Transfer Functions
Transfer Functions
Assume zero initial conditions.
• Transfer functions defined
N
• Examples
• System stability
dk y(t)
dtk
=
ak sk Y (s)
=
ak
k=0
• Pole-Zero Plots
N
• Sinusoidal steady-state analysis
Y (s)
• Transfer function synthesis
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ak sk
Transfer Functions
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J. McNames
Initial Conditions
Assume zero initial conditions.
N
k=0
N
dk y(t)
dtk
=
ak sk Y (s)
=
ak
k=0
M
k=0
M
bk
bk sk X(s)
= X(s)
k=0
ECE 222
M
dk x(t)
dtk
k=0
N
Y (s)
J. McNames
bk
k=0
k=0
• Linearity and time invariance defined
M
=
M
bk sk
k=0
M
k
k=0 bk s
N
k
k=0 ak s
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X(s) = H(s)X(s)
Transfer Functions
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Transfer Functions Continued
M
k
k=0 bk s
X(s) = H(s)X(s)
Y (s) = N
k
k=0 ak s
dk x(t)
dtk
• In the time domain, the relationship can be complicated
bk sk X(s)
• In the s domain, the relationship of Y (s) to X(s) of LTI systems
simplifies to a rational function of s
k=0
• All voltages and currents are due to independent sources
(superposition)
• H(s) is usually a rational ratio of two polynomials
• Energy stored in capacitors and inductors also act like independent
sources
• Specifically, the transfer function of an LTI system can be defined
as the ratio of Y (s) to X(s)
• H(s) is called the transfer function
• We will now focus a specific class of circuits
– Only one independent source (input)
– No energy stored in capacitors or inductors
• Usually denoted by H(s), sometimes G(s)
• Without loss of generality, usually aN 1
• Greatly simplifies analysis
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Transfer Functions
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Transfer Functions
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Example 1: Transfer Function vs. Impulse Response
Transfer Functions and the Impulse Response
Fill in the missing parts to determine how the transfer function of an
LTI system G(s) is related to the impulse response h(t)
x(t) = δ(t)
X(s) =
y(t) =
Y (s) =
L {h(t)} =
L−1 {G(s)} =
x(t)
h(t)
y(t)
H(s)
x(t)
y(t)
• Because of their relationship, both H(s) and h(t) completely
characterize the LTI system
• If the LTI system is a circuit, once you know either H(s) or h(t),
you have sufficient information to calculate the output
• You now have three different approaches to solve for the output of
an LTI circuit
– y(t) = x(t) ∗ h(t)
– Solve for H(s), X(s), and then y(t) = L−1 {H(s) X(s)}
– Use Laplace transform circuit analysis to solve for the outputs
of interest
• All three have limitations, advantages, and disadvantages
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Transfer Functions
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Continous-Time Convolution Tradeoffs
x(t)
h(t)
y(t)
x(t)
H(s)
y(t)
x(t)
6
Transfer Functions
h(t)
y(t)
H(s)
x(t)
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y(t)
• Advantages
– Reduces differential equation to an algebra problem
– Usually the easiest approach
– Easy to find the output for different input signals
• Disadvantages
– Cannot account for non-zero initial conditions, requires
complete x(t) and y(t)
– Can be difficult to write and solve integrals
– Can only be used for single-input single-output (SISO) systems
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Transfer Function: y(t) = L−1 {H(s) X(s)}
• Advantages
– Can find solution for all t, not just t > 0
– Can be approximated using discrete-time convolution
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Transfer Functions
Transfer Function Analysis Tradeoffs
Continuous-time Convolution: y(t) = x(t) ∗ h(t)
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• Disadvantages
– Can only solve for y(t) for t > 0
– Requires zero initial conditions
– Can only be used for SISO systems
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ECE 222
Transfer Functions
Laplace Transform Circuit Analysis Tradeoffs
h(t)
x(t)
y(t)
x(t)
H(s)
Example 6: Transfer Functions
R
y(t)
+
Laplace Transform Circuit Analysis
vs(t)
• Advantages
– Elegant method of handling non-zero initial conditions
– Can handle multiple sources (multiple inputs) & can solve for
multiple outputs (any voltage or current) — MIMO systems
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Transfer Functions
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vo(t)
-
Find the transfer function for the circuit above. The input is the
voltage source vs (t) and the output is labeled vo (t).
