Engineering 43
Fourier
Transfer Fcn
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Engineering-43: Engineering Circuit Analysis
1
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-06a_Fourier_XferFcn.pptx
Fourier Transform
 A Fourier Transform  A Conceptual Example
is an integral
• This Irregular Signal
transform that reexpresses a function
in terms of different
Sine/Cosine waves
• Is the SAME as the
of varying
Sum of these Sinusoids
amplitudes,
wavelengths, and
phases.
Engineering-43: Engineering Circuit Analysis
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-06a_Fourier_XferFcn.pptx
Fourier Transform
 John Baptiste
Joseph Fourier
investigated Time
Varying Heat-Flow
in a Metal Bar
 For a Given,
arbitrary Periodic
Function, f(t), The
Fourier Equivalents
 His great Insight:
ANY Periodic
Function Could be
Expressed as the
sum of Sinusoidal
Functions
f t   b0   Bm cosm 0 t   m 
Engineering-43: Engineering Circuit Analysis
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
m 1

f t   d 0   Dn sin n 0 t   n 
n 1
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-06a_Fourier_XferFcn.pptx
Example: Square Wave
 The SquareWave Shown at Bottom-Lt
can be described by a sum-of-sines
vsq 
4A

sin 0t  
4A
4A
4A
4A
sin 30t  
sin 50t  
sin 70t  
sin 90t   
3
5
7
9
Engineering-43: Engineering Circuit Analysis
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-06a_Fourier_XferFcn.pptx
Transfer Fuction, H(f)
iin
 Consider a “Black Box”
vin
that takes Input Power,
vin & iin Transforms this
Power into an Output, vout & iout
iout
vout
• A typical transformation would be to “FilterOut” certain electrical frequencies.
 For Phasor Voltages
Vin & Vout Define the
voltage
Transfer Function as
Engineering-43: Engineering Circuit Analysis
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Vout
Hf 
Vin
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-06a_Fourier_XferFcn.pptx
Transfer Function
Vout
Hf 
Vin
 Note that the Transfer Function
• Is a Function of FREQENCY ONLY
• Can Change (and usually does change)
the Magnitude and Phase-Angle of many
of the incoming, frequency-dependent,
electrical signals
 Measuring an Unknown “Black Box”
Apply Sinusoidal Vin
(Vin0°), Measure Vout
(Voutφ°) and Plot:
Vout / Vin and φ
Engineering-43: Engineering Circuit Analysis
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-06a_Fourier_XferFcn.pptx
H f 
Example Transfer Function
H  f  
f Hz 
f Hz 
Engineering-43: Engineering Circuit Analysis
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-06a_Fourier_XferFcn.pptx
Example Transfer Function
 Find vout for vin = 1.35Vcos(40∙2πt+65°)
H  f  
H f 
−25
Engineering-43: Engineering Circuit Analysis
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f Hz 
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-06a_Fourier_XferFcn.pptx
Example Transfer Function
 Then at 40 Hz
(40∙2π rads/sec)
Vout
H 40Hz   25  150 
Vin
 Using the Values
Taken from the H(f)
Mag & Phase
Graphs
 Recall vin
Vout  25  150 1.35V65
Vout  33.75V  85
 In Phasor for
 Or in the Time
Domain
vin  1.35V cos40  2t  65
Vin  1.35V65
vout t    33.75V cos40  2t  85
 Thus
Vout  H 40Hz  Vin
Engineering-43: Engineering Circuit Analysis
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-06a_Fourier_XferFcn.pptx
MultiFrequency Example 6.2
 Note the THREE
Frequencies
• 0 Hz
• 1000 Hz
– 1000∙2π
rad/sec
• 2000 Hz
– 2000∙2π
rad/sec
Engineering-43: Engineering Circuit Analysis
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-06a_Fourier_XferFcn.pptx
Ex6.2 Transfer Function
 Apply to vin the Transfer Function
 From the Transfer Function find
H 0  40 H 1000  330 H 2000  260
• Apply To components of vin
Engineering-43: Engineering Circuit Analysis
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-06a_Fourier_XferFcn.pptx
Example 6.2
 Using This H(f) Set find
H 0  40 H 1000  330 H 2000  260
Vout1  H 0 Vin1  40  30  120  12
Vout 2  H 1000 Vin2  330  20  630
Vout3  H 2000 Vin3  260 1  70  2  10
 Note that the above Phasors CanNOT
be added as they have DIFFERENT
Frequencies.
Engineering-43: Engineering Circuit Analysis
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-06a_Fourier_XferFcn.pptx
Example 6.2
 Because of Differing Frequencies
MUST add TIME-DOMAIN Voltages
Vout1  12 Vout 2  630 Vout3  2  10
vout1 t   12 vout 2 t   6 cos1000  2  30 vout3 t   2 cos2000  2  10
 Then vout(t)
is simply
the SUM
of the above
Engineering-43: Engineering Circuit Analysis
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-06a_Fourier_XferFcn.pptx
1st Order Lo-Pass Filter
 Consider the RC Ckt
Shown below
Vin
 In the Frequency
Domain the Cap
Impedance, Zc
Engineering-43: Engineering Circuit Analysis
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Vout
1
1
ZC 

