced
11.9 Obtain the readings when the two- wattmeter method is applied to the circuit of Problem 11.8.
11.10 A three- phase supply, with an e . ective line voltage 240 V, has an unbalanced - connected load
11.11 Obtain the readings of wattmeters placed in lines A and B of the circuit of Problem 11.10 ( Line C
11.12 A three- phase, four- wire, ABC system, with line voltage VBC ¼ 294: 2 08 V, has a Y- connected
11.13 The Y- connected load impedances ZA ¼ 10 08 , Z ¼ 15 308 , and ZC ¼ 10 308 , in Fig.
11.14 Obtain the total average power for the unbalanced, Y- connected load in Problem 11.13, and
11.15 A three- phase, three- wire, balanced, - connected load yields wattmeter readings of 1154W and
11.16 A balanced - connected load, with Z ¼ 30 308 , is connected to a three- phase, three- wire,

12 Frequency Response,Filters, and Resonance

12.1 FREQUENCY RESPONSE
12.2 HIGH- PASS AND LOW- PASS NETWORKS
12.3 HALF- POWER FREQUENCIES
12.4 GENERALIZED TWO- PORT, TWO- ELEMENT NETWORKS
12.5 THE FREQUENCY RESPONSE AND NETWORK FUNCTIONS
12.6 FREQUENCY RESPONSE FROM POLE- ZERO LOCATION
12.7 IDEAL AND PRACTICAL FILTERS
12.8 PASSIVE AND ACTIVE FILTERS
12.9 BANDPASS FILTERS AND RESONANCE
12.10 NATURAL FREQUENCY AND DAMPING RATIO
12.11 RLC SERIES CIRCUIT; SERIES RESONANCE
12.12 QUALITY FACTOR
12.13 RLC PARALLEL CIRCUIT; PARALLEL RESONANCE
12.14 PRACTICAL LC PARALLEL CIRCUIT
12.15 SERIES- PARALLEL CONVERSIONS
12.16 LOCUS DIAGRAMS
12.17 SCALING THE FREQUENCY RESPONSE OF FILTERS
12.1 In the two- port network shown in Fig. 12- 33, R1 ¼ 7 k and R2 ¼ 3k . Obtain the voltage
12.2 ( a) Find L2 in the high- pass circuit shown in Fig. 12- 34, if jHvð! Þj ¼ 0: 50 at a frequency of
12.3 A voltage divider, useful for high- frequency applications, can be made with two capacitors C1
12.4 Find the frequency at which jHvj ¼ 0: 50 for the low- pass RC network shown in Fig. 12- 35.
12.5 For the series RLC circuit shown in Fig. 12- 36, . nd the resonant frequency ! ¼ 2 f0. Also
12.6 Derive the Q of ( a) the series RLC circuit, ( b) the parallel RLC circuit.
12.7 A three- element series circuit contains R ¼ 10 , L ¼ 5 mH, and C ¼ 12: 5 mF. Plot the magnitude
12.8 Show that ! ¼ ! l! for the series RLC circuit. . . . . .
12.9 Compute the quality factor of an RLC series circuit, with R ¼ 20 , L ¼ 50 mH, and C ¼ 1 mF,
12.10 A coil is represented by a series combination of L ¼ 50 mH and R ¼ 15 . Calculate the quality
12.11 Convert the circuit constants of Problem 12.10 to the parallel form ( a) at 10 kHz, ( b) at 250 Hz.
12.12 For the circuit shown in Fig. 12- 40, ( a) obtain the voltage transfer function Hvð! Þ, and ( b) . nd the
12.13 Obtain the bandwidth for the circuit of Fig. 12- 40 and plot against the parameter
12.14 In the circuit of Fig. 12- 40, let R1 ¼ R2 ¼ 2 k , L ¼ 10 mH, and C ¼ 40 nF. Find the
12.15 For the circuit of Fig. 12- 40, R1 ¼ 5 k and C ¼ 10 nF. If V2= V1 ¼ 0: 8 08 at 15 kHz, calculate
12.16 Compare the resonant frequency of the circuit shown in Fig. 12- 43 for R ¼ 0 to that for
12.17 Measurements on a practical inductor at 10MHz give L ¼ 8: 0 mH and Qind ¼ 40. ( a) Find the
12.18 A lossy capacitor, in the series- circuit model, consists of R ¼ 25 and C ¼ 20 pF. Obtain the
12.19 A variable- frequency source of V ¼ 100 08 V is applied to a series RL circuit having R ¼ 20
12.20 The circuit shown in Fig. 12- 46 is in resonance for two values of C when the frequency of the
12.21 Show by locus diagrams that the magnitude of the voltage between points A and B in Fig. 12- 48 is

