1 chapter 7 first-order circuits 電路學 ( 一 ). 2 first-order circuits chapter 7 7.1the...

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Chapter 7Chapter 7

First-Order CircuitsFirst-Order Circuits

電路學電路學 (( 一一 ))

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First-Order CircuitsFirst-Order CircuitsChapter 7Chapter 7

7.1 The Source-Free RC Circuit7.2 The Source-Free RL Circuit7.3 Singularity Functions7.4 Step Response of an RC Circuit7.5 Step Response of an RL Circuit7.6 Applications

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7.1 The Source-Free7.1 The Source-FreeRC Circuit (1)RC Circuit (1)

• A first-order circuit is characterized by a first-order differential equation.

• Apply Kirchhoff’s laws to purely resistive circuit results in algebraic equations.

• Apply the laws to RC and RL circuits produces differential equations.

Ohms law Capacitor law

0 dt

dvC

R

v0 CR iiBy KCL

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7.1 The Source-Free 7.1 The Source-Free RC Circuit (2)RC Circuit (2)

• The natural response of a circuit refers to the behavior (in terms of voltages and currents) of the circuit itself, with no external sources of excitation.

• The time constant of a circuit is the time required for the response to decay by a factor of 1/e or 36.8% of its initial value.

• v decays faster for small and slower for large .

CRTime constantDecays more slowly

Decays faster

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The key to working with a source-free RC circuit is finding:

1. The initial voltage v(0) = V0 across the capacitor.

2. The time constant = RC.

/0)( teVtv CRwhere

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Example 1

In the figure, let vC(0) = 15 V. Find vC, vx, and ix for t > 0.

5

8

12

ix

0.1 F

+

vx

–

+

vC

–

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Example 2

Refer to the circuit below, determine vC, vx, and io for t ≥ 0.

Assume that vC(0) = 30 V.

• Please refer to lecture or textbook for more detail elaboration.Answer: vC = 30e–0.25t V ; vx = 10e–0.25t ; io = –2.5e–0.25t A

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Example 3

The switch in circuit has been closed for a long time, and it is opened at t = 0. Find v(t) for t ≥ 0. Calculate the initial energy stored in the capacitor.

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Example 4

The switch in circuit below is opened at t = 0, find v(t) for t ≥ 0.

• Please refer to lecture or textbook for more detail elaboration.Answer: V(t) = 8e–2t V

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7.2 The Source-Free 7.2 The Source-Free RL Circuit (1)RL Circuit (1)

• A first-order RL circuit consists of a inductor L (or its equivalent) and a resistor (or its equivalent)

0 RL vvBy KVL

0 iRdt

diL

Inductors law Ohms law

dtL

R

i

di LtReIti /

0 )(

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• The time constant of a circuit is the time required for the response to decay by a factor of 1/e or 36.8% of its initial value.

• i(t) decays faster for small and slower for large .• The general form is very similar to a RC source-free circuit.

/0)( teIti

R

L

A general form representing a RL

where

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/0)( teIti

R

L

A RL source-free circuit

where /0)( teVtv RC

A RC source-free circuit

where

Comparison between a RL and RC circuit

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The key to working with a source-free RL circuit is finding:

1. The initial voltage i(0) = I0 through the inductor.

2. The time constant = L/R.

/0)( teIti

R

Lwhere

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Example 5

Assume that i(0) = 10 A, calculate i(t) and ix(t) in the circuit.

.

1. Equivalent circuit

2. Loop analysis

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Example 6

Find i and vx in the circuit.

Assume that i(0) = 5 A.

• Please refer to lecture or textbook for more detail elaboration.

Answer: i(t) = 5e–53t A

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Example 7

The switch in the circuit has been closed for a long time. At t = 0, the switch is opened. Calculate i(t) for t > 0.

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Example 8

For the circuit, find i(t) for t > 0.


Answer: i(t) = 2e–2t A

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Example 9 Assume the switch in the circuit was open for a long time, find io, vo, and i for all time.

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7.3 Singularity Functions (1)7.3 Singularity Functions (1)• The singularity functions 奇異函數 are

functions that either are discontinuous or have discontinuous derivatives.

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7.3 Singularity Functions (2)7.3 Singularity Functions (2)• The unit step function 單位步階函數 u(t) is 0

for negative values of t and 1 for positive values of t.

0,1

0,0)(

t

ttu

o

oo tt

ttttu

,1

,0)(

o

oo tt

ttttu

,1

,0)(

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1. voltage source.

2. for current source:

Represent an abrupt change for:

7.3 Singularity Functions (3)7.3 Singularity Functions (3)

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7.3 Singularity Functions (4)7.3 Singularity Functions (4)• The unit impulse function 單位脈衝函數 (t) is

zero everywhere except at t = 0, where is undefined.

0,0

0 Undefined

0,0

)()(

t

t

t

tudt

dt

1)(0

0

dtt

btatfdttttfb

a 000 where)()()(

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7.3 Singularity Functions (5)7.3 Singularity Functions (5)• The unit ramp function單位斜率函數 r(t) is zero

for negative t and has a unit slope for positive values of t.

0,

0,0)(

tt

ttr

00

00 ,

,0)(

tttt

ttttr

dt

tdrtu

dt

tdut

)()(,

)()(

tt

dttutrdtttu )()(,)()(

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7.3 Singularity Functions (6)7.3 Singularity Functions (6)Example 10 Express the voltage pulse in the figure in terms of the unit step. Calculate its derivative and sketch it.

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7.3 Singularity Functions (7)7.3 Singularity Functions (7)Example 11 Express the current pulse in the figure in terms of the unit step. Find its integral and sketch it.

