Download Lecture 13

INC 111 Basic Circuit Analysis
Week 13
Frequency Response
Frequency Response
Frequency response is a study of how the change of frequency
affect the voltage or current.
+
R
Asin(ωt)
AC
-
+
vL(t)
-
L
What is the change of vL(t)
when the frequency of the
voltage source change?
+
R
Asin(ωt)
AC
-
Change to
phasor
+
vL(t)
-
jL
vL 
A
R  jL
1. When
L  R
R
+
L
A∟0
AC
-
+
vL(t)
-
jωL
To sketch graph, we look at 3 points
jL
jAL
vL 
A
R  jL
R
jL
jR
j
A
A
A  0.707 A45
2. When L  R vL 
R  jL
R  jR
1 j
3. When
L  R
jL
jL
vL 
A
A A
R  jL
jL
+
R
Asin(ωt)
AC
-
+
vL(t)
-
L
|vL| (magnitude)
A
ω
Linear Scale & Log Scale
Linear Scale
1
2
3
4
5
6
Log Scale
0.1
1
10
100
1000
1000
Log scale of frequency
|vL| (magnitude)
A
Log ω
+
R
Asin(ωt)
AC
-
+
vC(t)
-
C
What is the frequency response
of vc(t)
+
R
Asin(ωt)
AC
-
+
vC(t)
-
Change to
phasor
C
R
+
A∟0
AC
-
+
vC(t)
-
1/jωC
1
1
jC
vC 
A
 A To sketch graph, we look at 3 points
1
1  jRC
R
j C
1
1
vC 
A A A
1. When RC  1
1  jRC
1
2. When
3. When
RC  1
RC  1
1
1
vC 
A
 A  0.707 A  45
1  jRC
1 j
1
1
A
vC 
A
A
1  jRC
jRC
jRC
R
+
Asin(ωt)
AC
-
+
vC(t)
-
C
What is the frequency response
of vc(t)
|vc| (magnitude)
A
Log ω
Input & Output
Input
System
Output
Input is usually what we can control.
Output is usually what we are interested in.
+
R
Asin(ωt)
AC
-
+
vC(t)
-
C
What is the frequency response
of vc(t)
Input = voltage source
Output = vc(t)
System = RC circuit
Measure Frequency Response
Input
Freq 10Hz
Amplitude = 1
Freq 100Hz
Amplitude = 1
Freq 800Hz
Amplitude = 1
System
Output
Phase = 0
Freq 10Hz
Amplitude = 0.96 Phase = 12
Phase = 0
Freq 100Hz
Amplitude = 0.82 Phase = 44
Phase = 0
Freq 800Hz
Amplitude = 0.53 Phase = 56
Frequency Response Plot
Magnitude
Phase
Note: log frequency and log magnitude
The HP 35670A
Dynamic Signal
Analyzer obtains
frequency response
data from a
physical
system.
Frequency Domain
Frequency domain is another point of view of
things in the world.
Some analysis are easier done in frequency
domain than time domain.
Metaphor of Frequency Domain
Frequency Domain
Time Domain
(1,1,1)
(2,-3,0)
Time Domain
R
L
2
sin 2t  2
s 4
C
Frequency Domain
R
sL
1/sC
Transfer Function
Transfer function = ratio of output and input
in frequency domain
Input
System
H(s)
U(s)
Transfer function
Output
Y(s)
Y ( s)
H (s) 
U (s)
Note: Transfer function describes characteristic of a system.
Example: Obtaining Transfer
Function
input is v(t)
output is vc(t)
Change all components to Phasor
(frequency domain)
Kirchoff’s Voltage Law
I ( s)
V ( s )  LsI ( s )  RI ( s ) 
Cs
1
I (s) 
V ( s)
1
Ls  R 
Cs
1
VC ( s )  I ( s ) 
Cs
1
VC ( s)
LC

V (s) s 2  R s  1
L
LC
Transfer
function
Fourier Series
“Any periodic signal can be written in the sum of sine wave signals
at different frequency.”
Filter
+
2sin(t)+3sin(9t)
AC
AC
-
R
+
vC(t)
2sin(t)+3sin(9t) +
C
We want to get rid of the signal 3sin(9t)
|vc| (magnitude)
Cut-off frequency
ω=5
Log ω
Example
Find vc(t)
1Ω
+
vC(t)
2sin(t)+2sin(9t) +
AC
0.2f
Use superposition to consider the effect of two different frequencies
1Ω
+
2sin(t)
AC
-
+
vC(t)
-
1Ω
+
0.2f
AC
2sin(9t)
-
+
vC(t)
-
0.2f
For 2sin(t)
1
+
2∟0
AC
-
+
vC(t)
-
-j5
1
1
5

   j5
jC j 1 0.2 j
 j5
5  90


Vc 
 20 

2

0
 j5  1
5.1  78.69
 1.96  11.31
vC (t )  1.96 sin( t  11.31 )
For 3sin(9t)
1
+
2∟0
AC
-
+
vC(t)
-
-j0.556
1
1
0.556


  j 0.556
jC j  9  0.2
j
 j 0.556
0.556  90


Vc 
 20 

2

0
 j 0.556  1
1.144  29.07 
 0.972  60.93
vC (t )  0.972 sin( 9t  60.93 )
Finally, we add vc(t) from both frequencies
vC (t )  1.96 sin( t  11.31 )  0.972 sin( 9t  60.93 )
AC
1Ω
+
vC(t)
2sin(t)+2sin(9t) +
0.2f
Other topics
• Three-phase circuits
• Magnetically Coupled circuits
• Transformer
Power Line Systems
3 phases systems have three wire called “line” which are
sine wave with phase shift 120 from each other
Line 1
220 2120
Line 2
220 20
Line 3
220 2240
AC
AC
AC
Power Plant
3 phases, 3 wires

Power Plant
Line 1
220 2120
Line 2
220 20
Line 3
220 2240

AC
AC
AC
Neutral
3 phases, 4 wires
L1-N = 220Vrms,
L2-N = 220Vrms
L3-N = 220Vrms
L1-L2 = 381Vrms,
L2-L3 = 381Vrms,
L1-L3 = 381Vrms
Star Connection ( Y-connection)
AC
Z
Z
AC
Z
AC
Load
Each load get 220 Vrms
Delta Connection ( Δ-connection)
AC
AC
Z
Z
AC
Z
Load
Each load get 381 Vrms