Chapter 7curt.nelson/engr228/lecture/chap7.pdf · Engr228 -Chapter 7, Nilsson 10E 12 General RC...

Engr228 - Chapter 7, Nilsson 10E 1

Chapter 7

Engr228

Circuit Analysis

Dr Curtis Nelson

Chapter 7 Objectives

• Be able to determine the natural response of both RL and RC circuits;

• Be able to determine the step response of both RL and RCcircuits.


Review: DC Characteristics of an Inductor

i(t)

L1V1Ω

i(t)

1V1Ω

dttdiLtv L

L)()( =

When iL(t) is constant, diL(t) = 0 thus, vL(t) = 0.In other words, the inductor can be replaced with a short circuit.

Review: DC Characteristics of a Capacitor

i(t)

1V1Ω

C

i(t)

1V1Ω

dttdvCti C

C)()( =

When vC(t) is constant, dvC(t) = 0 thus, iC(t) = 0.In other words, the capacitor can be replaced with an open circuit.


• Transient response, natural response, homogeneous solution (temporary position change)

– Fades to zero over time.• Forced response, steady-state response, particular solution

(permanent position change)– Follows the input;– Independent of time passed.

Types of First-Order Responses

I am holding a ball with a rope attached. What is the movement of the ball if I move my hand to another point?

Two movements:

1. Oscillation

2. Forced position change

Mechanical Analogue


Forced response Natural response

Mechanical Analogue

Source Free RL Circuits

LR+

-

-

+

i(t)

Inductor L has energy stored so initial current is I0

Similar to a pendulum that is at a height h where the potential energy is nonzero.

height


0)()(

0)()(

=+

=+

tiLR

dttdi

dttdiLtRi

There are 2 ways to solve this first-order differential equation.

LR+

-

-

+

i(t)

Source Free RL Circuits

Method 1Assume solution is of the form where A and s are the constants that need to solved for.

stAeti =)(

Substitute into the equation:stAeti =)( 0)()(=+ ti

LR

dttdi

0)(

0

=+

=+

st

stst

AeLRs

AeLRAse

LRs -=

tLR

Aeti-

=)(

Solving Source Free RL Circuits


Initial condition: 0)0( Ii =tLR

Aeti-

=)(from

AIAeI

==

0

00

ThereforetLR

eIti-

= 0)(

Solving Source Free RL Circuits - continued

Method 2: Direct integration

òò -=

-=

-=

=+

tti

I

dtLR

titdi

dtLR

titdi

tiLR

dttdi

tiLR

dttdi

0

)(

0)()(

)()(

)()(

0)()(

tLR

tti

I

eIti

tLRIti

tLRti

-=

--=-

-=

0

0

0

)(

)(

)0(ln)(ln

)(ln0

Solving Source Free RL Circuits


Time Constant

The ratio L/R is called the time constant and is denoted by the symbol τ (tau).

RL

=t

One time constant is defined as the amount of time required for the output to go from its initial value I0 to 36.8% of its initial value.

ttt

LR

eIeIti--

== 00)(

Units: seconds

368.01 =-e

Time Constant Graph


1st Order Response Observations

• The voltage on a capacitor or the current through an inductor is the same prior to and after a switch at t = 0 seconds because these quantities cannot change instantaneously.

• Resistor voltage (or current) prior to the switch v(0-) can be different from the voltage (or current) after the switch v(0+).

• All voltages and all currents in an RC or RL circuit follow the same natural response e-t/τ.

General RL Circuits

The time constant of a single inductor circuit will be τ = L/Reqwhere Req is the resistance seen by the inductor.

Example: Req=R3+R4+R1R2 / (R1+R2)


Example First-Order RL Circuit

Assume the switch is in the closed position for a long time for t < 0. Find the voltage v(t) at t = 200 ms.

v(t) = -12.99 volts at t = 200 ms

Textbook Problem 7.2 – Nilsson 10th

The switch is opened at t = 0 seconds.a) Find i0(t) for t ≥ 0 seconds.b) Find v0(t) for t ≥ 0+ seconds.

io(t) = 0.6e-750t Ampsv0(t) = -7.20e-750t Volts


As you might expect, source-free RC circuits are an analogue of source-free RL circuits. The derivation for the capacitor voltage is now a node equation rather than a loop equation.

