1. 12. Variable frequency network performance

2. Learning Goals Variable-Frequency Response Analysis
Network performance as function of frequency.
Transfer function
Sinusoidal Frequency Analysis
Bode plots to display frequency response data
Resonant Circuits
The resonance phenomenon and its characterization
Scaling
Impedance and frequency scaling
Filter Networks
Networks with frequency selective characteristics:
low-pass, high-pass, band-pass

3. Variable frequency-response analysis
In AC steady state analysis the frequency is assumed constant (e.g., 60Hz).
Here we consider the frequency as a variable and examine how the performance
varies with the frequency.
Variation in impedance of basic components
Resistor

4. Inductor

5. Capacitor

6. Driving Point functions
Series RLC network
Simplification in notation
Driving Point functions

7. Pulse represented by sum of sinusoidal signal

8. Network functions Nomenclature
When voltages and currents are defined at different terminal pairs we define the ratios as Transfer Functions
INPUT
OUTPUT
TRANSFER FUNCTION
SYMBOL
Voltage
Voltage Gain
Gv(s)
Current
Transimpedance
Z(s)
Current Gain
Gi(s)
If voltage and current are defined at the same terminals we define
Driving Point functions (Impedance/Admittance)
Network functions for basic components
General form

9. Driving point function
Nomenclature
Port #1
Port #2
Circuit
terminals
terminals
V2 or I2
Transfer function
Driving point function

10. Example 12.1
MATLAB can be effectively used to compute frequency response characteristics

11. Some Matlab command
A= [2,3,1; 4, 5, 7]
A= [1:2:9]
A= A’

12. Matlab example clear; % clear all variables.
clf; % clear current figure.
w = [1:1:1.e3];
b=[15*2.53*1e-3 0];
a=[0.1*2.53*1e-3 15*2.53*1e-3 1];
num = polyval(b,i*w);
den = polyval(a,i*w);
H = num./den;
mag = abs(H);
plot(w,mag);

13. Example
Find the pole and zero locations and the value of K0 for the voltage gain
Zeros = roots of numerator
Poles = roots of denominator
For this case the gain was shown to be

14. clear;
clf;
w = logspace(1,6,200);
b = [1.e0 * 40e3*pi, 0];
a = [1, 40.1e3*pi, 4e6*pi*pi];
num = polyval(b,i*w);
den = polyval(a,i*w);
H = num./den;
%arg = angle(H)*180/pi;
mag = abs(H);
plot(w,mag);
%semilogx(w,mag);
%freqs(num, den,w);

15. Poles and Zeros (More nomenclature) ; Arbitrary network function
Using the roots, every (monic) polynomial can be expressed as a
product of first order terms
The network function is uniquely determined by its poles and zeros
and its value at some other value of s (to compute the gain)

16. Positive real function
The coefficients of the polynomial are related to the values of R, L, C and they are all real and positive.
At the same time, the real part of the poles are all negative if the circuit contains only passive components.

17. Sinusoidal frequency analysis
Circuit represented by
network function

18. Transfer function
H(s) +
Vin - +
Vout -

19. Drawing transfer function is difficult!
Bode plot
Drawing transfer function is difficult!
Magnitude :
Phase :

20. History of the Decibel
Originated as a measure of relative (radio) power
By extension
Using log scales the frequency characteristics of network functions
have simple asymptotic behavior.
The asymptotes can be used as reasonable and efficient approximations

21. Standard form of a transfer function
On applying to log function, terms containing poles and zeroes can be separated. Then, it becomes easier to draw the transfer function.

22. How to draw Bode plot of simple zero or poles
Bode plots can be drawn simply following the procedures below.
Draw first two extreme regions where and
Connect those two regions with a smooth curve.
Mark the exact point where
Magnitude
Phase

23. Bode plot of quadratic poles
Drawing a transfer function containing quadratic polynomial deserves a special care. Depending on the value of ζ, the shape of the transfer function curves changes significantly.
p1 and p2 are real numbers and have different values.
p1 and p2 are real numbers and have the same value.
p1 and p2 are complex number and have different values.

24. When the poles are real valued
(1) p1, p2 same
(2) p1, p2 distinct

25. When the poles are complex valued
With smaller

26. Standard form of quadratic functions
Instead of ζ, Q is more widely used and is called ‘Quality factor’. The Q factor is related with the power loss of a passive component.
p1 and p2 are real numbers and have different values.
p1 and p2 are real numbers and have the same value.
p1 and p2 are complex number and have different values.

27. General form of a network function
Poles/zeros at the origin
Frequency independent
Quadratic terms for
complex conjugate poles/zeros
First order terms
Display each basic term separately and add the results to obtain final answer
Let’s examine each basic term

28. a. Constant Term
b. Poles/Zeros at the origin

29. c. Simple pole or zero Behavior in the neighborhood of the corner
Asymptote for phase
High freq. asymptote
Low freq. Asym.

