RLC and Series RLC Circuits - Pitt

410

9. The Complete Response of Circuits with Two Energy Storage Elements

Table 9.13-1 Natural Frequencies of Parallel RLC and Series RLC Circuits

PARALLEL RLC

Circuit i(t)

R

C

L

Differential equation Characteristic equation Damping coefficient, rad/s Resonant frequency, rad/s Damped resonant frequency, rad/s Natural frequencies: overdamped case

Natural frequencies: critically damped case Natural frequencies: underdamped case

d2 dt2

i?t?

?

1 RC

d dt

i?t?

?

1 LC

i?t?

?

0

s2

?

1 RC

s

?

1 LC

?

0

a

?

1 2RC

1

o0

?

pffiffiffiffiffiffi LC

sffiffiffiffiffiffiffi1ffiffiffiffiffiffiffiffiffi2ffiffiffiffiffiffiffiffiffiffi1ffiffiffiffi od ? 2RC ? LC

s1; s2

?

?

1 2RC

?

sffiffiffiffiffiffiffi1ffiffiffiffiffiffiffiffiffi2ffiffiffiffiffiffiffiffiffiffi1ffiffiffi 2RC ? LC

rffiffiffi

when

R

<

1 2

L C

rffiffiffi

s1

?

s2

?

1 ? 2RC

when R

?

1 2

L C

s1; s2

?

?

1 2RC

?

jsLffiffi1ffiCffiffiffiffiffi?ffiffiffiffiffiffiffiffi2ffiffiRffi1ffiffiCffiffiffiffiffiffiffi2ffi

rffiffiffi

when

R

>

1 2

L C

SERIES RLC

L

+

R

C v(t)

?

d2 dt2

v?t?

?

R L

d dt

v?t?

?

1 LC

v?t?

?

0

s2

?

R L

s

?

1 LC

?

0

R a ? 2L

1

o0

?

pffiffiffiffiffiffi LC

sffiffiffiffiffiRffiffiffiffiffiffiffiffi2ffiffiffiffiffiffiffiffiffi1ffiffiffiffi od ? 2L ? LC

s1;

s2

?

?

R 2L

?

sffiffiffiffiffiRffiffiffiffiffiffiffiffi2ffiffiffiffiffiffiffiffiffi1ffiffiffiffi 2L ? LC

rffiffiffi

when R > 2

L C

rffiffiffi

s1

?

s2

?

R ? 2L

when

R

?

2

L C

s1;

s2

?

?

R 2L

?

jsffiLffi1ffiCffiffiffiffiffi?ffiffiffiffiffiffiffiffi2ffiRffiLffiffiffiffiffiffi2ffiffi

rffiffiffi

when R < 2

L C

Table 9.13-2 Natural Response of Second-Order Circuits

CASE

Overdamped Critically damped Underdamped

NATURAL FREQUENCIES

s1; s2 ? ?a ? pffiaffiffi2ffiffiffi?ffiffiffiffiffioffiffiffi20ffiffi s1; s2 ? ?a s1; s2 ? ?a ? jpffioffiffiffi20ffiffiffi?ffiffiffiffiffiaffiffi2ffiffi ? ?a ? jod

NATURAL RESPONSE, xn

A1es1t ? A2es2t (A1?A2t)e?at (A1 cos odt?A2 sin odt)e?at

Table 9.13-3 Forced Response of Second-Order Circuits

INPUT, f(t)

Constant Ramp Sinusoid Exponential

K Kt K cos ot, K sin ot, or K cos (ot?y) Ke?bt

FORCED RESPONSE, xf

A A?Bt A cos ot ? B sin ot Ae?bt

Problems

411

PROBLEMS

Problem available in WileyPLUS at instructor's discretion.

Section 9.2 Differential Equation for Circuits with Two Energy Storage Elements

Hint: Use the direct method.

P 9.2-1 Find the differential equation for the circuit shown

in Figure P 9.2-1 using the direct method.

t = 0

2

1 mH

R1

i(t)

R3

vs

+ ?

Figure P 9.2-1

100

10 ?F

+ ?

Vs

+

L

v(t)

C

R2

?

P 9.2-2 Find the differential equation for the circuit shown in Figure P 9.2-2 using the operator method.

Answer:

d2 dt2

iL?t?

?

11; 000

d dt

iL?t?

?

1:1

?

108iL?t?

?

108is?t?

Figure P 9.2-4

P 9.2-5 The input to the circuit shown in Figure P 9.2-5 is the voltage of the voltage source, vs. The output is the capacitor voltage v(t). Represent the circuit by a second-order differential equation that shows how the output of this circuit is related to the input for t > 0.