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7.3 Singularity Functions (8)7.3 Singularity Functions (8)Example 12 Express the sawtooth function in the figure in terms of the singularity functions.

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7.3 Singularity Functions (9)7.3 Singularity Functions (9)Example 13 Given the signal

express g(t) in terms of step and ramp functions.

3, 0

( ) 2, 0 1

2 4, 1

t

g t t

t t

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7.3 Singularity Functions (10)7.3 Singularity Functions (10)Example 14 Evaluate the following integrals involving the impulse function:

10 2

0( 4 2) ( 2)

[ ( 1) cos ( 1) sin ]t t

t t t dt

t e t t e t dt

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• Initial condition: v(0-) = v(0+) = V0

• Applying KCL,

or

• Where u(t) is the unit-step function

7.4 The Step-Response 7.4 The Step-Response of a RC Circuit (1)of a RC Circuit (1)

• The step response of a circuit is its behavior when the excitation is the step function, which may be a voltage or a current source.

0)(

R

tuVv

dt

dvc s

RC

tuVv

dt

dv s )(

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• Integrating both sides and considering the initial conditions, the solution of the equation is:

0)(

0)(

/0

0

teVVV

tVtv

tss

Final value at t -> ∞

Initial value at t = 0

Source-free Response

Complete Response = Natural response + Forced Response (stored energy) (independent source)

= V0e–t/τ + Vs(1–e–t/τ)

完整響應自然響應激勵響應

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Three steps to find out the step response of an RC circuit:

1. The initial capacitor voltage v(0).2. The final capacitor voltage v() — DC voltage

across C.3. The time constant .

/ )]( )0( [ )( )( tevvvtv steady-state

response穩態響應

transient response暫態響應

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Example 15

The switch has been in position A for a long time. At t = 0, the switch move to B. Find v(t) for t > 0 in the circuit and calculate v(t) at t = 1 s and 4 s.


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Example 16

Find v(t) for t > 0 in the circuit in below. Assume the switch has been open for a long time and is closed at t = 0.

Calculate v(t) at t = 0.5.



Answer: and v(0.5) = 0.5182V515)( 2 tetv

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Example 17

In the figure, switch has been close for a long time and is opened at t = 0. Find i and v for all time.


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7.5 The Step-response 7.5 The Step-response of a RL Circuit (1)of a RL Circuit (1)

• The step response of a circuit is its behavior when the excitation is the step function, which may be a voltage or a current source.

• Initial currenti(0-) = i(0+) = Io

• Final inductor current i(∞) = Vs/R

• Time constant = L/R

t

so

s eRV

IRV

ti

)()(

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7.5 The Step-Response 7.5 The Step-Response of a RL Circuit (2)of a RL Circuit (2)

Three steps to find out the step response of an RL circuit:

1. The initial inductor current i(0) at t = 0+.2. The final inductor current i().

3. The time constant .

Note: The above method is a short-cut method. You may also determine the solution by setting up the circuit formula directly using KCL, KVL , ohms law, capacitor and inductor VI laws.

/ )]( )0( [ )( )( teiiiti

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Example 18

Find i(t) in the circuit for t > 0. Assume that the switch has been closed for a long time.


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Example 19

The switch in the circuit shown below has been closed for a long time. It opens at t = 0.

Find i(t) for t > 0.



Answer: teti 102)(

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Example 20

At t = 0, switch 1 is closed, and switch 2 is closed 4 s later. Find i(t) for t > 0. Calculate i for t = 2 s and t = 5 s.


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7.6 First-Order Op Amp Circuits 7.6 First-Order Op Amp Circuits (1)(1)

Example 21

For the op amp circuit, find vo for t > 0, given that that v(0) = 3 V. Let Rf = 80 k, R1 = 20 k, and C = 5 μF.

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Example 22

Determine v(t) and vo(t) in the circuit.

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Example 23

Find the step response vo(t) for t > 0 in the op amp. Let vi = 2u(t) V, R1 = 20 k, Rf = 50 k, R2 = R3 = 10 k, and C = 2 μF.

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7.7 Applications (1)7.7 Applications (1)Delay Circuits

Example 24

Consider the delay circuit, and assume that R1 = 1.5 M, 0 < R2 < 2.5 M. (a) Calculate the extreme limits of the time constant of the circuit. (b) How long does it take for the lamp to glow for the first time after the switch is closed? Let R2 assume its largest value.

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7.7 Applications (2)7.7 Applications (2)Photoflash Unit

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7.7 Applications (3)7.7 Applications (3)Example 25

An electric flashgun has a current-limiting 6-k resistor and 2000-μF electrolytic capacitor charged to 240 V. If the lamp resistor is 12 , find (a) the peak charging current, (b) the time required for the capacitor to fully charge, (c) the peak discharging current, (d) the total energy stored in the capacitor, and (e) the average power dissipated by the lamp.

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7.7 Applications (4)7.7 Applications (4)Relay Circuits

Example 26

The coil of a certain relay is operated by a 12-V battery. If the coil has a resistance of 150 and an inductance of 30 mH and the current needed to put in is 50 mA, calculate the relay delay time.

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7.7 Applications (5)7.7 Applications (5)Automobile Ignition Circuit

Example 27

A solenoid with resistance 4 and an inductance 6 mH is used in an automobile ignition circuit. If the battery supplies 12-V, determine: the final current through the solenoid when the switch is closed, the energy stored in the coil, and the voltage across the air gap, assume that the switch takes 1 μs to open.

1 chapter 7 first-order circuits 電路學 ( 一 ). 2 first-order circuits chapter 7 7.1the...

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