13 Two- Port Networks

13.1 TERMINALS AND PORTS
13.2 Z- PARAMETERS
13.3 T- EQUIVALENT OF RECIPROCAL NETWORKS
13.4 Y- PARAMETERS
13.5 PI- EQUIVALENT OF RECIPROCAL NETWORKS
13.6 APPLICATION OF TERMINAL CHARACTERISTICS
13.7 CONVERSION BETWEEN Z- AND Y- PARAMETERS
13.8 h- PARAMETERS
13.9 g- PARAMETERS
13.10 TRANSMISSION PARAMETERS
13.11 INTERCONNECTING TWO- PORT NETWORKS
13.12 CHOICE OF PARAMETER TYPE
13.13 SUMMARY OF TERMINAL PARAMETERS AND CONVERSION
13.1 Find the Z- parameters of the circuit in Fig. 13- 16( a).
13.2 The Z- parameters of a two- port network N are given by
13.3 Find the Z- parameters of the two- port network in Fig. 13- 18.
13.4 Find the Z- parameters of the two- port network in Fig. 13- 19 and compare the results with those
13.5 Find the Y- parameters of Fig. 13- 19 using its Z- parameters.
13.6 Find the Y- parameters of the two- port network in Fig. 13- 20 and thus show that the networks of
13.7 Apply the short- circuit equations ( 10) to . nd the Y- parameters of the two- port network in Fig.
13.8 Apply KCL at the nodes of the network in Fig. 13- 21 to obtain its terminal characteristics and Yparameters.
13.9 Z- parameters of the two- port network N in Fig. 13- 22( a) are Z11 ¼ 4s, Z12 ¼ Z21 ¼ 3s, and
13.10 Two two- port networks a and b, with open- circuit impedances Za and Zb, are connected in series
13.11 Two two- port networks a and b, with short- circuit admittances Ya and Yb, are connected in
13.12 Find ( a) the Z- parameters of the circuit of Fig. 13- 23( a) and ( b) an equivalent model which uses
13.13 ( a) Obtain the Y- parameters of the circuit in Fig. 13- 23( a) from its Z- parameters. ( b) Find
13.14 Referring to the network of Fig. 13- 23( b), convert the voltage source and its series resistor to its
13.15 The h- parameters of a two- port network are given. Show that the network may be modeled by
13.16 Find the h- parameters of the circuit in Fig. 13- 25.
13.17 Find the h- parameters of the circuit in Fig. 13- 25 from its Z- parameters and compare with results
13.18 The simpli . ed model of a bipolar junction transistor for small signals is shown in the circuit of
13.19 h- parameters of a two- port device H are given by
13.20 The device H of Problem 13- 19 is placed in the circuit of Fig. 13- 29( a). Replace H by its model
13.21 A load ZL is connected to the output of a two- port device N ( Fig. 13- 30) whose terminal

14 Mutual Inductance and Transformers

14.1 MUTUAL INDUCTANCE
14.2 COUPLING COEFFICIENT
14.3 ANALYSIS OF COUPLED COILS
14.4 DOT RULE
14.5 ENERGY IN A PAIR OF COUPLED COILS
14.6 CONDUCTIVELY COUPLED EQUIVALENT CIRCUITS
14.7 LINEAR TRANSFORMER
14.8 IDEAL TRANSFORMER
14.9 AUTOTRANSFORMER
14.10 REFLECTED IMPEDANCE
14.1 When one coil of a magnetically coupled pair has a current 5.0A the resulting . uxes and
14.2 Two coupled coils have self- inductances L1 ¼ 50 mH and L2 ¼ 200 mH, and a coe . cient of
14.3 Apply KVL to the series circuit of Fig. 14- 18.
14.4 In a series aiding connection, two coupled coils have an equivalent inductance LA; in a series
14.5 ( a) Write the mesh current equations for the coupled coils with currents i1 and i2 shown in Fig.
14.6 Obtain the dotted equivalent circuit for the coupled circuit shown in Fig. 14- 20, and use it to . nd
14.7 Obtain the dotted equivalent for the circuit shown in Fig. 14- 22 and use the equivalent to . nd the
14.8 ( a) Compute the voltage V for the coupled circuit shown in Fig. 14- 24. ( b) Repeat with the
14.9 Obtain the equivalent inductance of the parallel- connected, coupled coils shown in Fig. 14- 25.
14.10 For the coupled circuit shown in Fig. 14- 26, show that dots are not needed so long as the second
14.11 For the coupled circuit shown in Fig. 14- 27, . nd the ratio V2= V1 which results in zero current I1.
14.12 In the circuit of Fig. 14- 28, . nd the voltage across the 5 reactance with the polarity shown.
14.13 Obtain The ´ venin and Norton equivalent circuits at terminals ab for the coupled circuit shown in
14.14 Obtain a conductively coupled equivalent circuit for the magnetically coupled circuit shown in
14.15 For the transformer circuit of Fig. 14- 11( b), k ¼ 0: 96, R1 ¼ 1: 2 , R2 ¼ 0: 3 , X1 ¼ 20 ,
14.16 For the linear transformer of Problem 14.15, calculate the input impedance at the terminals where
V1 is applied.
14.17 In Fig. 14- 33, three identical transformers are primary wye- connected and secondary delta- connected.
14.18 For the ideal autotransformer shown in Fig. 14- 34, . nd V2, Icb, and the input current I1.
14.19 In Problem 14.18, . nd the apparent power delivered to the load by transformer action and that
14.20 In the coupled circuit of Fig. 14- 35, . nd the input admittance Y1 ¼ I1= V1 and determine the
14.21 Find the input impedance Z1 ¼ V1= I1 in the coupled circuit of Fig. 14- 36.