? +

? +

10

is

100

10 ?F

1 mH

iL

Figure P 9.2-2

P 9.2-3 Find the differential equation for iL(t) for t > 0 for the circuit of Figure P 9.2-3.

iL R1

is

Hint: Use the direct method.

t = 0 R1

vs

i(t)

L R2

+ C v(t)

?

Figure P 9.2-5

P 9.2-6 The input to the circuit shown in Figure P 9.2-6 is the voltage of the voltage source, vs. The output is the inductor current i(t). Represent the circuit by a second-order differential equation that shows how the output of this circuit is related to the input for t > 0.

L t = 0

Hint: Use the direct method.

vs

+ ?

+

R2

t = 0 R1

C

vc

?

R2

+

Figure P 9.2-3

vs

C

v(t)

P 9.2-4 The input to the circuit shown in Figure P 9.2-4 is

L

i(t)

?

the voltage of the voltage source, Vs. The output is the inductor current i(t). Represent the circuit by a second-order differential

equation that shows how the output of this circuit is related to

the input for t> 0.

Figure P 9.2-6

412

9. The Complete Response of Circuits with Two Energy Storage Elements

P 9.2-7 The input to the circuit shown in Figure P 9.2-7 is the voltage of the voltage source, vs. The output is the inductor current i2(t). Represent the circuit by a second-order differential equation that shows how the output of this circuit is related to the input for t > 0.

Hint: Use the operator method.

R2

P 9.2-10 The input to the circuit shown in Figure P 9.2-10 is the voltage of the voltage source, vs. The output is the capacitor voltage v(t). Represent the circuit by a second-order differential equation that shows how the output of this circuit is related to the input for t > 0.

Hint: Find a Thevenin equivalent circuit.

ia

t = 0

i(t)

t = 0 R1

vs

L1

i1(t)

L2

i2(t)

R1

vs

bia

L

+

R2

C

v(t)

?

R3

Figure P 9.2-10

? +

? +

? + ? +

Figure P 9.2-7

P 9.2-8 The input to the circuit shown in Figure P 9.2-8 is the voltage of the voltage source, vs. The output is the capacitor voltage v2(t). Represent the circuit by a second-order differential equation that shows how the output of this circuit is related to the input for t > 0.

Hint: Use the operator method.

t = 0

R1 vs C1

R2

+

v1(t)

C2

?

+ v2(t)

?

P 9.2-11 The input to the circuit shown in Figure P 9.2-11 is the voltage of the voltage source, vs(t). The output is the voltage v2(t). Derive the second-order differential equation that shows how the output of this circuit is related to the input.

Hint: Use the direct method.

R1

?

+

+

+ ?

vs(t)

v1(t)

C1

?

Figure P 9.2-11

R2 C2

+ v2(t)

?

R3

Figure P 9.2-8

P 9.2-9 The input to the circuit shown in Figure P 9.2-9 is the voltage of the voltage source, vs. The output is the capacitor voltage v(t). Represent the circuit by a second-order differential equation that shows how the output of this circuit is related to the input for t > 0. Hint: Use the direct method.

P 9.2-12 The input to the circuit shown in Figure P 9.2-12 is the voltage of the voltage source, vs(t). The output is the voltage vo(t). Derive the second-order differential equation that shows how the output of this circuit is related to the input. Hint: Use the operator method.

C2

t = 0

? v2(t) +

vs

R1

R2

L

i(t)

+

C

v(t)

?

R1

C1

R2

?

+ v1(t) ?

+

+

+ ?

vs(t)

vo(t)

?

Figure P 9.2-9

Figure P 9.2-12

Problems

413

P 9.2-13 The input to the circuit shown in Figure P 9.2-13 is the voltage of the voltage source, vs(t). The output is the voltage vo(t). Derive the second-order differential equation that shows how the output of this circuit is related to the input.

Hint: Use the direct method.

t = 0

R1

+ ?

vs(t)

? v(t) +

C ? +

Figure P 9.2-13

i(t)

L

+

R2 vo(t)

?

Section 9.3 Solution of the Second-Order Differential Equation--The Natural Response P 9.3-1 Find the characteristic equation and its roots for the circuit of Figure P 9.2-2.

P 9.3-2 Find the characteristic equation and its roots for the circuit of Figure P 9.3-2.

Answer: s2 ? 400s ? 3 ? 104 ? 0 roots: s ? ?300; ?100

100 mH

iL +

is

40

vc

1 3 mF

?

Figure P 9.3-2

P 9.3-3 Find the characteristic equation and its roots for the circuit shown in Figure P 9.3-3.

