We can thus plot it as
a surface on the complex plane.
Note: at
s
=
λ
, the
transfer function goes
to
∞
, like a
pole
pointing up to the sky!
22
Prof. C.K. Tse: Dynamic circuits:
Frequency domain analysis
Meaning of transfer function
So, what do we actually mean by transfer function?
More precisely, it is the ratio of
V
2
to
V
1
at a given frequency
s
, in the steady state.

V
2
/
V
1

23
Prof. C.K. Tse: Dynamic circuits:
Frequency domain analysis
Frequency response
In particular, if we put
s
=
j
ω
, the input is a sinusoidal driving source
.
In other words,
we are looking at the cross section of the surface when it is
cut along the imaginary axis (yaxis).

V
2
/
V
1

24
Prof. C.K. Tse: Dynamic circuits:
Frequency domain analysis
Frequency response vs. transient
Consider a simple firstorder system.
The frequency response has a pole at s =
λ
.
The transient is
A
exp (
λ
t
).
So, there is correspondence between frequency response and transient.

V
2
/
V
1

v
2
(
t
) =
Ae
λ
t
1
s
+
λ
t
25
Prof. C.K. Tse: Dynamic circuits:
Frequency domain analysis
More examples
time domain
frequency domain
Recall: Laplace transform pairs
26
Prof. C.K. Tse: Dynamic circuits:
Frequency domain analysis
Firstorder lowpass frequency response
The transfer function of a firstorder lowpass
RC filter is
G s
V
s
V s
sC
R
sC
sCR
( )
( )
( )
/
/
=
=
+
=
+
2
1
1
1
1
1
Hence,
1.
this transfer function has a
pole
at
p
CR
=
−
1
2.
for large magnitudes of
s
(i.e., 
s\
→
∞
),
G
(
s
) goes to 0.
(At this frequency,
G
(
s
) goes to
∞
.)
From this, we can roughly sketch
G
(
s
) on the complex plane.
27
Prof. C.K. Tse: Dynamic circuits:
Frequency domain analysis
Rubber sheet analogy
Pole:
–1/CR
magnitude goes to
∞
.
All surroundings go to 0,
for large s.
28
Prof. C.K. Tse: Dynamic circuits:
Frequency domain analysis
Firstorder lowpass frequency response
30
Prof. C.K. Tse: Dynamic circuits:
Frequency domain analysis
Frequency response closeup
Observations:
If the pole is further away from the
origin, then the frequency response
starts to drop off at a higher
frequency.
At
ω
= 1/CR rad/s, the response drops
to 0.7071 of the dc value, which is
exactly 3 dB below the dc value.
3dB corner frequency
=1/CR
G j
j CR
C R
(
)
ω
ω
ω
=
+
=
+
1
1
1
1
2
2
2
Exact formula:
31
Prof. C.K. Tse: Dynamic circuits:
Frequency domain analysis
Example
Consider a first order circuit having the
following transfer function:
The eigenvalue is –10000.
The time domain response of the freeoscillating
system is:
y
(
t
) =
A e
–10000
t
In the frequency domain, the ratio Y/X is 10 at dc,
and starts to drop at around 10000 rad/s (or 1.59
kHz) which is the 3dB corner frequency.
Y s
X s
s
( )
( )
=
+
10
1
10000
10000 rad/s
or
1.59 kHz
10
7.071
2.929
Y/X
32
Prof. C.K. Tse: Dynamic circuits:
Frequency domain analysis
Secondorder lowpass frequency response with
complex pole pair
Consider a second order circuit having the
following transfer function:
Suppose the eigenvalues are complex.
The response depends on
ζ
.
G s
s
s
n
n
( )
=
+
+
1
1
2
2
2
ς
ω
ω
s
j
n
n
=
−
±
−
ςω
ω
ς
1
2
Here,
ζ
is the damping factor
ω
n
is the resonant frequency
complex poles
ζ
< 0.7071 :
light damping — freq response shows a peaking
ζ
> 0.7071 :
heavy damping — freq response shows no peaking
ω
n
is roughly where the corner frequency is.
33
Prof. C.K. Tse: Dynamic circuits:
Frequency domain analysis
Complex pole pair seen on the splane
The complex frequency plane is often called the
splane.
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 Summer '16
 Martin Chow
 Signal Processing, Complex number, Complex Plane, Prof. C.K. Tse