INITIAL AND FINAL CONDITION FOR CIRCUIT

ContentsExplain the transient response of a RC circuit

As the capacitor stores energy when there is: a transition in a unit step function source, u(t-to) or a voltage or current source is switched into the

circuit.Explain the transient response of a RL circuit

As the inductor stores energy when there is: a transition in a unit step function source, u(t-to) or a voltage or current source is switched into the

circuit.Also known as a forced response to an independent source

RC CircuitIC = 0A when t < to

VC = 0V when t < to

Because I1 = 0A (replace it with an open circuit).

RC CircuitFind the final conditionof the voltage across the capacitor.

Replace C with an opencircuit and determine the voltage across the terminal.

IC = 0A when t ~ ∞ s VC = VR = I1R when t ~ ∞ s

RC CircuitIn the time between to and t = ∞ s, the capacitor stores energy and currents flow through R and C.

RCeRItV

IdtdVC

RV

IIIRVI

dtdVCI

VV

tt

C

CC

CR

RR

CC

RC

1)(

0

0

0

1

1

1

RL Circuit

RL Circuit (con’t)Initial condition is not important as the

magnitude of the voltage source in the circuit is equal to 0V when t ≤ to. Since the voltage source has only been turned

on at t = to, the circuit at t ≤ to is as shown below. As the inductor has not stored any energy because

no power source has been connected to the circuit as of yet, all voltages and currents are equal to zero.

RL CircuitSo, the final condition of the inductor current

needs to be calculated after the voltage source has switched on.Replace L with a short circuit and calculate

IL(∞).

Final Condition

RVI

IIVV

R

RL

L

1

)(0)(

RL Circuit

/)(1

1

1

1)(

0

0

ottL

LL

RL

eRVtI

LVI

LR

dtdI

VRIdtdI

RL

dtdILV

RVIIVVV

LL

RRL

RL

/01

Electronic ResponseTypically, we say that the currents and

voltages in a circuit have reached steady-state once 5 have passed after a change has been made to the value of a current or voltage source in the circuit.In a circuit with a forced response, percentage-

wise how close is the value of the voltage across a capacitor in an RC circuit to its final value at 5?

Complete ResponseIs equal to the natural response of the circuit

plus the forced responseUse superposition to determine the final

equations for voltage across components and the currents flowing through them.

Example -1Suppose there were two unit step function

sources in the circuit.

Example -1 (con’t)The solution for Vc would be the result of

superposition where:I2 = 0A, I1 is left on

The solution is a forced response since I1 turns on at t = t1

I1 = 0A, I2 is left on The solution is a natural response since I2 turns off

at t = t2

Example -1 (con’t)

1

)t-(t-

1

1

when e-1 )(

when 0)(1

ttRItV

ttVtV

RCC

C

Example -1 (con’t)

2

)t-(t-

2

22

when e)(

when )(2

ttRItV

ttRItV

RCC

C

Example -1 (con’t)If t1 < t2

2

)t-(t-

2

)t-(t-

1

212

)t-(t-

1

12

when e e-1 )(

when e1)(

hen w 0)(

21

1

ttRIRItV

tttRIRItV

ttRIVtV

RCRCC

RCC

C

General Equations

RL

eIILtV

eIIItI

RC

eVVCtI

eVVVtV

tLLL

tLLLL

tCCC

tCCCC

/

)0()()(

)()0()()(

)0()()(

)()0()()(

/

/

/

/

When a voltage or

current source

changes its magnitude

at t= 0s in a simple RC or RL circuit.Equations for a simple RC circuit

Equations for a simple RL circuit

SummaryThe final condition for:

the capacitor voltage (Vo) is determined by replacing the capacitor with an open circuit and then calculating the voltage across the terminals.

The inductor current (Io) is determined by replacing the inductor with a short circuit and then calculating the current flowing through the short.

The time constant for:an RC circuit is RC and an RL circuit is L/R

The general equations for the forced response of:the voltage across a capacitor is the current through an inductor is

o

ttoL

ott

oC

tteItI

tteVtVo

o

when 1)(

when 1)(/)(

/)(

SummaryGeneral equations when the magnitude of a

voltage or current source in the circuit changes at t = 0s for the:voltage across a capacitor iscurrent through an inductor is

Superposition should be used if there are multiple voltage and/or current sources that change the magnitude of their output as a function of time.

/

/

)()0()()(

)()0()()(t

LLLL

tCCCC

eIIItI

eVVVtV

References E draw max (for drawing)Electrical circuit