P 9.2-14 The input to the circuit shown in Figure P 9.2-14 is the voltage of the voltage source, vs(t). The output is the voltage v2(t). Derive the second-order differential equation that shows how the output of this circuit is related to the input. Hint: Use the direct method.

R2

+ v1(t) ?

R1

+ ?

vs(t)

C1 ?

+

R3 +

C2

v2(t)

?

Figure P 9.2-14

P 9.2-15 Find the second-order differential equation for i2 for the circuit of Figure P 9.2-15 using the operator method.

Recall that the operator for the integral is 1=s.

Answer:

3

d2 i2 dt2

?

4

di2 dt

?

2

i2

?

d2 vs dt2

1

vs

+ ?

iL

2

+ vc ? 1 mH

10 ?F

Figure P 9.3-3

P 9.3-4 German automaker Volkswagen, in its bid to make more efficient cars, has come up with an auto whose engine saves energy by shutting itself off at stoplights. The stop?start system springs from a campaign to develop cars in all its world markets that use less fuel and pollute less than vehicles now on the road. The stop?start transmission control has a mechanism that senses when the car does not need fuel: coasting downhill and idling at an intersection. The engine shuts off, but a small starter flywheel keeps turning so that power can be quickly restored when the driver touches the accelerator.

A model of the stop?start circuit is shown in Figure P 9.3-4. Determine the characteristic equation and the natural frequencies for the circuit.

Answer: s2 ? 20s ? 400 ? 0 s ? ?10 ? j17:3

1

2

vs

+ ?

i1

1 H i2

12F

10u(t) V

+ ?

10

1 2H

+ ?

7u(t) V

5 mF

Figure P 9.2-15

Figure P 9.3-4 Stop?start circuit.

414

9. The Complete Response of Circuits with Two Energy Storage Elements

Section 9.4 Natural Response of the Unforced Parallel RLC Circuit

P 9.4-1 Determine v(t) for the circuit of Figure P 9.4-1 when L ? 1 H and vs ? 0 for t ! 0. The initial conditions are v(0) ? 6 V and dv=dt(0) ? ?3000 V/s.

Answer: v?t? ? ?2e?100t ? 8e?400t V

L

vs(t)

+ ?

80

+ v(t)

?

25? F

the change in the voltage v(t) activates a light at the flight attendant's station. Determine the natural response v(t). Answer: v?t? ? ?1:16e?2:7t ? 1:16e?37:3t V

Sensor

t = 0 1 A

Light bulb 0.4 H 1

1 40 F

+ v(t)

?

Figure P 9.4-5 Smoke detector.

Figure P 9.4-1

P 9.4-2 An RLC circuit is shown in Figure P 9.4-2, in which v(0) ? 2 V. The switch has been open for a long time before closing at t ? 0. Determine and plot v(t).

+

t = 0

1 3F

v(t)

34

1 H

?

Section 9.5 Natural Response of the Critically Damped Unforced Parallel RLC Circuit

P 9.5-1 Find vc(t) for t > 0 for the circuit shown in Figure P 9.5-1.

Answer: vc?t? ? ?3 ? 6000t?e?2000t V

25 mH

100

30u(?t) mA vc+? 10 mF

Figure P 9.4-2

P 9.4-3 Determine i1(t) and i2(t) for the circuit of Figure P 9.4-3 when i1(0) ? i2(0) ? 11 A.

2 H

i1 1

i2

3 H

2

Figure P 9.5-1

P 9.5-2 Find vc(t) for t > 0 for the circuit of Figure P 9.5-2. Assume steady-state conditions exist at t ? 0?.

Answer: vc?t? ? ?8te?2t V

10

t = 0

Figure P 9.4-3

P 9.4-4 The circuit shown in Figure P 9.4-4 contains a switch that is sometimes open and sometimes closed. Determine the damping factor a, the resonant frequency o0, and the damped resonant frequency od of the circuit when (a) the switch is open and (b) the switch is closed.

40 20 V

10

i(t)

2 H

+

50

v(t)

?

5 mF

? +

Figure P 9.4-4

P 9.4-5 The circuit shown in Figure P 9.4-5 is used in airplanes to detect smokers who surreptitiously light up before they can take a single puff. The sensor activates the switch, and

20 V

+ ?

1H

1

1 4 F + vc

?

Figure P 9.5-2

P 9.5-3 Police often use stun guns to incapacitate potentially dangerous felons. The handheld device provides a series of high-voltage, low-current pulses. The power of the pulses is far below lethal levels, but it is enough to cause muscles to contract and put the person out of action. The device provides a pulse of up to 50,000 V, and a current of 1 mA flows through an arc. A model of the circuit for one period is shown in Figure P 9.5-3. Find v(t) for 0 < t < 1 ms. The resistor R represents the spark gap. Select C so that the response is critically damped.