15 Circuit Analysis Using Spice and PSpice

15.1 SPICE AND PSPICE
15.2 CIRCUIT DESCRIPTION
15.3 DISSECTING A SPICE SOURCE FILE
15.4 DATA STATEMENTS AND DC ANALYSIS
15.5 CONTROL AND OUTPUT STATEMENTS IN DC ANALYSIS
15.7 OP AMP CIRCUITS
15.8 AC STEADY STATE AND FREQUENCY RESPONSE
15.9 MUTUAL INDUCTANCE AND TRANSFORMERS
15.10 MODELING DEVICES WITH VARYING PARAMETERS
15.11 TIME RESPONSE AND TRANSIENT ANALYSIS
15.12 SPECIFYING OTHER TYPES OF SOURCES
15.13 SUMMARY
15.1 Use PSpice to . nd Vð3; 4Þ in the circuit of Fig. 15- 23.
15.2 Write the source . le for the circuit of Fig. 15- 24 and . nd I in R4.
15.3 Find the three loop currents in the circuit of Fig. 15- 25 using PSpice and compare your solution
15.4 Using PSpice, . nd the value of Vs in Fig. 15- 4 such that the voltage source does not supply any
15.5 Perform a dc analysis on the circuit of Fig. 15- 26 and . nd its The ´ venin equivalent as seen from
15.6 Perform an ac analysis on the circuit of Fig. 15- 27( a). Find the complex magnitude of V2 for f
15.7 Perform dc and ac analysis on the circuit in Fig. 15- 28. Find the complex magnitude of V2 for f
15.8 Plot resonance curves for the circuit of Fig. 15- 29( a) for R ¼ 2, 4, 6, 8, and 10 .
15.9 Use . TRAN and . PROBE to plot VC across the 1- mF capacitor in the source- free circuit of Fig.
15.10 Plot the voltages between the two nodes of Fig. 15- 31( a) in response to a 1- mA step current
15.11 Find the The ´ venin equivalent of Fig. 15- 32 seen at the terminal AB:
15.12 Plot the frequency response VAB= Vac of the open- loop ampli . er circuit of Fig. 15- 33( a).
15.13 Model the op amp of Fig. 15- 34( a) as a subcircuit and use it to . nd the frequency response of
15.14 Referring to the RC circuit of Fig. 15- 22, choose the height of the initial pulse such that the
15.15 Plot the voltage across the capacitor in the circuit in Fig. 15- 35( a) for R ¼ 0: 01 and 4: 01 .