30. Simple zero
Simple pole

31. d. Quadratic pole or zero
Corner/break frequency
Resonance frequency
These graphs are inverted for a zero
Magnitude for quadratic pole
Phase for quadratic pole

32. Example Generate magnitude and phase plots Draw asymptotes
for each term
Draw composites

34. Example Generate magnitude and phase plots Draw asymptotes for each
Form composites

35. Example A function with complex conjugate poles Put in standard form
Draw composite asymptote
Behavior close to corner of conjugate pole/zero
is too dependent on damping ratio.
Computer evaluation is better

36. Using MATLAB to compute Magnitude & Phase information
MATLAB commands required to display magnitude
and phase as function of frequency
NOTE: Instead of comma (,) one can use space to
separate numbers in the array
EXAMPLE
Missing coefficients must
be entered as zeros
» num=[15*2.53*1e-3,0];
» den=[0.1*2.53*1e-3,15*2.53*1e-3,1];
» freqs(num,den)
» num=[15*2.53*1e-3 0];
» den=[0.1*2.53*1e-3 15*2.53*1e-3 1];
» freqs(num,den)
This sequence will also
work. Must be careful not
to insert blanks elsewhere

38. clear;
clf;
w = logspace(1,7,200);
num = [1.e0 * 40e3*pi, 0];
den = [1, 40.1e3*pi, 4e6*pi*pi];
freqs(num, den,w);

39. Evaluation of frequency response using MATLAB
Using default options
» num=[25,0]; %define numerator polynomial
» den=conv([1,0.5],[1,4,100]) %use CONV for polynomial multiplication
den =
» freqs(num,den)

40. Determining Transfer function from Bode plot
This is the inverse problem of determining frequency characteristics. We will use only the composite asymptotes plot of the magnitude to postulate a transfer function. The slopes will provide information on the order
A. different from 0dB.
There is a constant Ko A B C D E
B. Simple pole at 0.1
C. Simple zero at 0.5
D. Simple pole at 3
E. Simple pole at 20
If the slope is -40dB we assume double real pole. Unless we are given more data

41. Example Determine a transfer function from the composite
magnitude asymptotes plot
A. Pole at the origin.
Crosses 0dB line at 5 C E A
B. Zero at 5 D B
C. Pole at 20
D. Zero at 50
E. Pole at 100

42. 12.3 Resonant circuits H(s) + Vin - + Vout -
A circuit that selects one frequency is called a resonant circuit.

43. Series resonance
H(s) +
Vout -

44. Transfer function of a resonator
clear;
clf;
w0=10^2;
Q=10;
H0=1;
w = logspace(1,4,200);
num = [1/(Q*w0), 0];
den = [1/(w0*w0), 1/(Q*w0),1];
freqs(num, den,w);
Spectrum analyzer

45. Half Power Band-width
Power 가 1/2이 되는 지점.
Half power band-width

46. HPBW vs. Q 3dB-BW With large Q, BW gets narrow. Q : Quality factor
On a log scale, a point where |H| is lowered by 3dB from the peak value is called a half power point.
3dB-BW
With large Q, BW gets narrow.
Q : Quality factor

47. Parallel resonance
H(s)

48. Resonant circuits
These are circuits with very special frequency characteristics. And resonance is a very important physical phenomenon
The frequency at which the circuit becomes purely resistive is called
the resonance frequency

49. Properties of resonant circuits
At resonance the impedance/admittance is minimal
Current through the serial circuit/
voltage across the parallel circuit can
become very large (if resistance is small)
Given the similarities between series and parallel resonant circuits,
we will focus on serial circuits

50. Properties of resonant circuits
At resonance the power factor is unity
Phasor diagram for parallel circuit
Phasor diagram for series circuit

51. Example 12.7 Determine the resonant frequency, the voltage across each
element at resonance and the value of the quality factor

52. Example 12.8
Given L = 0.02H with a Q factor of 200, determine the capacitor
necessary to form a circuit resonant at 1000Hz
What is the rating for the capacitor if the
circuit is tested with a 10V supply?
The reactive power on the capacitor
exceeds 12kVA

53. Energy transfer in resonant circuits
Normalization
factor

54. Quality factor in terms of Energy
Energy dissipated in one cycle
Energy stored in L

55. Example 12.10 A series RLC circuit as the following properties:
Determine the values of L,C.
1. Given resonant frequency and bandwidth determine Q.
2. Given R, resonant frequency and Q determine L, C.

56. Example 12.12 The Tacoma Narrows Bridge Opened: July 1, 1940
Collapsed: Nov 7, 1940
Likely cause: wind
varying at frequency
similar to bridge
natural frequency

57. Displacement vs. voltage
0.44’
1.07’

58. Example 12.15 Increasing selectivity by cascading low Q circuits
Single stage tuned amplifier

59. 12.5 Filter networks High-pass filter Low-pass filter
Networks designed to have frequency selective behavior
COMMON FILTERS
High-pass filter
Low-pass filter
We focus first on
PASSIVE filters
Band-reject filter
Band-pass filter

60. Examples : voice signal spectrum
Time domain signal
Frequency domain spectrum

61. Example : ADSL signal spectrum

62. Filter
H(s) +
Vin - +
Vout -
1-st order
Low-pass filter

63. Prototype low pass filter

64. LPF-to-HPF transformation
LPF→HPF

65. LPF→HPF

66. LPF-to-BPF transformation
LPF→BPF

68. LPF-to-BRF transformation
LPF→BPF

70. Example 12.19
A simple notch filter to eliminate 60Hz interference

71. Active filter Passive filters have several limitations
1. Cannot generate gains greater than one
2. Loading effect makes them difficult to interconnect
3. Use of inductance makes them difficult to handle
Using operational amplifiers one can design all basic filters, and more,
with only resistors and capacitors
The linear models developed for operational amplifiers circuits are valid, in a
more general framework, if one replaces the resistors by impedances
These currents are zero
Ideal Op-Amp

72. Basic Inverting Amplifier
Linear circuit equivalent

73. Basic Non-inverting amplifier
Due to the internal op-amp circuitry, it has
limitations, e.g., for high frequency and/or
low voltage situations. The Operational
Transductance Amplifier (OTA) performs
well in those situations

74. Example 12.40 “BASS-BOOST” AMPLIFIER DESIRED BODE PLOT
(non-inverting op-amp)
OPEN SWITCH

75. Example 12.41 TREBLE BOOST Desired boost Original player response
Proposed boost circuit
Non-inverting amplifier