10 mH

104 V

+ ?

t = 0 C

+ v R = 106

?

Figure P 9.5-3

Problems

415

P 9.5-4 Reconsider Problem P 9.4-1 when L ? 640 mH and the other parameters and conditions remain the same.

Answer: v?t? ? ?6 ? 1500t?e?250t V

P 9.5-5 An automobile ignition uses an electromagnetic trigger. The RLC trigger circuit shown in Figure P 9.5-5 has a step input of 6 V, and v(0) ? 2 V and i(0) ? 0. The resistance R must be selected from 2 V < R < 7 V so that the current i(t) exceeds 0.6 A for greater than 0.5 s to activate the trigger. A critically damped response i(t) is required to avoid oscillations in the trigger current. Select R and determine and plot i(t).

1 H

1 4F

Trigger

+? v(t)

6 u(t) V

+ ?

iR

Figure P 9.5-5

Section 9.6 Natural Response of an Underdamped Unforced Parallel RLC Circuit

P 9.6-1 A communication system from a space station uses short pulses to control a robot operating in space. The transmitter circuit is modeled in Figure P 9.6-1. Find the output voltage vc(t) for t > 0. Assume steady-state conditions at t ? 0?.

Answer: vc?t? ? e?400t?3 cos 300t ? 4 sin 300t V

250 t = 0

0.8 H

250

+ vc ? 5 ? 10-6 F

+ ?

6 V

Figure P 9.6-1

P 9.6-2 The switch of the circuit shown in Figure P 9.6-2 is opened at t ? 0. Determine and plot v(t) when C ? 1=4 F. Assume steady state at t ? 0?. Answer: v?t? ? ?4e?2t sin 2t V

t = 0 3

6 V

+ ?

1

12H

+ C v(t)

?

Figure P 9.6-2

P 9.6-3 A 240-W power supply circuit is shown in Figure P 9.6-3a. This circuit employs a large inductor and a large capacitor. The model of the circuit is shown in Figure P 9.6-3b. Find iL(t) for t > 0 for the circuit of Figure P 9.6-3b. Assume steady-state conditions exist at t ? 0?.

Answer: iL?t? ? e?2t??4 cos t ? 2 sin t? A

# Courtesy of R.S.R. Electronics, Inc.

(a)

4 H iL

t = 0

14F

2 4

8 7 A

(b)

Figure P 9.6-3 (a) A power supply. (b) Model of the power supply circuit.

P 9.6-4 The natural response of a parallel RLC circuit is measured and plotted as shown in Figure P 9.6-4. Using this chart, determine an expression for v(t).

Hint: Notice that v(t) ? 260 mV at t ? 5 ms and that v(t) ? ?200 mV at t ? 7.5 ms. Also, notice that the time between the first and third zero-crossings is 5 ms.

Answer: v?t? ? 544e?276t sin 1257t V

600

500

400

300

200

v(t) (mV)

100

0

? 100

? 200

? 300

? 400 0

5

10 15 20 25 30

Time (ms)

Figure P 9.6-4 The natural response of a parallel RLC circuit.

416

9. The Complete Response of Circuits with Two Energy Storage Elements

P 9.6-5 The photovoltaic cells of the proposed space station shown in Figure P 9.6-5a provide the voltage v(t) of the circuit shown in Figure P 9.6-5b. The space station passes behind the shadow of earth (at t ? 0) with v(0) ? 2 V and i(0) ? 1/10 A. Determine and sketch v(t) for t > 0.

P 9.7-2 Determine the forced response for the capacitor voltage vf for the circuit of Figure P 9.7-2 when (a) vs ? 2 V, (b) vs ? 0.2t V, and (c) vs ? 1e?30t V.

7

0.1 H

vs u(t) V

+ ?

+ 833.3 ?F v

?

Photocells

Figure P 9.7-2

P 9.7-3 A circuit is described for t > 0 by the equation

d2v dt2

?

5

dv dt

?

6v

?

vs

Find the forced response vf for t > 0 when (a) vs ? 8 V, (b) vs ?

3e?4t V, and (c) vs ? 2e?2t V.

Answer:

(a)

vf

?

8=6

V

(b)

vf

?

3 2

e?4t

V

(c)

vf

?

2te?2t

V

Section 9.8 Complete Response of an RLC Circuit

P 9.8-1 Determine i(t) for t > 0 for the circuit shown in Figure

P 9.8-1.

i

iL

11 mA

t = 0

2 k +

1 k

vc

?