16 The Laplace Transform Method

16.1 INTRODUCTION
16.2 THE LAPLACE TRANSFORM
16.3 SELECTED LAPLACE TRANSFORMS
16.4 CONVERGENCE OF THE INTEGRAL
16.5 INITIAL- VALUE AND FINAL- VALUE THEOREMS
16.6 PARTIAL- FRACTIONS EXPANSIONS
16.7 CIRCUITS IN THE s- DOMAIN
16.8 THE NETWORK FUNCTION AND LAPLACE TRANSFORMS
16.1 Find the Laplace transform of e cos ! t, where a is a constant.
16.2 If l f show that l e f Apply this result to Problem 16.1. ½ ðtÞ ¼ FðsÞ, ½ ðtÞ ¼ Fðs þ aÞ.
16.3 Find the Laplace transform of f ðtÞ ¼ 1 e where a is a constant.
16.4 Find
16.5 Find
16.6 In the series RC circuit of Fig. 16- 5, the capacitor has an initial charge 2.5 mC. At t ¼ 0, the
16.7 In the RL circuit shown in Fig. 16- 6, the switch is in position 1 long enough to establish steadystate
16.8 In the series RL circuit of Fig. 16- 7, an exponential voltage v 50e ( V) is applied by closing ¼
16.9 The series RC circuit of Fig. 16- 8 has a sinusoidal voltage source v ¼ 180 sin ð2000t þ Þ ( V) and
16.10 In the series RL circuit of Fig. 16- 9, the source is v ¼ 100 sin ð500t þ Þ ( V). Determine the
16.11 Rework Problem 16.10 by writing the voltage function as
16.12 In the series RLC circuit shown in Fig. 16- 10, there is no initial charge on the capacitor. If the
16.13 In the two- mesh network of Fig. 16- 11, the two loop currents are selected as shown. Write the sdomain
16.14 In the two- mesh network of Fig. 16- 13, . nd the currents which result when the switch is closed.
16.15 Apply the initial- and . nal- value theorems in Problem 16.14.
16.16 Solve for i1 in Problem 16.14 by determining an equivalent circuit in the s- domain.
16.17 In the two- mesh network shown in Fig. 16- 15 there is no initial charge on the capacitor. Find the
16.18 Referring to Problem 16.17, obtain the equivalent impedance of the s- domain network and
16.19 In the network of Fig. 16- 18 the switch is closed at t ¼ 0 and there is no initial charge on either of
16.20 Apply the initial- and . nal- value theorems to the s- domain current of Problem 16.19.

17 Fourier Method of Waveform Analysis

17.1 INTRODUCTION
17.2 TRIGONOMETRIC FOURIER SERIES
17.3 EXPONENTIAL FOURIER SERIES
17.4 WAVEFORM SYMMETRY
17.5 LINE SPECTRUM
17.6 WAVEFORM SYNTHESIS
17.7 EFFECTIVE VALUES AND POWER
17.8 APPLICATIONS IN CIRCUIT ANALYSIS
17.9 FOURIER TRANSFORM OF NONPERIODIC WAVEFORMS
17.10 PROPERTIES OF THE FOURIER TRANSFORM
17.11 CONTINUOUS SPECTRUM
17.1 Find the trigonometric Fourier series for the square wave shown in Fig. 17- 18 and plot the line
17.2 Find the trigonometric Fourier series for the triangular wave shown in Fig. 17- 20 and plot the line
17.3 Find the trigonometric Fourier series for the sawtooth wave shown in Fig. 17- 22 and plot the line
17.4 Find the trigonometric Fourier series for the waveform shown in Fig. 17- 24 and sketch the line
17.5 Find the trigonometric Fourier series for the half- wave- recti . ed sine wave shown in Fig. 17- 26
17.6 Find the trigonometric Fourier series for the half- wave- recti . ed sine wave shown in Fig. 17- 28,
17.7 Obtain the trigonometric Fourier series for the repeating rectangular pulse shown in Fig. 17- 29
17.8 Find the exponential Fourier series for the square wave shown in Figs. 17- 18 and 17- 31, and
17.9 Find the exponential Fourier series for the triangular wave shown in Figs. 17- 20 and 17- 33 and
17.10 Find the exponential Fourier series for the half- wave recti . ed sine wave shown in Figs. 17- 26 and
17.11 Find the average power in a resistance R ¼ 10 , if the current in Fourier series form is
17.12 Find the average power supplied to a network if the applied voltage and resulting current are
17.13 Obtain the constants of the two- element series circuit with the applied voltage and resultant
17.14 The voltage wave shown in Fig. 17- 35 is applied to a series circuit of R ¼ 2k and L ¼ 10 H.
17.15 The current in a 10- mH inductance has the waveform shown in Fig. 17- 37. Obtain the trigonometric

APPENDIX A Complex Number System

A1 COMPLEX NUMBERS
A2 COMPLEX PLANE
A3 VECTOR OPERATOR j
A4 OTHER REPRESENTATIONS OF COMPLEX NUMBERS
A5 SUM AND DIFFERENCE OF COMPLEX NUMBERS
A6 MULTIPLICATION OF COMPLEX NUMBERS
A7 DIVISION OF COMPLEX NUMBERS
A8 CONJUGATE OF A COMPLEX NUMBER

APPENDIX b Matrices and Determinants

B1 SIMULTANEOUS EQUATIONS AND THE CHARACTERISTIC MATRIX
B2 TYPES OF MATRICES
B3 MATRIX ARITHMETIC
B4 DETERMINANT OF A SQUARE MATRIX
B5 EIGENVALUES OF A SQUARE MATRIX

INDEX