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
Electric Circuits (Fall 2016) Pingqiang Zhou
Lecture 7
- RC/RL First-Order Circuits
10/12/2016
Reading: Chapter 6&7
Lecture 5 2
Electric Circuits (Fall 2016) Pingqiang Zhou
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]
Electric Circuits (Fall 2016) Pingqiang Zhou
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]
Electric Circuits (Fall 2016) Pingqiang Zhou
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]
Electric Circuits (Fall 2016) Pingqiang Zhou
Outline
• Capacitors and inductors
• Response of a circuit
• Natural response of RC/RL circuits
• Step response of RC/RL circuits
Lecture 5 6
Electric Circuits (Fall 2016) Pingqiang Zhou
Capacitors
7Lecture 5
Passive element that stores energy in electric field
Parallel plate capacitor
[Source: Berkeley]
Electric Circuits (Fall 2016) Pingqiang Zhou
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]
Electric Circuits (Fall 2016) Pingqiang Zhou
Example
9Lecture 5
Electric Circuits (Fall 2016) Pingqiang Zhou
V-I Relationship of Capacitors
10Lecture 5[Source: Berkeley]
Electric Circuits (Fall 2016) Pingqiang Zhou
Stored Energy
• The instantaneous power delivered to the capacitor is
• The energy stored in a capacitor is:
Lecture 5 11
dvi C
dt
Electric Circuits (Fall 2016) Pingqiang Zhou
Capacitor Response
12Lecture 5
C =
[Source: Berkeley]
Electric Circuits (Fall 2016) Pingqiang Zhou
Important Property of Capacitors
dvi C
dt
Lecture 5 13
Electric Circuits (Fall 2016) Pingqiang Zhou
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]
Electric Circuits (Fall 2016) Pingqiang Zhou
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]
Electric Circuits (Fall 2016) Pingqiang Zhou
V-I Relationship of Inductors
Lecture 5 16
l
ANL
2
[Source: Berkeley]
Electric Circuits (Fall 2016) Pingqiang Zhou
Energy Stored in An Inductor
• The power delivered to the inductor is:
• The energy stored is:
div L
dt
Lecture 5 17
Electric Circuits (Fall 2016) Pingqiang Zhou
Inductor Response
18Lecture 5[Source: Berkeley]
Electric Circuits (Fall 2016) Pingqiang Zhou
Important Property of Inductors
div L
dt
Lecture 5 19
Electric Circuits (Fall 2016) Pingqiang Zhou
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
Electric Circuits (Fall 2016) Pingqiang Zhou
Summary of Capacitors and Inductors
Lecture 5 21[Source: Berkeley]
Electric Circuits (Fall 2016) Pingqiang Zhou
Example
22Lecture 5
Electric Circuits (Fall 2016) Pingqiang Zhou
Outline
• Capacitors and inductors
• Response of a circuit
• Natural response of RC/RL
• Step response of RC/RL circuits
Lecture 5 23
Electric Circuits (Fall 2016) Pingqiang Zhou
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).
24Lecture 5[Source: Berkeley]
Electric Circuits (Fall 2016) Pingqiang Zhou
Response of a Circuit
• Transient response
Behavior when voltage or current source are suddenly applied to
or removed from the circuit due to switching.
Temporary behavior
25Lecture 5[Source: Berkeley]
Electric Circuits (Fall 2016) Pingqiang Zhou
Response of a Circuit
• Steady-state response (aka. forced response)
Response that persists long after transient has decayed
26Lecture 5[Source: Berkeley]
Electric Circuits (Fall 2016) Pingqiang Zhou
Outline
• Capacitors and inductors
• Response of a circuit
• Natural response of RC/RL circuits
• Step response of RC/RL circuits
Lecture 5 27
Electric Circuits (Fall 2016) Pingqiang Zhou
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
28Lecture 5[Source: Berkeley]
Electric Circuits (Fall 2016) Pingqiang Zhou
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
Electric Circuits (Fall 2016) Pingqiang Zhou
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+.
Electric Circuits (Fall 2016) Pingqiang Zhou
Natural Response of a Charged Capacitor
31Lecture 5
Electric Circuits (Fall 2016) Pingqiang Zhou
Natural Response of RC
32Lecture 5
Time constant: 𝜏 = 𝑅𝐶
Electric Circuits (Fall 2016) Pingqiang Zhou
Time Constant 𝜏 (= 𝑹𝑪)
• A circuit with a small time constant has a fast response
and vice versa.
33Lecture 5
Electric Circuits (Fall 2016) Pingqiang Zhou
Example
• In the circuit below, let 𝑣𝐶 0 = 15V. Find 𝑣𝐶 , 𝑣𝑥, and 𝑖𝑥 for
𝑡 > 0.
34Lecture 5
Electric Circuits (Fall 2016) Pingqiang Zhou
Natural Response of the RL Circuit
35Lecture 5
Electric Circuits (Fall 2016) Pingqiang Zhou
Natural Response of the RL Circuit
36Lecture 5
Electric Circuits (Fall 2016) Pingqiang Zhou
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
Electric Circuits (Fall 2016) Pingqiang Zhou
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
Electric Circuits (Fall 2016) Pingqiang Zhou
Outline
• Capacitors and inductors
• Response of a circuit
• Natural response of RC/RL
• Step response of RC/RL circuits
Lecture 5 39
Electric Circuits (Fall 2016) Pingqiang Zhou
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
Electric Circuits (Fall 2016) Pingqiang Zhou
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
Electric Circuits (Fall 2016) Pingqiang Zhou
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
Electric Circuits (Fall 2016) Pingqiang Zhou
Example
• Express the voltage pulse below in terms of the unit step.
43Lecture 5
Electric Circuits (Fall 2016) Pingqiang Zhou
Step Response of the RC Circuit
44
𝑣 0−
= 𝑣 0+
= 𝑣0
Lecture 5
Electric Circuits (Fall 2016) Pingqiang Zhou
Step Response of the RC Circuit
45
𝑣 0−
= 𝑣 0+
= 𝑣0
Lecture 5
Electric Circuits (Fall 2016) Pingqiang Zhou
Step Response of the RC Circuit
• 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
Electric Circuits (Fall 2016) Pingqiang Zhou
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
Electric Circuits (Fall 2016) Pingqiang Zhou
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
Electric Circuits (Fall 2016) Pingqiang Zhou
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
Electric Circuits (Fall 2016) Pingqiang Zhou
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
Electric Circuits (Fall 2016) Pingqiang Zhou
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
Electric Circuits (Fall 2016) Pingqiang Zhou
Example
• Find 𝑖(𝑡) in the circuit for 𝑡 > 0. Assume that the switch
has been closed for a long time.
52Lecture 5
Electric Circuits (Fall 2016) Pingqiang Zhou
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.
53Lecture 5[Source: Berkeley]
Electric Circuits (Fall 2016) Pingqiang Zhou
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.
54Lecture 5[Source: Berkeley]
Electric Circuits (Fall 2016) Pingqiang Zhou
Response Form of Basic First-Order Circuits
55Lecture 5[Source: Berkeley]
Electric Circuits (Fall 2016) Pingqiang Zhou
New Content for the Discussion Session
• Parallel/Series of
Capacitors
Inductors
• Mutual inductance
• RC Op-Amp circuits
56Lecture 5