Step Response: 1st Order Circuits - Virginia TechLiaB/Lectures/Ch_7/Slides/RC and RL... · Explain the transient response of a RC circuit ... an RC circuit is RC and an RL circuit

1st Order Circuits

Objective of the Lecture Explain 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 Circuit

IC = 0A when t < to

VC = 0V when t < to

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

RC Circuit Find the final condition

of the voltage across the

capacitor.

Replace C with an open

circuit 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

Idt

dVC

R

V

III

R

VI

dt

dVCI

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 Circuit So, 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

R

VI

II

VV

R

RL

L

1

)(

0)(

RL Circuit

/)(1

1

1

1)(

0

0

ott

L

LL

RL

eR

VtI

L

VI

L

R

dt

dI

VRIdt

dI

R

L

dt

dILV

RVII

VVV

LL

RRL

RL

/

01

Electronic Response Typically, 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 Response Is equal to the natural response of the circuit plus the

forced response

Use superposition to determine the final equations for voltage across components and the currents flowing through them.

Example #1 Suppose 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

eIIL

tV

eIIItI

RC

eVVC

tI

eVVVtV

t

LLL

t

LLLL

t

CCC

t

CCCC

/

)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

Needed to Complete HW # 21

How to renew your MatLAB licensehttp://swat.eng.vt.edu/Matlabtutorial.html

If you would like to contact them directly, SWAT is located at:

2080 Torgersen HallOffice Hours: 12:00pm to 4:00pm Monday - FridayPhone: (540) 231-7815E-mail: [email protected]

Introductory Tutorials MathWorks (www.mathworks.com) has

On-line tutorials including A Very Elementary MATLAB Tutorial

http://www.mathworks.com/academia/student_center/tutorials/intropage.html

Videos (look at the ones below Getting Started)http://www.mathworks.com/products/matlab/demos.html

Worked examples (further down the demos page)http://www.mathworks.com/products/matlab/demos.html

Textbook has a MatLAB tutorial in Appendix E.

Summary The 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

tt

oL

o

tt

oC

tteItI

tteVtV

o

o

when 1)(

when 1)(

/)(

/)(

Summary General equations when the magnitude of a voltage or

current source in the circuit changes at t = 0s for the:

voltage across a capacitor is

current 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

t

CCCC

eIIItI

eVVVtV