jC j 2f C
 Notice the Limits of
Behavior
1
lim ZC  lim

f 0
f 0 j 2f C
1
lim Z C  lim
0
f 
f  j 2f C
 A cap is
• OPEN to Low-Freq
• SHORT to Hi-Freq
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-06a_Fourier_XferFcn.pptx
1st Order Lo-Pass Filter
 Thus the Behavior
of a Cap-Based
Impedance
ZR  R
Vin
• At LO-Frequencies a
Cap acts as an
OPEN Circuit
• At HI-Frequencies a
Cap Acts as a
SHORT Circuit
 Now use Phasor VDivider on RC ckt
Engineering-43: Engineering Circuit Analysis
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ZC 
Vout
1
j 2f C
Vout
ZC
1  j 2f C 

Vin 
Vin
Z tot
R  1  j 2f C 
 Multiplying Top&Bot
by j2πfC
Vout
1

Vin
1  j 2f RC
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-06a_Fourier_XferFcn.pptx
1st Order Lo-Pass Filter
 Then the Transfer
Function
Vout
1
H f 

Vin 1  j 2f RC
 ReWriting
 fB is the “Break
point” Frequency at
which H(f) falls to
70.7% of its Original
Magnitude Value.
 Note The Mag & Ph
1
1
Hf 

of H(f) in terms of fB :
1  jf 2RC  1  jf 1 f B 
1
1  j f f B 


H
f

Hf 
2
2


1 f fB
1  f fB 
 Where
1
fB 
2RC
Engineering-43: Engineering Circuit Analysis
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H  f    arctan  f f B 
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-06a_Fourier_XferFcn.pptx
Lo-Pass Filter
Vin
Vout
 The LoPass Filter Transfer Function
 fB : is also call the Half-Power-Frequency
• Recall Full Power to a Resistor: I 2 R or V 2 R
• Then HALF Power: I
Engineering-43: Engineering Circuit Analysis
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
2
2 R or
V
2

2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-06a_Fourier_XferFcn.pptx
R
LR (LowPass) Filter
 Find the Transfer
Function for LR Ckt
Z L  j 2fL
Vin
I

Vout

 Use Ohm Find The
Single Loop Current
Vin
Vin
I

Z L  R j 2fL  R
Engineering-43: Engineering Circuit Analysis
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 Then also by Ohm
 Vin
 1
Vin
Vout  I  R  



j 2fL R  1
 j 2fL  R  1
R
 ReWriting
Vout 
Vin

j 2fL R  1 j
Vin
Vin

f
f
1 1 j
 fB 
 R 
 2fL 


 Arrive at Xfer Fcn very
similar to RC Ckt
Hf 
Vout
1

Vin 1  jf f B
where : f B  R 2L 
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-06a_Fourier_XferFcn.pptx
The deciBel (dB)
 Named after
Alexander Graham
Bell, the deciBel
(dB) relates two
Power Levels
P2
LdB  10 log
P1
 SomeTimes The
Power Level is
Referenced to a
Standard Value, P0
Engineering-43: Engineering Circuit Analysis
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 In this case
P
LdB  10 log
P0
 ReCall a Current or
Voltage delivering
Power to a Resistor
Pv  V 2 R
Pi  I 2 R
 Then the dB in
Current or Voltage
Ratios
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-06a_Fourier_XferFcn.pptx
The deciBel (dB)
 dB In Terms of
Voltage Ratios
 V22 R 
P2
LdB  10 log
 10 log  2 
P1
 V1 R 
2
 V22 
 V2 
 V2 




 10 log  2   10 log    20 log  
 V1 
 V1 
 V1 
 Or dB for Currents
 I12 R 
P2
LdB  10 log
 10 log  2 
P1
 I2 R 
I
 10 log 
I
2
2
2
1
2

I 
I 
  10 log  2   20 log  2 

 I1 
 I1 
Engineering-43: Engineering Circuit Analysis
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 Now we Defined
H f  
Vout
Vin 
2
H  f   Vout Vin
 Since |H(f)| is a
Voltage Ratio,
define
H  f  dB  20 log  H  f  
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-06a_Fourier_XferFcn.pptx
dB Plots (SemiLog) Plot
 Plotting H(f) on the logarithmic dB Scale
makes it easier to distinguish Very
Large (104 vs 105) or Very Small (10−4
vs 10−5) Points on the Plots
 85db  20 log H  f   H  f   1085 20  104.25  0.0000562
Engineering-43: Engineering Circuit Analysis
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-06a_Fourier_XferFcn.pptx
Cascaded NetWork Gain
 Consider the
Transfer Function of
the “BlackBox” at
Right   Vout
H f 
Vin
 Looking inside the
BlackBox find
Vout Vout 2 Vout 2
Vout 2 Vout1