• Disadvantages
– Can only solve for y(t) for t > 0
– Cannot account for full history, x(t) for t < 0. Requires this
effect to be captured in the initial conditions
– Can be tedious
– Specific to application (circuits), we did not discuss
generalization to other types of systems
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C
9
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Example 7: Transfer Functions
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C
C
R
+
+
vs(t)
vo(t)
vo(t)
-
-
Find the transfer function for the circuit above. Do you recognize this
function?
J. McNames
Transfer Functions
Example 8: Transfer Functions
R
vs(t)
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Transfer Functions
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Find the transfer function for the circuit above. Do you recognize this
function?
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Transfer Functions
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Example 9: Transfer Functions
Example 9: Workspace
CB
CA
RB
RA
+
vs(t)
vo(t)
RL
-
Find the transfer function for the circuit above.
J. McNames
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Transfer Functions
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Example 10: Transfer Functions
+ vR(t)
R
vs(t)
-
+ vL(t)
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Transfer Functions
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Example 10: Workspace
-
L
+
C
vC(t)
-
Find the transfer function from the input voltage to an output voltage
across each element of the three passive elements in a series RLC
circuit.
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Transfer Functions
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Transfer Functions
Poles and Stability
Pole-Zero Plots
Assume all of the poles in a transfer function H(s) are unique. Then
H(s) can be written as follows using partial fraction expansion:
N
N (s) k
H(s) =
=
D(s)
s − p
H(s) =
• Zeros: roots of N (s)
• Poles: roots of D(s)
=1
L−1 {H(s)}
= h(t) =
N
N (s)
D(s)
• Poles must be in the left half plane for the system to be stable
k e+p t u(t)
• As the poles get closer to the boundary, the system becomes less
stable
=1
• Note the expansion is in terms of the poles, rather than −p
∞
• If −∞ |h(t)| dt < ∞, the LTI system is bounded-input
bounded-output (BIBO) stable
• Pole-Zero Plot: plot of the zeros and poles on the complex s
plane
• You will use these throughout the junior sequence (ECE 32x)
• That is |h(t)| < α < ∞ for all t
• h(t) is bounded if Re{p } < 0 for all • The system is BIBO stable if and only if all the poles are in the
left half of the complex plane
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Transfer Functions
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Example 11: Pole-Zero Plots
s3
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Pole−Zero Plot
1.5
s2 − 1
+ 4s2 + 6s + 4
1
Imaginary Axis
Using the MATLAB, we can quickly find the roots
H(s) =
Transfer Functions
Example 11: Pole-Zero Plot
Use MATLAB to generate a Pole-Zero plot for a system with the
following transfer function
H(s) =
ECE 222
(s + 1)(s − 1)
(s + 2)(s + 1 − j)(s + 1 + j)
Is the system stable? The pole-zero plot, impulse response, and step
response are shown on the following slides.
