Pulse Response - First Order
Circuits
Unit Step functions, Pulse Sources,
and SPICE
Kevin D. Donohue, University of Kentucky
1
The Unit Step Function
The unit step function models the behavior of a switch
(i.e. off-on) and can be used to describe more complex
circuit sources.
0 for t < 0
u (t ) = 
1 for t > 0
ØMultiplying u(t) by a constant changes the value of u(t) for t > 0.
ØSubtracting a positive number from the argument of u(t) shifts the step in
the positive t direction.
ØAdding step functions together results in piece-wise steps with the
resultant values equal to the sum of step values in each t interval.
Kevin D. Donohue, University of Kentucky
2
Example
Determine the unit step response for io(t) the circuit
below with input vs(t) = u(t):
io(t)
10 Ω
0.1 mF
vs(t)
40 Ω
Step Response
Show:
iostep (t ) =
20
18
(0.8 − 0.8 exp(−1250 t ))u (t ) amps
40
16
14
12
10
ma
8
6
4
2
0
0
Kevin D. Donohue, University of Kentucky
1
2
ms
3
4
3
The Unit Pulse Function
The unit pulse function is defined as :
1.4
Pulse Function for T = 5
1
0.8
Volts
 0 for t < 0

v(t; T ) = 1 for 0 < t < T
0 for t > T

1.2
0.6
0.4
0.2
0
-10
-5
0
5
10
15
Time
It can be expressed in terms of a combination of unit
step functions:
v (t ) == [u (t ) − u (t − T )]
Kevin D. Donohue, University of Kentucky
4
Example
Determine the pulse
response for io(t) the
circuit below with input
vs(t) = 40v(t) V, where
pulse duration T is 2 ms:
io(t)
10 Ω
vs(t)
0.1 mF
40 Ω
Show:
i0pulse (t ) = [(0.8 − 0.8 exp( −1250t ))u (t ) − (0.8 − 0.8 exp( −1250( t − 2m )))u (t − 2m )]amps
iopulse (t ) = 0.8[u (t ) − u(t − 2m) ]− 0.8 exp( −1250 t )[u (t ) − 12.1825u (t − 2m )]amps
Pulse Response
0.8
0.7
0.6
ma
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
4
ms
5
6
7
8
Kevin D. Donohue, University of Kentucky
5
Plotting Transient Responses
Ø
Ø
Programs written for Matlab can be created either
as functions or scripts. For plotting transient
responses it is convenient to write a function that
acts like the unit step.
A function acts like a subroutine in the Matlab
workspace. Arguments must be passed to it and it
assigns its output to variables in the workspace.
Intermediate variable use by the function are
cleared.
Kevin D. Donohue, University of Kentucky
6
Creating a Function
Ø
Ø
Ø
Ø
A function is a text file (can use Notepad or Matlab’s
editor. If using a word processor, you must store the as a
text file.)
The first line begins with the word ‘function’ and is
followed by the syntax of the function you wish to create.
The following lines can be comments, other Matlab
functions, and commands to process the input arguments to
create the output arguments.
The file must be saved with a based name the same as the
function and a ‘.m’ extension in the current or working
directory.
Kevin D. Donohue, University of Kentucky
7
Example
Ø
Create a function called ‘unit’ such that for an input array it will output a corresponding
array of zeros and ones as the unit step function would do. Must save as text file called
unit.m
function out = unit(t)
% This function is the unit step:
%
% sig = unit(t), where SIG and T are vectors of the same dimension.
%
% elements of out are either 0 or 1 corresponding to elements of T
% being negative or non-negative, respectively.
%
%
written by Kevin D. Donohue 2/93
out = zeros(size(t));
% Initialize output array to zeros
ind_of_positive = find(t >= 0); % Find index array of positive values on t-axis
out(ind_of_positive) = 1;
% Assign points corresponding to postive values to 1
Kevin D. Donohue, University of Kentucky
8
Create Plot in Matlab
i0pulse (t ) = [(0.8 − 0.8 exp(−1250t ) )u (t ) − (0.8 − 0.8 exp(−1250(t − 2m)))u (t − 2 m)]amps
>> tau = 1/1250; % Define time constant
>> t = [-0.1:(tau/50):(10*tau+2e-3)]; % Create time axis
>> i0 = (.8-.8*exp(-t/tau)).*unit(t)-(.8-.8*exp(-(t-2e-3)/tau)).*unit(t-2e-3);
>> plot(t,i0)
>> xlabel(‘Seconds’)
>> ylabel(‘Amps’)
>> title(‘Transient Response’)
Kevin D. Donohue, University of Kentucky
9
SPICE Simulation with Step Source
Do a SPICE Simulation to determine the pulse response for io(t) the
Previous circuit with input vs(t) = 40v(t) V, where pulse duration T is 2 ms:
Ø
To solve you must do a transient analysis, define the voltage source as piece-wise
linear, and describe its transient properties.
R
10
(Amp) +0.000e+000
+2.000m
+4.000m
Time
+6.000m
R0
40
tranex -Transient -14
C
0.0001
+8.000m
V
0
tranex-Transient-14-Table
TIME
I(VAM)
(s)
(Amp)
+0.000e+000
+0.000e+000
+10.000n
+99.999n
+10.840n
+109.104n
:
:
:
:
:
:
+7.285m
+972.534u
+7.445m
+795.710u
+7.605m
+651.035u
+7.765m
+532.665u
+7.925m
+435.817u
VAm
(s)
+8.000m
+500.000m
+396.630u
+0.000e+000
I(VAM)
Kevin D. Donohue, University of Kentucky
10
SPICE Simulation with Switch
Do a SPICE Simulation to determine the pulse response for io(t) the
circuit below with input vs(t) = 40 V, where pulse duration T is 2 ms:
Ø
To solve you must do a transient analysis, define auxiliary circ uits for voltage
sources and meters that control the opening and closing of the switches.
R
10
SwitchV0
VAm
R0
40
C
0.0001
SwitchV1
V
40
tranexsw-Transient-0
(Amp)+0.000e+000
+2.000m
+4.000m
Time (s)
+6.000m
+8.000m
+600.000m
(Amp)
+400.000m
V1
0
V0
0
IVm
IVm0
+200.000m
+0.000e+000
I(VAM)
Kevin D. Donohue, University of Kentucky
11