Response of First-OrderRL and RC CircuitsRL and RC Circuits

Text book:

Electric CircuitsJames W. Nilsson & Susan A. Riedel8th Edition.8 Edition.

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Obj tiObjectives

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

Be able to determine the step response of both RL and RC circuits.

Know how to analyze circuits with sequential switching.

OutlinesOutlinesThe natural response of an RL circuit & an RC circuit

The step response of RL & RC circuits

Sequential switching

Unbounded response

General ConceptsGeneral Concepts

The natural response:The response that arise when stored energy in an inductor or capacitor is suddenly released.

The step response:The step response:The response that arise when energy is being acquired by an inductor or capacitor due to sudden application of a dc voltage or current sourcecurrent source.

First order circuits (RL or RC circuits):First order circuits (RL or RC circuits):Circuits where voltages and currents are described by first-order differential equations.

Four possible first order circuitsFour possible first order circuits

L or C connected to a Thevenin equivalent

L or C connected to a Norton equivalent

The natural response of an RL circuitThe natural response of an RL circuit

The switch is closed for a long time and opened at t = 0

0=dit ≤ 0 v = 0 (short circuit)0dt

( )

All the source current I0 appears in the inductive branch

t ≥ 0 Apply KVL:

0=+ RiddiL (first order differential equation)dt

The natural response of an RL circuitThe natural response of an RL circuit

Since the current cannot change instantaneously in an inductorSince the current cannot change instantaneously in an inductor

( ) ( )( ) ( )tLRIti

Iii/

000−

+− ==

t ≥ 0( ) ( )tLReIti /0= t ≥ 0

The voltage across the resistor using Ohm’s law

( )

( ) ( ) RI

tIiRv tLR /0

000

0 Re ≥==+

+−

The energy delivered to the resistor during any interval of time after the switch has been opened

The power dissipated in the resistor

( ) ( ) RIvv 00 00 == +−

( ) +− ≥== 0Re /22 tIivp tLR( ) ≥== 0 Re0 tIivp

The time constant (τ)The time constant (τ)

The time constant ( )The time constant (τ)

Interpretation of the time constant of the RL circuitwhen τ = t i = I0

Summary:

1) Find the initial current, I0 , through the inductor

2) Find the time constant of the circuit, τ = L/R

3) Use I0e- τ/t , to generate i(t) from I0 and τ.

E l 1Example 1

E l 1 S l tiExample 1 - Solution

E l 1 S l ti ( t)Example 1 – Solution (cont)

E l 1 S l ti ( t)Example 1 – Solution (cont)

E l 2Example 2

a) Find i1 i2 and i3a) Find i1 , i2 and i3 .b) Calculate the initial energy stored in the parallel inductors.c) Calculate the energy stored in the inductor as t ∞d) Show that the total energy delivered to the resistive network equals

to the difference between the result obtained in (b) and (c).

E l 2 S l tiExample 2 – Solution

E l 2 S l ti ( t)Example 2 – Solution (cont)

E l 2 S l ti ( t)Example 2 – Solution (cont)

The natural response of an RC circuitThe natural response of an RC circuit

Assume the switch has been in position a for a long time:

0=dvt ≤ 0 i = 0 (open circuit)0dt

( )

vC = Vg

t ≥ 0 Apply node voltage technique:

( ) 0/ ≥= − teVtv t τ( ) 0 0 ≥= teVtv

The natural response of an RC circuitThe natural response of an RC circuit

The current goes through the resistor

( ) Vtv

The power dissipated in the resistor

( ) ( ) ( ) +− ≥== 0 /0 teRV

Rtvti t τ

The power dissipated in the resistor

( ) +− ≥== 0 /22

0 teR

Vvip t τ

The energy delivered to the resistor

R

12Vtt( ) ( )( ) 0 1

21 /22

00

/22

0

0

≥−=== −−∫∫ teCVdteR

Vpdtw t

tt

tττ

Example 3Example 3Find:

E l 3 ( t)Example 3 (cont)

E l 4Example 4

E l 4 ( t)Example 4 (cont)

b) Calculate the initial energy stored in the capacitor C1 and C2

c) Calculate how much energy is stored in the Capacitors as t ∞

d) Show that the total energy delivered to the 250 kΩ resistor is the difference between the results obtained in (b) and (c)between the results obtained in (b) and (c)

Th t f RL i itThe step response of an RL circuit

The step response of an RL circuitThe step response of an RL circuit

Th t f RL i itThe step response of an RL circuit

Example 5Example 5

Example 5 (cont)Example 5 (cont)

Th t f RC i itThe step response of an RC circuit

Apply KCL:

sC IRv

ddvC =+ sRdt

( ) ( ) 0 , /0 ≥−+= − teRIVRItv RCt

ssC

( ) ( ) RCts

C eRC

RIVCdt

dvCti /

01 −⎟⎠⎞

⎜⎝⎛ −−==

( ) +− ≥⎟⎠⎞

⎜⎝⎛ −= 0 , /0 te

RV

Iti RCts

⎠⎝ R

Example 6Example 6

E l 6 ( t)Example 6 (cont)

E l 6 ( t)Example 6 (cont)

A l l ti f t l & tA general solution for natural & step responses

A general solution for natural & step responsesA general solution for natural & step responses

( ) ( )[ ] ⎟⎠⎞

⎜⎝⎛ −

−

−+= τ0

0

tt

ff extxxtx

x(t) the unknown variable as a function of time

xf the final value of the variable

x(t0) the initial value of the variable

t0 time of switching

Τ time constant

1) Identify the variable of interest of the circuit. For RC circuits, it is best to choose vC ; for RL circuit, it is best to choose iL.

Procedure:

for RL circuit, it is best to choose iL.

2) Determine the initial value of the variable.(vc(t0) in case of RC circuit and iL(t0) in case of RL circuits)

3) Calculate the final value of the variable (value at t = ∞)3) Calculate the final value of the variable (value at t )

4) Calculate the time constant for the circuit.

S ti l S it hiSequential Switching

Switching occurs more than once in a circuit.

The time reference for switching cannot be t = 0The time reference for switching cannot be t = 0.

Procedure for sequential switching problem(1) Obtain the initial value x(t )(1) Obtain the initial value x(t0)

(2) Apply the techniques described previously to find current and voltage value.

(3) Redraw the circuit that pertains to each time interval and repeat step (1).( ) p p p ( )

Note: Since inductive current IL and capacitive voltage VC can change instantaneously at the time of switching, these value should be solved firstfor sequential switching problem.

Example 8Example 8

Example 8 (cont)Example 8 (cont)

Example 8 (cont)Example 8 (cont)

Example 8 (cont)Example 8 (cont)

U b d d RUnbounded Response

A circuit response may grow, rather than decay, exponentially with time.

This type of response is called an unbounded response.

It may happen when the circuit contains dependent source.

I thi th Th i i l t ith t t th t i l f ithIn this case, the Thevenin equivalent with respect to the terminals of either an inductor or a capacitor may be negative, which resulting in a negative time constant.

T l th i it hi h h b d d d t d i thTo solve the circuit which have unbounded response, we need to derive the differential equation that describes the circuit containing the negative Rth.

Example 9Example 9