0.5
0
−0.5
−1
−1.5
−3
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Transfer Functions
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−2.5
−2
−1.5
−1
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−0.5
0
Real Axis
ECE 222
0.5
1
Transfer Functions
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2
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Example 11: MATLAB Code for Pole-Zero Plot
Example 11: Impulse Response
sys = tf([1 0 -1],[1 4 6 4]);
Impulse Response
figure;
[p,z] = pzmap(sys);
h = plot(real(p),imag(p),’bx’,real(z),imag(z),’ro’);
set(h,’LineWidth’,1.2);
set(h,’MarkerSize’,5);
hold on;
plot([0 0],[-2 2],’k:’,[-3 2],[0 0],’k:’);
hold off;
xlabel(’Real Axis’);
ylabel(’Imaginary Axis’);
title(’Pole-Zero Plot’);
axis([-3 2 -1.5 1.5]);
1
0.8
h(t)
0.6
0.4
0.2
0
−0.2
−0.4
0
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J. McNames
1
2
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Example 11: MATLAB Code for Impulse Response
3
4
Time (seconds)
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5
6
Transfer Functions
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Example 11: Step Response
sys = tf([1 0 -1],[1 4 6 4]);
Step Response
figure;
t = 0:0.01:7;
[h,t] = impulse(sys,t);
h = plot(t,h);
set(h,’LineWidth’,1.5);
hold on;
plot([0 0],[-2 2],’k:’,[0 max(t)],[0 0],’k:’);
hold off;
axis([0 max(t) -0.5 1.2]);
xlabel(’Time (seconds)’);
ylabel(’h(t)’);
title(’Impulse Response’);
0.1
0
y(t)
−0.1
−0.2
−0.3
−0.4
−0.5
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Transfer Functions
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0
1
2
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3
4
Time (seconds)
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6
Transfer Functions
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Example 11: MATLAB Code for Step Response
Steady-State Sinusoidal Analysis
Assume a system H(s) is BIBO stable. Consider the case where the
source is sinusoidal:
sys = tf([1 0 -1],[1 4 6 4]);
figure;
t = 0:0.01:7;
[h,t] = step(sys,t);
h = plot(t,h);
set(h,’LineWidth’,1.5);
hold on;
plot([0 0],[-2 2],’k:’,[0 max(t)],[0 0],’k:’);
hold off;
axis([0 max(t) -0.5 0.2]);
xlabel(’Time (seconds)’);
ylabel(’y(t)’);
title(’Step Response’);
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ECE 222
x(t)
= A cos φ cos ωt − A sin φ sin ωt
A (s cos φ − ω sin φ)
X(s) =
s2 + ω 2
Y (s) = H(s)X(s)
A (s cos φ − ω sin φ)
Y (s) = H(s)
s2 + ω 2
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Steady-State Sinusoidal Analysis Continued
Y (s)
=
2|k| cos(ωt + ∠k)u(t) +
N
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t→∞
• If the input to an LTI system is sinusoidal,
– The steady-state output is sinusoidal at the same frequency
– The amplitude and phase of y(t) differ from that of x(t)
k e−pt u(t)
• We applied this idea when we did phasor analysis
• But how is k related to H(s), A, and φ?
= yss (t) + ytr (t)
=
Transfer Functions
yss (t) = lim y(t) = 2|k| cos(ωt + ∠k)
=1
yss (t)
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If x(t) = A cos(ωt + φ),
A (s cos φ − ω sin φ)
s2 + ω 2
N
k
k
k∗
+
+
s − jω s + jω
s + p
=1
y(t)
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Steady-State Sinusoidal Analysis Comments
= H(s)
=
= A cos(ωt + φ)
lim y(t)
t→∞
= 2|k| cos(ωt + ∠k)
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Transfer Functions
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Solving for the Complex Residue
Sinusoidal Steady-State Output
Since H(jω) is complex, we can write it in polar form as
A (s cos φ − ω sin φ)
Y (s) = H(s)
s2 + ω 2
N
k
k
k∗
=
+
+
s − jω s + jω
s + p
=1
A (s cos φ − ω sin φ) k = H(s)
s + jω
s=+jω
=
=
=
J. McNames
H(jω) = |H(jω)| ej∠H(jω)
Then using the results of the previous slide, we have
k = 12 H(jω)A ejφ
|k| = 12 |H(jω)|A
A(jω cos φ − ω sin φ)
2jω
A(cos φ + j sin φ)
H(jω)
2
jφ
1
H(jω)A
e
2
= 12 |H(jω)|A ej(φ+∠H(jω))
∠k = φ + ∠H(jω)
H(jω)
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Transfer Functions
yss (t)
H(s)
2|k| cos(ωt + ∠k)
= |H(jω)|A cos (ωt + φ + ∠H(jω))
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Sinusoidal Steady-State Output
x(t)
=
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Transfer Functions
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Example 12: Steady-State Sinusoidal Analysis
R
y(t)
+
x(t)
yss (t)
vs(t)
= A cos(ωt + φ)
= |H(jω)|A cos (ωt + φ + ∠H(jω))
C
vo(t)
-
• The input is sinusoidal
Find the steady-state sinusoidal response to an input voltage of
vs (t) = cos(ωt).