6.25 H 1 ?F

+ ?

4 V

Figure P 9.8-1

(a)

i

5

2 H

+

1 10 F

v

?

P 9.8-2 Determine i(t) for t > 0 for the circuit shown in Figure

P 9.8-2.

Hint:

Show

that

1

?

d2 dt2

i?t?

?

5

d dt

i?t?

?

5i?t?

for t > 0

Answer: i?t? ? 0:2 ? 0:246 e?3:62t ? 0:646 e?1:38t A for t > 0.

Space station electric motors

The photovoltaic cells connected

in parallel

(b)

Figure P 9.6-5 (a) Photocells on space station. (b) Circuit with photocells.

Section 9.7 Forced Response of an RLC Circuit

P 9.7-1 Determine the forced response for the inductor current if when (a) is ? 1 A, (b) is ? 0.5t A, and (c) is ? 2e?250t A for the circuit of Figure P 9.7-1.

is u(t) A Figure P 9.7-1

100 65

10 mH

i

1 mF

2u(t) ? 1 V

+ ?

1

+ v(t) ?

4 0.25 F 4 H i(t)

Figure P 9.8-2

P 9.8-3 Determine v1(t) for t > 0 for the circuit shown in Figure P 9.8-3.

Answer: v1?t? ? 10 ? e?2:4?104 t ? 6 e?4?103 t V for t > 0

1 k

1 k

10 V

+ ?

+

+

v1(t) 1/6 ?F v2(t) 1/16 ?F

?

?

t = 0

Figure P 9.8-3

P 9.8-4 Find v(t) for t > 0 for the circuit shown in Figure P 9.8-4 when v(0) ? 1 V and iL(0) ? 0.

Problems

417

Answer:

v ? 25e?3t

?

1 17

[429e?4t

?

21

cos

t

?

33

sin

t

V

1

1

5 cos t V

+ ?

0.5 H iL

+ 1 12 F ? v

? 1 4 u(t) + 1 2 A

4

8

2 H

1 4F

iL(t)

+

vC(t) ?

Figure P 9.8-4

P 9.8-5 Find v(t) for t > 0 for the circuit of Figure P 9.8-5.

Answer:

v?t?

?

????1166ee??t ??t?21?6?e?136t e??38?t?u?2?t??

? 8 u?t

?

2?

V

1 3F

Figure P 9.8-8

P 9.8-9 In Figure P 9.8-9, determine the inductor current i(t) when is ? 5u(t) A. Assume that i(0) ? 0, vc(0) ? 0. Answer: i(t) ? 5 + e?2t [?5 cos 5t ? 2 sin 5t] A

+

2[u(t) ? u(t ? 2)] A

4 v

1 H

?

is

2

8 29 H

1 8F

i

Figure P 9.8-5

P 9.8-6 An experimental space station power supply system is modeled by the circuit shown in Figure P 9.8-6. Find v(t) for t > 0. Assume steady-state conditions at t ? 0?.

(10 cos t)u(t) V

t = 0

?+

+

0.125 F v(t)

2

4

4 H

?

+ ?

5 V

Figure P 9.8-9

P 9.8-10 Railroads widely use automatic identification of railcars. When a train passes a tracking station, a wheel detector activates a radio-frequency module. The module's antenna, as shown in Figure P 9.8-10a, transmits and receives a signal that bounces off a transponder on the locomotive. A

Vehicle-mounted

(a)

transponder tag

i(t) Figure P 9.8-6

P 9.8-7 Find vc(t) for t > 0 in the circuit of Figure P 9.8-7 when (a) C ? 1=18 F, (b) C ? 1=10 F, and (c) C ? 1=20 F.

Answers:

(a) vc?t? ? 8e?3t ? 24te?3t ? 8 V (b) vc?t? ? 10e?t ? 2e?5t ? 8 V (c) vc?t? ? e?3t?8 cos t ? 24 sin t? ? 8 V

8

2u(t) A

+

4 C v(t)

2 H

?

a

i(t)

Figure P 9.8-7

P 9.8-8 Find vc(t) for t > 0 for the circuit shown in Figure

P 9.8-8.

Hint:

2

?

d2 dt2

vc?t?

?

6

d dt

vc?t?

?

2vc?t?

for t > 0

Answer: vc?t? ? 0:123e?5:65t ? 0:877e?0:35t ? 1 V for t > 0.

Wheel detector input

Antenna

+v?

(b)

L

0.5 F

i

1.5

1

is

0.5

Figure P 9.8-10 (a) Railroad identification system. (b) Transponder circuit.

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