Transcript Systems Concepts (7-1 to 7-2)

Systems Concepts
Dr. Holbert
March 19, 2008
Lect15
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Introduction
• Several important topics today, including:
– Transfer function
– Impulse response
– Step response
– Linearity and time invariance
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Transfer Function
• The transfer function, H(s), is the ratio of some
output variable (y) to some input variable (x)
Y(s) Output
H(s) 

X(s) Input
• The transfer function is portrayed in block
diagram form as
X(s) ↔ x(t)
Input
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System
H(s) ↔ h(t)
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Y(s) ↔ y(t)
Output
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Common Transfer Functions
• The transfer function, H(s), is bolded because it
is a complex quantity (and it’s a function of
frequency, s = jω)
• Since the transfer function, H(s), is the ratio of
some output variable to some input variable, we
may define any number of transfer functions
–
–
–
–
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ratio of output voltage to input voltage (i.e., voltage gain)
ratio of output current to input current (i.e., current gain)
ratio of output voltage to input current (i.e., transimpedance)
ratio of output current to input voltage (i.e., transadmittance)
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Finding a Transfer Function
• Laplace transform the circuit (elements)
– When finding H(s), all initial conditions are
zero (makes transformation step easy)
• Use appropriate circuit analysis methods
to form a ratio of the desired output to the
input (which is typically an independent
source); for example:
Output Vout (s)
H(s) 

Input
Vin (s)
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Transfer Function Example
vin(t)
+
–
+
R
C
vout(t)
Vin(s)
–
Time Domain
+
–
+
R
Vout(s)
1/sC
–
Frequency Domain
Using voltage division, we find the transfer function
1 /( sC )
1
Vout ( s)  Vin ( s)
 Vin ( s)
R  1 /( sC )
sRC  1
Vout ( s)
1 /( RC )
1/
H( s) 


Vin ( s) s  1 /( RC ) s  1 / 
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Transfer Function Use
• We can use the transfer function to find the
system output to an arbitrary input using simple
multiplication in the s domain
Y(s) = H(s) X(s)
• In the time domain, such an operation would
require use of the convolution integral:




y(t)   h( ) x(t -  ) d   h(t -  ) x( ) d
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Impulse Response
• Let the system input be the impulse function:
x(t) = δ(t); recall that X(s) = L [δ(t)] = 1
• Therefore:
Y(s) = H(s) X(s) = H(s)
• The impulse response, designated h(t), is the
inverse Laplace transform of transfer function
y(t) = h(t) = L -1[H(s)]
• With knowledge of the transfer function or
impulse response, we can find the response of a
circuit to any input
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(Unit) Step Response
• Now, let the system input be the unit step
function:
x(t) = u(t)
• We recall that X(s) = 1/s
• Therefore:
1
Y(s)  H(s) X(s)  H(s)
s
• Using inverse Laplace transform skills, and a
specific H(s), we can find the step response, y(t)
 H(s) 
y(t)  L1 [ Y(s)]  L1 [H(s) X(s)]  L1 
 s 
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Step Response from Convolution
• We could also use the convolution integral in
combination with the impulse response, h(t), to
find the system response to any other input




y(t)   h( ) x(t -  ) d   h(t -  ) x( ) d
• Either form of the convolution integral above can
be used, but generally one expression leads to a
simpler, or more interpretable, result
• We shall use the first formulation here
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Impulse – Step Response Relation
• The step input function is
1   t
x(t -  )  u(t -  )  
0 t  
• The convolution integral becomes

t


y(t)   u(t -  ) h( ) d   h( ) d
• We observe that the step response is the time
integral of the impulse response
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(Unit) Ramp Response
• Besides the impulse and step responses,
another common benchmark is the ramp
response of a system (because some
physical inputs are difficult to create as
impulse and step functions over small t)
• The unit ramp function is
t·u(t)
which has a Laplace transform of 1/s2
• The ramp response is the time integral of
the unit step response
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Pole-Zero Plot
• For a pole-zero plot
place "X" for poles
and "0" for zeros
using real-imaginary
axes
• Poles directly indicate
the system transient
response features
• Poles in the right half
plane signify an
unstable system
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• Consider the following
transfer function
( s  3)( s  3.5)( s 2  4s  5)
H ( s) 
( s  5)( s 2  4)( s  1.5)
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Im
Re
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Linearity
• Linearity is a property of superposition
αx1(t) + βx2(t) → αy1(t) + βy2(t)
• A system with a constant (additive) term is
nonlinear; this aspect results from another
property of linear systems, that is, a zero input to
a linear system results in an output of zero
• Circuits that have non-zero initial conditions are
nonlinear
• An RLC circuit initially at rest is a linear system
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Time-Invariant Systems
• In broad terms, a system that does not change
with time is a time-invariant system; that is, the
rule used to compute the system output does not
depend on the time at which the input is applied
• The coefficients to any algebraic or differential
equations must be constant for the system to be
time-invariant
• An RLC circuit initially at rest is a time-invariant
system
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Class Examples
• Drill Problems P7-1, P7-2, P7-4
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