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Electric Circuits (Fall 2016) Pingqiang Zhou

Announcements

• Midterm exam 1

When: Oct. 26, 2016, Wednesday, in class (10:15am – 12:15pm)

Where: Teaching Center 201

• Review class before exam

Oct. 24, 2016, Monday in class

1Lecture 5


Lecture 7

- RC/RL First-Order Circuits

10/12/2016

Reading: Chapter 6&7

Lecture 5 2


Temporal Behavior of Circuit Responses

• Till now we discussed static analysis of a circuit

Responses at a given time depend only on inputs at that time.

Circuit responds to input changes infinitely fast.

3Lecture 5[Anant Agarwal 2005]


Temporal Behavior of Circuit Responses

• From now on we start to discuss dynamic circuit

Time-varying sources and responses

4Lecture 5

We have to introduce capacitors

and inductors to explain such

temporal behavior.

[Anant Agarwal 2005]


Parasitic Modeling of Interconnect

• Two inverters communicating over a long interconnect

Interconnect has potential difference with surroundings

– Electric field E, modeled as capacitor

Current flows through the interconnect loop

–Magnetic flux density B linked by the loop, modeled as inductor

5Lecture 5[Anant Agarwal 2005]


Outline

• Capacitors and inductors

• Response of a circuit

• Natural response of RC/RL circuits

• Step response of RC/RL circuits

Lecture 5 6


Capacitors

7Lecture 5

Passive element that stores energy in electric field

Parallel plate capacitor

[Source: Berkeley]


Fulid-Flow Analogy

8Lecture 5

Does DC current flow through a capacitor?

Does AC current flow through a capacitor?

A reservoir

[Allan R. Hambley 2011]


Example

9Lecture 5


V-I Relationship of Capacitors

10Lecture 5[Source: Berkeley]


Stored Energy

• The instantaneous power delivered to the capacitor is

• The energy stored in a capacitor is:

Lecture 5 11

dvi C

dt


Capacitor Response

12Lecture 5

C =

[Source: Berkeley]


Important Property of Capacitors

dvi C

dt

Lecture 5 13


Practical (Imperfect) Capacitors

• A real capacitor has parasitic effects, leading to a slow

loss of the stored energy internally.

Lecture 514

𝑅𝑠: resistivity of pate materials

𝐿𝑠:current through C creates magnetic field

𝑅𝑝: dielectric not perfect insulator (~100 MΩ)

[Allan R. Hambley 2011]


Inductors

• A passive element that stores energy in magnetic field.

They have applications in power supplies, transformers, radios,

TVs, radars, and electric motors.

• Any conductor has inductance, but the effect is typically

enhanced by coiling the wire up.

Lecture 5 15[Allan R. Hambley 2011]


V-I Relationship of Inductors

Lecture 5 16

l

ANL

2

[Source: Berkeley]


Energy Stored in An Inductor

• The power delivered to the inductor is:

• The energy stored is:

div L

dt

Lecture 5 17


Inductor Response



Important Property of Inductors

div L

dt

Lecture 5 19


Practical (Imperfect) Inductors

• Like the ideal capacitor, the ideal inductor does not

dissipate energy stored in it.

• In reality, inductors do have internal resistance due to the

wiring used to make them.

A winding resistance in series with it.

A small winding capacitance due to the closeness of the windings

These two characteristics are typically small, though at high

frequencies, the capacitance may matter.

Lecture 5 20


Summary of Capacitors and Inductors

Lecture 5 21[Source: Berkeley]


Example

22Lecture 5


Outline



• Natural response of RC/RL


Lecture 5 23


Response of a Circuit

• Circuit (dynamic) response

the reaction of a certain voltage or current in the circuit to

change, such as the adding of a new source, the elimination of a

source, in the circuit configuration.

• Natural response

Behavior (i.e., current and voltage) when stored energy in the

inductor or capacitor is released to the resistive part of the

network (containing no independent sources).




• Transient response

Behavior when voltage or current source are suddenly applied to

or removed from the circuit due to switching.

Temporary behavior




• Steady-state response (aka. forced response)

Response that persists long after transient has decayed



Outline



• Natural response of RC/RL circuits


Lecture 5 27


RC and RL Circuits

• A circuit that contains only

sources, resistors and a

capacitor is called an RC

circuit.

• A circuit that contains only

sources, resistors and an

inductor is called an RL

circuit.

–+

vs L

R

i–+

vs C

R

i



Natural Response

RL Circuit

• Inductor current cannot change instantaneously

• In steady state, an inductor behaves like a short circuit.

RC Circuit

• Capacitor voltage cannot change instantaneously

• In steady state, a capacitor behaves like an open circuit

R

i

L

+

v

–

RC


Natural Response of a Charged Capacitor

30Lecture 5

(a) t = 0− is the instant just before the

switch is moved from terminal 1 to

terminal 2;

(b) t = 0 is the instant just after it was

moved, t = 0 is synonymous with t =

0+.


