ES250: Electrical Science HW8: Complete Response of RL and RC
Circuits
Slide 2
RL and RC circuits are called first-order circuits. In this
chapter we will do the following: develop vocabulary that will help
us talk about the response of a first-order circuit analyze
first-order circuits with inputs that are constant after some
particular time, t 0, typically t 0 = 0 analyze first-order
circuits that experience more than one abrupt change, e.g., when a
switch opens or closes introduce the step function and use it to
determine the step response of a first-order circuit
Introduction
Slide 3
Circuits that contain only one inductor or capacitor can be
represented by a first-order differential equation these circuits
are called first-order circuits Thvenin and Norton equivalent
circuits simplify the analysis of first-order circuits by
permitting us to represent all first-order circuits as one of two
possible simple equivalent first-order circuits, as shown:
First-Order Circuits
Slide 4
Slide 5
Consider the first-order circuit with input voltage v s (t);
the output, or response, is the voltage across the capacitor:
Assume the circuit is in steady state before the switch is closed
at time t = 0, then closing the switch disturbs the circuit;
eventually, the disturbance dies out and the resulting circuit
assumes a new steady state condition, as shown on the next slide
First-Order Circuits
Slide 6
Slide 7
When the input to a circuit is sinusoidal, the steady-state
response is also sinusoidal; furthermore, the frequency of the
response sinusoid must be the same as the frequency of the input
sinusoid If the prior circuit is at steady state before the switch
is closed, the capacitor voltage will be of the form: The switch
closes at time t = 0, the capacitor voltage is: After the switch
closes, the response will consist of two parts: a transient part
that eventually dies out and a steady- state part, as shown:
First-Order Circuits
Slide 8
The steady-state part of the circuit response to a sinusoidal
input will also be sinusoidal at the same frequency as the input,
while the transient part of the response of a first- order circuit
is exponential of the form Ke t/ Note, the transient part of the
response goes to zero as t becomes large; when this part of the
response dies out, the steady-state response remains, e.g., M
cos(1000t + ) The complete response of a first-order circuit can be
represented in several ways, e.g.: First-Order Circuits
Slide 9
Alternatively, the complete response can be written as: The
natural response is the part of the circuit response solely due to
initial conditions, such as a capacitor voltage or inductor
current, when the input is zero; while the forced response is the
part of the circuit response due to a particular input, with zero
initial conditions, e.g.: In the case when the input is a constant
or a sinusoid, the forced response is the same as the steady-state
response and the natural response is the same as the transient
response First-Order Circuits
Slide 10
Steps to find the complete response of first-order circuits:
Step 1: Find the forced response before the disturbance, e.g., a
switch change; evaluate this response at time t = t 0 to obtain the
initial condition of the energy storage element Step 2: Find the
forced response after the disturbance Step 3: Add the natural
response = Ket/ to the forced response to get the complete
response; use the initial condition to evaluate the constant K
First-Order Circuits
Slide 11
Questions?
Slide 12
Find the complete response of a first-order circuit shown below
for time t 0 > 0 when the input is constant: Complete Response
to a Constant Input
Slide 13
Note, the circuit contains a single capacitor and no inductors,
so its response is first order in nature Assume the circuit is at
steady state before the switch closes at t 0 = 0 disturbing the
steady state condition for t 0 < 0 Closing the switch at t 0 = 0
removes resistor R 1 from the circuit; after the switch closes the
circuit can be represented with all elements except the capacitor
replaced by its Thvenin equivalent circuit, as shown: Complete
Response to a Constant Input
Slide 14
The capacitor current is given by: The same current, i(t),
passes through the resistor R t, Appling KVL to the circuit yields:
Combining these results yields the first-order diff. eqn.: What is
v(0 - )=v(0 + )? Complete Response to a Constant Input
Slide 15
Find the complete response of a first-order circuit shown below
for time t 0 > 0 when the input is constant: What is i(0 - )=i(0
+ )? Complete Response to a Constant Input
Slide 16
Closing the switch at t 0 = 0 removes resistor R 1 from the
circuit; after the switch closes the circuit can be represented
with all elements except the capacitor replaced by its Norton
equivalent circuit, as shown: Complete Response to a Constant
Input
Slide 17
Both of these circuits have eqns. of the form: where the
parameter is called the time constant Separating the variables and
forming an indefinite integral, we have: where D is a constant of
integration Performing the integration and solving for x yields:
where A = e D, which is determined from the IC x(0) To find A, let
t = 0, then: Complete Response to a Constant Input
Slide 18
Therefore, we obtain: where the parameter is called the time
constant Since the solution can be written as: Complete Response to
a Constant Input
Slide 19
The circuit time constant can be measured from a plot of x(t)
versus t, as shown: Complete Response to a Constant Input
Slide 20
Applying these results to the RC circuit yields the solution:
Complete Response to a Constant Input
Slide 21
Applying these results to the RL circuit yields the solution:
Complete Response to a Constant Input
Slide 22
The circuit below is at steady state before the switch opens;
find the current i(t) for t > 0: Note: Example 8.3-5:
First-Order Circuit
Slide 23
The figures below show circuit after the switch opens (left)
and its the Thvenin equivalent circuit (right): The parameters of
the Thvenin equivalent circuit are: Solution
Slide 24
The time constant is: Substituting these values into the
standard RC solution: where t is expressed in units of ms Now that
the capacitor voltage is known, node voltage applied to node a at
the top of the circuit yields: Substituting the expression for the
capacitor voltage yields: Solution
Slide 25
Solving for v a (t) yields: Finally, we calculating i(t) using
Ohm's law yields: Solution
The application of a constant source, e.g., a battery, by means
of switches may be considered equivalent to a source that is zero
up to t 0 and equal to the voltage V 0 thereafter, as shown below:
We can represent voltage v(t) using the unit step, as shown: The
Unit Step Source
Slide 29
Where the unit step forcing function is defined as: The Unit
Step Source Note: That value of u(t 0 ) is undefined
Slide 30
Consider the pulse source v(t) = V 0 u(t t 0 )V 0 u(t t 1 )
defined: We can represent voltage v(t) using the unit step, as
shown: The Unit Step Source
Slide 31
The pulse source v(t) can schematically as: Recognize that the
unit step function is an ideal model. No real element can switch
instantaneously at t = t 0 ; However, if it switches in a very
short time (say, 1 ns), we can consider the switching as
instantaneous for medium-speed circuits As long as the switching
time is small compared to the time constant of the circuit, it can
be ignored. The Unit Step Source
Slide 32
Consider the application of a pulse source to an RL circuit as
shown below with t 0 = 0, implying a pulse duration of t 1 sec:
Assume the pulse is applied to the RL circuit when i(0) = 0; since
the circuit is linear, we may use the principle of superposition,
so that i = i 1 + i 2 where i 1 is the response to V 0 u(t) and i 2
is the response to V 0 u(t t 1 ) Ex: Pulse Source Driving an RL
Circuit
Slide 33
We recall that the response of an RL circuit to a constant
forcing function applied at t = t n with i(0) = 0, I sc = V 0 /R,
and where = L/R is given by: Consequently, we may add the two
solutions to the two- step sources, carefully noting t 0 = 0 and t
1 as the start of each response, respectively, as shown: Ex: Pulse
Source Driving an RL Circuit
Slide 34
Adding the responses provides the complete response of the RL
circuit shown: Ex: Pulse Source Driving an RL Circuit