Source-Free RC Circuits

Source-Free RC Circuits

0)()(

0)()(

=+

=+

RCtv

dttdv

dttdvC

Rtv

• To be consistent with the direction of assigned voltage:

• Comparing the last equation to that of the RL circuit, it becomes obvious that the form of the solution is the same with the time constant τ = RC rather than L/R

0)()(=+ ti

LR

dttdi


Comparison Between Source-free RL and RC

t

t

t

CRCt

CC

t

L

tLR

LL

eVeVtv

eIeIti--

--

==

==

00

00

)(

)(

RCRL

C

L

=

=

t

t

• Transient response equations:

• Time constants:

RC Natural Response

The time constant is τ = RC for a first-order RC circuit.


General RC Circuits

The time constant of a single-capacitor circuit will be τ = ReqCwhere Req is the resistance seen by the capacitor.

Example: Req=R2+R1R3 / (R1+R3)

The Source Free RC Circuit

Assume the switch is in the closed position for a long time for t < 0. Find the voltage v(t) at t = 200µS.

v(t) = 321mV at t = 200µS


(a) Find vC(t) for all time in the circuit below.(b) At what time is vC = 0.1vC(0)?

Example Problem

vC(0) = 192VvC = 0.1vC(0) when t = 18.42mS

tCC evtv 125)0()( -=

• Many RL and RC circuits are driven by a DC or an AC source. The complete response of a driven RL or RC circuit is the sum of the transient response and the forced response:

Driven RL and RC Circuits

)()()( tforcedttranti +=


The Step Function u(t)

First, we will define a Step Function:• A Step Function is often used to drive circuits.• The forcing function of 1u(t) represents a function that has

zero value up until t = 0, and a value of 1 forever after.

t=0 t=0

R R1V 1V

t

v(t)

1V

t

v(t)

1V

0V 0V

Unit Step function

Step Function Examples


Suppose the voltage source changes abruptly from 1V to 2V.Does the current change abruptly as well?

I = 1A I = 2A

I

L1V1Ω

I

L2V1Ω

Circuit Example

Current

time

1A

2A

Transient Response + Forced Response

Voltage

time

1V

2V

Example - continued


)()()( tvdttdiLtRi S=+

0)()( =+dttdiLtRi

Complete Response

• The differential equation for the circuit above now becomes

• The transient part of the complete solution is determined by setting the forcing function vS(t) = 0:

Complete Solution for RL Circuits The complete solution is found on page 225 of your text book:

!" # = !%(∞) + !" 0 − !%(∞) +,-. /

!" # = 012 + !" 0 − 012 +,

3" /


Complete Solution for RC Circuits The complete solution is found on page 229 of your text book:

!" # = !%(∞) + !" 0 − !%(∞) +,-./

!" # = 012 + !" 0 − 012 +,-./

Procedure for Solving First-Order Circuits

1. Identify the variable of interest for the circuit. For RC circuits, it is most convenient to choose the capacitive voltage - for RL circuits, it is best to choose the inductor current.

2. Determine the initial value of the variable at t0. Note that if you choose capacitor voltage or inductive current, it is not necessary to distinguish between t = 0- and t = 0+ because these are continuous variables.

3. Calculate the final value of the variable as t → ∞.4. Calculate the time constant of the circuit.


Example Problem

t=0

R=1Ω

2VL=1H

1V

i(t)

The switch is in the position shown for a long time before t = 0. Find i(t).

teti -+=1)(

2A

1Ateti -+=1)(2)( =ti

)(ti

Example Graph


Example Problem 7.52 – Nilsson 10th

vC(t)=10 + (6 – 10)e-160t V

Find vc(t).

Example: RL Circuit with Step Input

Find i(t).

i(t)=25+25(1-e-t/2)u(t) A


Chapter 7 Summary

• Showed how to determine the natural response of both RLand RC circuits;

• Showed how to determine the step response of both RLand RC circuits.

Chapter 7curt.nelson/engr228/lecture/chap7.pdf · Engr228 -Chapter 7, Nilsson 10E 12 General RC...

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