Natural Response of a Charged Capacitor

31Lecture 5


Natural Response of RC

32Lecture 5

Time constant: 𝜏 = 𝑅𝐶


Time Constant 𝜏 (= 𝑹𝑪)

• A circuit with a small time constant has a fast response

and vice versa.

33Lecture 5


Example

• In the circuit below, let 𝑣𝐶 0 = 15V. Find 𝑣𝐶 , 𝑣𝑥, and 𝑖𝑥 for

𝑡 > 0.

34Lecture 5


Natural Response of the RL Circuit

35Lecture 5


Natural Response of the RL Circuit

36Lecture 5


Example

• The switch in the circuit below has been closed for a long

time. At 𝑡 = 0, the switch is opened. Calculate 𝑖(𝑡) for 𝑡 >0.

37

When 𝑡 < 0 When 𝑡 > 0

Lecture 5


Natural Response Summary

RL Circuit

• Inductor current cannot change instantaneously

• time constant

RC Circuit

• Capacitor voltage cannot change instantaneously

• time constant

R

L

/)0()(

)0()0(

teiti

ii

R

i

L

+

v

–

RC

/)0()(

)0()0(

tevtv

vv

RC


Outline



• Natural response of RC/RL


Lecture 5 39


Step Response of RC Circuit

• When a DC source is suddenly applied to a RC circuit, the

source can be modeled as a step function.

The circuit response is known as the step response.

40Lecture 5


The Unit Step u(t)

• A step function is one that maintains a constant value

before a certain time and then changes to another constant

afterwards.

41

0, 0

1, 0

tu t

t

switching time may be shifted to 𝑡 = 𝑡0 by

0

0

0

0,

1,

t tu t t

t t

Lecture 5


Equivalent Circuit of Unit Step

• The unit step function has an equivalent circuit to represent

when it is used to switch on a source.

42Lecture 5


Example

• Express the voltage pulse below in terms of the unit step.

43Lecture 5


Step Response of the RC Circuit

44

𝑣 0−

= 𝑣 0+

= 𝑣0

Lecture 5



45

𝑣 0−

= 𝑣 0+

= 𝑣0

Lecture 5



• This is known as the complete response, or total

response.46

0

/

0

, 0

0t

s s

V tv t

V V V e t

Lecture 5


Forced Response

• The complete response

can be written as:

47

n fv v v

0

/

0

, 0

0t

s s

V tv t

V V V e t

𝑣 0−

= 𝑣 0+

= 𝑣0

Lecture 5


Another Perspective

• Another way to look at the response is to break it up into

the transient response and the steady state response:

𝑣 𝑡 = 𝑣 ∞

steady 𝑣𝑠𝑠

+ 𝑣 0 − 𝑣(∞) 𝑒−𝑡/𝜏

transient 𝑣𝑡

48

0

/

0

, 0

0t

s s

V tv t

V V V e t

Lecture 5

𝑣 0−

= 𝑣 0+

= 𝑣0


Example

• The switch has been in position A for a long time. At 𝑡 = 0,

the switch moves to B. Find 𝑣(𝑡).

49Lecture 5


Step Response of the RL Circuit

• We will use the transient and steady

state response approach.

• We know that the transient response will

be an exponential:

• After a sufficiently long time, the current

will reach the steady state:

50

/t

ti Ae

Lecture 5

sss

Vi

R


Step Response of RL Circuit

• This yields an overall response of:

51

/t sVi Ae

R

00 0i i I

Lecture 5

0sV

A IR

/

0

ts sV Vi t I e

R R


Example

• Find 𝑖(𝑡) in the circuit for 𝑡 > 0. Assume that the switch

has been closed for a long time.

52Lecture 5


Procedure for Finding RC/RL Response

1. Identify the variable of interest

• For RL circuits, it is usually the inductor current iL(t).

• For RC circuits, it is usually the capacitor voltage vc(t).

2. Determine the initial value (at t = t0- and t0

+) of the

variable

• Recall that iL(t) and vc(t) are continuous variables:

iL(t0+) = iL(t0

) and vc(t0+) = vc(t0

)

• Assuming that the circuit reached steady state before t0 , use the

fact that an inductor behaves like a short circuit in steady state or

that a capacitor behaves like an open circuit in steady state.



Procedure (cont’d)

3. Calculate the final value of the variable (its value as

t ∞)

• Again, make use of the fact that an inductor behaves like a

short circuit in steady state (t ∞) or that a capacitor behaves

like an open circuit in steady state (t ∞).

4. Calculate the time constant for the circuit

t = L/R for an RL circuit, where R is the Thévenin equivalent

resistance “seen” by the inductor.

t = RC for an RC circuit where R is the Thévenin equivalent

resistance “seen” by the capacitor.



Response Form of Basic First-Order Circuits



New Content for the Discussion Session

• Parallel/Series of

Capacitors

Inductors

• Mutual inductance

• RC Op-Amp circuits

56Lecture 5

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