12. Variable frequency network performance

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

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

Inductor

Capacitor

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

Pulse represented by sum of sinusoidal signal

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

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

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

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

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);

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

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);

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)

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.

Sinusoidal frequency analysis Circuit represented by network function

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

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

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

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.

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

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.

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

When the poles are complex valued With smaller

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.

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

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

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

Simple zero Simple pole

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

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

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

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

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

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);

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 = 1.0000 4.5000 102.0000 50.0000 » freqs(num,den)

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

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

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

Series resonance H(s) + Vout -

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

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

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

Parallel resonance H(s)

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

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

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

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

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

Energy transfer in resonant circuits Normalization factor

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

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.

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

Displacement vs. voltage 0.44’ 1.07’

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

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

Examples : voice signal spectrum Time domain signal Frequency domain spectrum

Example : ADSL signal spectrum

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

Prototype low pass filter

LPF-to-HPF transformation LPF→HPF

LPF→HPF

LPF-to-BPF transformation LPF→BPF

LPF-to-BRF transformation LPF→BPF

Example 12.19 A simple notch filter to eliminate 60Hz interference

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

Basic Inverting Amplifier Linear circuit equivalent

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

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

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