1 

Vin
Vin1
Vin1
Vin1 Vout1
 Note that with
Vout1 = Vin2
Engineering-43: Engineering Circuit Analysis
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Hf 
Vout Vout 2 Vout1 Vout 2 Vout1




or
Vin
Vin1 Vout1 Vin1 Vin2
so : H  f  
Vout1 Vout 2

 H1  f  H 2  f 
Vin1 Vin2
 Or in dB form
H  f   H1  f  H 2  f  
20 log H  f   20 log H1  f  H 2  f  
20 log H  f   20 log H1  f   20 log H 2  f  
H  f  dB  H1  f  dB  H 2  f  dB
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-06a_Fourier_XferFcn.pptx
Reading Logarithmic Scales
 Tools Needed
• Ruler
• Scientific Calculator
 To Find a Value of a Pt
Between Decades m & n
• Use Ruler to Measure
– Decade Distance, dd
– Distance from Pt to Lower
Decade (Decade m), dp
• Then The Value at the Pt
V  10
d p dd
10 m
Engineering-43: Engineering Circuit Analysis
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-06a_Fourier_XferFcn.pptx
10
10
-30
-31

d d  21.1 mm
10
-32
V  10
15.4 21.1
10
d p  15.4 mm
10
32
 10
0.730
10
32
 5.37 10
32
-33
400
405
410
415
Engineering-43: Engineering Circuit Analysis
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420
425
x
430
435
440
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-06a_Fourier_XferFcn.pptx
445
450
Octave
 An octave is the interval between two
points where the frequency at the
second point is twice the frequency of
the first.
 Given Frequencies f1 & f2
N oct

OR
N oct

 f1 
log 2  
 f2 
log  f1 f 2 
log 2 
Engineering-43: Engineering Circuit Analysis
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MUSICAL Octaves
Octave
1
2
3
4
5
6
7
8
Frequency (Hz)
63
125
250
500
1k
2k
4k
8k
Wavelength in air
(70oF, 21oC) (ft)
17.92
9.03
4.52
2.26
1.129
0.56
0.28
0.14
Wavelength in air
(70oF, 21oC) (m)
5.46
2.75
1.38
0.69
0.34
0.17
0.085
0.043
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-06a_Fourier_XferFcn.pptx
WhiteBoard Work
 Let’s This Nice
Problem 

vin t 
vout t 

 Find the OutPut
Voltage for For
this Input
vin t   17V  23V cos1000  2t   31V cos12000  2t 
Engineering-43: Engineering Circuit Analysis
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-06a_Fourier_XferFcn.pptx
All Done for Today
79.5 MHz
Notch
Filter
Engineering-43: Engineering Circuit Analysis
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-06a_Fourier_XferFcn.pptx
Engineering 43
Appendix
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Engineering-43: Engineering Circuit Analysis
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-06a_Fourier_XferFcn.pptx
Logarithm Change
of Base Proof
Engineering-43: Engineering Circuit Analysis
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-06a_Fourier_XferFcn.pptx
Engineering-43: Engineering Circuit Analysis
30
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-06a_Fourier_XferFcn.pptx
Engineering-43: Engineering Circuit Analysis
31
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-06a_Fourier_XferFcn.pptx
Engineering-43: Engineering Circuit Analysis
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-06a_Fourier_XferFcn.pptx
White Board RL Filter Problem
Engineering-43: Engineering Circuit Analysis
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-06a_Fourier_XferFcn.pptx
LR Filter Transfer Function
1
0.9
f = 0:10:20e3
HfB = 1./sqrt(1+(f/fB).^2);
plot(f,HfB,'LineWidth',3), grid, xlabel('f
(Hz)'), ylabel('|H(f)')
disp('showing fB plot - hit ANY KEY to
continue')
pause
fB = 2700/(2*pi*68e-3)
Hf = abs(2700./(2700 + j*2*pi*f*68e-3));
plot(f,Hf,'LineWidth',3), grid, xlabel('f
(Hz)'), ylabel('|H(f)')
0.8
|H(f)
0.7
0.6
0.5
0.4
0
0.2
0.4
0.6
0.8
1
f (Hz)
1.2
Engineering-43: Engineering Circuit Analysis
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1.4
1.6
1.8
2
4
x 10
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-06a_Fourier_XferFcn.pptx
P5.57 Graphics
Engineering-43: Engineering Circuit Analysis
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-06a_Fourier_XferFcn.pptx
P5.81 Graphics
Engineering-43: Engineering Circuit Analysis
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Lec-06a_Fourier_XferFcn.pptx