• The steady-state signal yss (t) is also a sinusoid
– Same frequency as x(t): ω
– Amplitude is scaled by |H(jω)|
– The phase is shifted by ∠H(jω)
• If we know H(s), we can easily find the steady-state solution for
any sinusoidal input signal
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Transfer Functions
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Example 12: Workspace
Steady-State Sinusoidal Analysis Comments
• We will study this in depth shortly
• There is analytical significance to how the magnitude and phase of
H(s) vary with s = jω
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Transfer Functions
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LTI Systems
x(t)
h(t)
y(t)
; ;
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Example 13: Transfer Function Analysis
H(s)
x(t)
y(t)
b
k
m
x(t)
• If we know the transfer function, we have sufficient information to
calculate the output for any input
y(t)
• This enables us to treat the circuit more abstractly as H(s)
Find the transfer function for the linear system shown above. The
external force x(t) is the input to the system and the displacement
y(t) is the output. Find the transfer function.
• The transfer function may be for another type of system:
mechanical, chemical, hydraulic, etc.
• Mathematically they are treated the same
• Field-specific analysis is used only to find H(s)
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Transfer Functions
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Transfer Function Synthesis
x(t)
h(t)
y(t)
H(s)
x(t)
Cascade Transfer Function Synthesis
x(t)
y(t)
• Thus far we have talked only about circuit analysis
H1(s)
H(s) =
H2(s)
HP(s)
y(t)
N (s)
= H1 (s) × H2 (s) × · · · × HP (s)
D(s)
• We now know several ways to solve for the output of a given
system
• There are many approaches to transfer function synthesis
• If there are zero initial conditions, then we can find the transfer
function H(s) of a given circuit
• Will discuss how to specify H(s) to meet the requirements for a
given application later this term
• Now we will discuss how to design a circuit that implements a
given H(s)
• The most common (and perhaps easiest) approach to synthesis is
to break H(s) up into 1st (real poles) or 2nd (complex poles)
order components
• This is called transfer function synthesis
• Thus each component, Hi (s) has either a 1st or 2nd order
polynomial in the numerator and denominator
• There are many circuits that have the same transfer function
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Transfer Functions
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Cascade Transfer Function Synthesis Continued
x(t)
H1(s)
H2(s)
HP(s)
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Summary
• Circuits with a single input (independent source) and zero initial
conditions can be represented generically by their transfer
functions
y(t)
• There are robust, standard circuits for implementing these
low-order components
• H(s) is the Laplace transform of the system impulse response
• The output of the system is y(t) = L−1 {H(s)X(s)} for any
causal input signal (x(t) = 0 for t < 0)
• The output of each transfer function is generated by an
operational amplifier
• For sinusoidal inputs, the output is also sinusoidal at the same
frequency but amplified by |H(jω)| and shifted in phase by
∠H(jω)
• This is essential for the cascade synthesis to work (will explain
later)
• Some of these 1st and 2nd order components are discussed in the
text (Chapter 15)
• Thus, transfer functions make sinusoidal steady-state analysis easy
• Others can be found in more advanced analog circuits texts
• Transfer function analysis used for all types of LTI systems, not
just circuits
• Generalization of phasors
• You will use cascade synthesis in the first lab for ECE 203
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• Can synthesize a transfer function using a cascade of 1st and 2nd
order active circuits
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