1.1
Transfer Functions:
Note in the last example of the previous unit, H ( j ) is entirely dependent on the
circuit elements and defines the relationship between the specified input vs and
output vo. Therefore, if the input magnitude and phase changes, the output can be
determined by multiplying the magnitude of the input by .33 ( H ( j2 ) ) and adding
-9.46 ( H ( j2) ) to the input phase. In this sense, H ( j ) transfers the input value to
the output value.
Definition:
A transfer function (TF) is a complex-valued function of , such that the TF
magnitude indicates the scaling between the magnitudes of the circuit input and
output, and the TF phase indicates the shifting between the phases of the circuit
input and output. The transfer function defines this relationship for every possible
input frequency.
To find a transfer function, convert to impedance circuit, but leave the impedance
values as functions of j=p. Then solve for the ratio of the phasor output divided by
the phasor input.
1.2
Transfer Function Example:
Determine the transfer function for the circuit below, where the input is vi(t) and the
output is io(t).
R1
vi(t)
io(t)
C
Show transfer function is given by:
 1 


0
CR
R

I
1 2
H ( p)  
Vi  R1  R2 

p
 CR1 R2 
R2
1.3
Transfer Function Example:
Determine the transfer function for the circuit below, where the input is vs(t) and
the output is io(t).
1H
10 
vs(t)
Show transfer function is given by:
4 2
p
0
I
5
H ( p) 

Vs p 3  6 p 2  14 p  4
5
2H
io(t)
3H
0.5 F
1.4
Transfer Function Example:
Determine the transfer function for the circuit below, where the input is vi(t) and the
output is vo(t).
C
+
Rh
-
vi(t)
R1
Show transfer function is given by:
 Rf 
1   p

Ri 

V
H ( p)  0 
1
Vi
p
Rh C
Rf
+
vo(t)
-
1.5
Determining Transfer Functions in SPICE:
If the input has unit amplitude and zero phase, then the ratio of the phasor output
over phasor input equals the phasor output. Thus, set input equal to the source:
Phasor Magnitude
>> V1 1
0
AC 1
0
Source Type
Phasor Phase
(optional, default =
0)
Node Location
Source Label
>> (Insert rest of circuit description)
Command SPICE to evaluate the phasor analysis over a frequency range (in Hz):
>>.AC {Point spacing =>LIN or DEC} { # of evaluation points} {start point 0} {end point}
Command SPICE to print a result to a table:
>>.PRINT AC VM(node), VP(node), VDB(node)
Voltage Magnitude
Voltage Phase
Voltage Magnitude in Decibels
Command SPICE to plot result:
>>.PLOT AC VM(node), VP(node), VDB(node)
1.6
SPICE Example:
Determine the transfer function using SPICE:
10 F
+
50 k
vi(t)
-
+
vo(t)
-
45 k
5 k
Hint: Use to the following model for Op Amps:
+
vd
-
1
2
+
6
-
4
for the A741 op-amp A0=2x105, Ri=2M, R0=75,
RL=10k, CL=1.1519 F
+
vo
3
1
+
vd
-
Ri
2
vd
RL
RL
5
CL
6
Ro
+
vL
-
+
4
A0 vL
+
vo
-
1.7
10F
45k
1
5
2
50k
+
vd
_
2M
vd
10k
6
1.1F
3
0
5k
0
0
* SPICE Code for OP amp circuit Example in
* Unit 1 to create a Transfer Function
* Circuit Components Outside of OP Amp
* Define input with unit magnitude and 0 phase
* so output will be identically equal to the
* transfer function
V1 1
0
AC 1
0
C1 1
2
10UF
R1 2
0
50K
R3 3
0
5K
R4 3
4
45K
4
10k
Vi=10
0
75
+
vL
_
0.2M vL
+
vo
_
1.8
* Circuit Components for OP Amp Model
R2
G1
R5
C2
E1
R6
2
0
5
5
0
6
3
5
0
0
6
4
2MEG
2
3
10K
1.519UF
5
0
75
.1M
2E5
* Analysis and Output Statements
.AC DEC 150
.PRINT
AC
.PROBE
.END
.01 150k
VM(4)
VP(4)
* Select Trace on Probe
* menu to add trace dB(abs(V(4))
* to plot TF magnitude.
* Then for a new plot add trace
* 180*atan(V(4) / 3.14159
* to plot TF phase.
1.9
Poles and Zeros:
The transfer function of a linear system can be written as a ratio of polynomials:




m
m 1
b
p

b
p
... b1 p  bo
m
m

1
H ( p)  n
p  an1 p n1  an2 p n2 ... a0
For all real-time systems m  n.
The order of the denominator polynomial defines the order of the system.
The roots of the numerator polynomial are referred to as zeros of the system.
The roots of the denominator polynomial are referred to as poles of the system.
The transfer function can be rewritten as:
G ( p  pz1 )( p  p z 2 )...( p  pzm )
H ( p)  0
( p  p p1 )( p  p p2 )...( p  p pn )
Identify the poles and zeros for the above system with the transfer function shown
above.
1.10
A pole-zero diagram of a transfer function is the complex number plane where the
zero positions are indicated by the 0 symbol and pole positions by the X symbol.
1.11
Pole-Zero Diagram Example:
Pole-zero diagrams are useful for rough sketches of the transfer function
magnitude.
 For values of  where the j-axis is near a pole, a relative maximum occurs.
 For values of  where the j-axis is near a zero, a relative minimum occurs.
 The closer the pole or zero is to the j-axis, the sharper the max or min point.
Determine the pole-zero diagram and sketch the transfer function magnitude.
H ( p) 
IM
1
-1
1
-1
RE
10 p
p2  2 p  2