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Fundamentals of Electric Circuits Chapter 14 Copyright © 2017 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. Overview • This chapter will introduce the idea of the transfer function: a means of describing the relationship between the input and output of a circuit. • Bode plots and their utility in describing the frequency response of a circuit will also be introduced. • The concept of resonance as applied to LRC circuits will be covered as well • Finally, frequency filters will be discussed. 2 Frequency Response • Frequency response is the variation in a circuit’s behavior with change in signal frequency. • This is significant for applications involving filters. • Filters play critical roles in blocking or passing specific frequencies or ranges of frequencies. • Without them, it would be impossible to have multiple channels of data in radio communications. 3 Transfer Function • One useful way to analyze the frequency response of a circuit is the concept of the transfer function H(ω). • It is the frequency dependent ratio of a forced function Y(ω) to the forcing function X(ω). H Y X 4 Transfer Function • There are four possible input/output combinations: H Voltage gain H Current gain Vo Vi I o I i H Transfer impedance H Transfer admittance Vo I i I o Vi 5 Zeros and Poles • To obtain H(ω), we first convert to frequency domain equivalent components in the circuit. • H(ω) can be expressed as the ratio of numerator N(ω) and denominator D(ω) polynomials. N H D • Zeros are where the transfer function goes to zero. • Poles are where it goes to infinity. • They can be related to the roots of N(ω) and D(ω) 6 Decibel Scale • We will soon discuss Bode plots. • These plots are based on logarithmic scales. • The transfer function can be seen as an expression of gain. • Gain expressed in log form is typically expressed in bels, or more commonly decibels (1/10 of a bel) GdB 10 log10 P2 P1 7 Bode Plots • One problem with the transfer function is that it needs to cover a large range in frequency. • Plotting the frequency response on a semilog plot (where the x axis is plotted in log form) makes the task easier. • These plots are referred to as Bode plots. • Bode plots either show magnitude (in decibels) or phase (in degrees) as a function of frequency. 8 Standard Form • The transfer function may be written in terms of factors with real and imaginary parts. For example: K j 1 j / z 1 j 2 / j / 1 H 2 1 1 k 1 j / p1 1 j 2 2 / n j / n k 2 • This standard form may include the following seven factors in various combinations: – A gain K – A pole (jω)-1 or a zero (jω) – A simple pole 1/(1+jω/p1) or a simple zero (1+jω/z1) – A quadratic pole 1/[1+j22ω/ ωn+ (jω/ ωn)2] or zero 1/[1+j21ω/ ωn+ (jω/ ωk)2] 9 Bode Plots • In a bode plot, each of these factors is plotted separately and then added graphically. • Gain, K: the magnitude is 20log10K and the phase is 0°. Both are constant with frequency. • Pole/zero at the origin: For the zero (jω), the slope in magnitude is 20 dB/decade and the phase is 90°. For the pole (jω)-1 the slope in magnitude is -20 dB/decade and the phase is -90° 10 Bode Plots • Simple pole/zero: For the simple zero, the magnitude is 20log10|1+jω/z1| and the phase is tan-1 ω/z1. • Where: H dB 20 log10 1 j z1 20 log10 as z1 • This can be approximated as a flat line and sloped line that intersect at ω=z1. • This is called the corner or break frequency 11 Bode Plots • The phase can be plotted as a series straight lines • From ω=0 to ω≤z1/10, we let =0 • At ω=z1 we let =45° • For ω≥10z1, we let = 90° • The pole is similar, except the corner frequency is at ω=p1, the magnitude has a negative slope 12 Bode Plots • Quadratic pole/zero: The magnitude of the quadratic pole 1/[1+j22ω/ ωn+ (jω/ ωn)2] is 20log10 [1+j22ω/ ωn+ (jω/ ωn)2] • This can be approximated as: H dB 0 as 0 40 log10 as n • Thus the magnitude plot will be two lines, one with slope zero for ω<ωn and the other with slope -40dB/decade, with ωn as the corner frequency 13 Bode Plots • The phase can be expressed as: 0 0 2 2 / n tan 1 90 n 2 2 1 / n 180 • This will be a straight line with slope of 90°/decade starting at ωn/10 and ending at 10 ωn. • For the quadratic zero, the plots are inverted. 14 Bode Plots 15 Bode Plots 16 Resonance • The most prominent feature of the frequency response of a circuit may be the sharp peak in the amplitude characteristics. • Resonance occurs in any system that has a complex conjugate pair of poles. • It enables energy storage in the firm of oscillations • It allows frequency discrimination. • It requires at least one capacitor and inductor. 17 Series Resonance • A series resonant circuit consists of an inductor and capacitor in series. • Consider the circuit shown. • Resonance occurs when the imaginary part of Z is zero. • The value of ω that satisfies this is called the resonant frequency 0 1 rad/s LC 18 Series Resonance • At resonance: – – – – The impedance is purely resistive The voltage Vs and the current I are in phase The magnitude of the transfer function is minimum. The inductor and capacitor voltages can be much more than the source. • There are two frequencies above and below resonance where the dissipated power is half the max: 2 R 1 R 1 2L 2 L LC 2 R 1 R 2 2L 2 L LC 19 Quality Factor • The “sharpness” of the resonance is measured quantitatively by the quality factor, Q. • It is a measure of the peak energy stored divided by the energy dissipated in one period at resonance. Q 0 L R 1 0CR • It is also a measure of the ratio of the resonant frequency to its bandwidth, B B R 0 L Q 20 Parallel Resonance • The parallel RLC circuit shown here is the dual of the series circuit shown previously. • Resonance here occurs when the imaginary part of the admittance is zero. • This results in the same resonant frequency as in the series circuit. 21 Series Resonance • The relevant equations for the parallel resonant circuit are: 2 1 1 1 1 2 RC 2 RC LC 1 B RC 2 1 1 1 2 2 RC 2 RC LC R Q 0 RC 0 L 22 Passive Filters • A filter is a circuit that is designed to pass signals with desired frequencies and reject or attenuate others. • A filter is passive if it consists only of passive elements, R, L, and C. • They are very important circuits in that many technological advances would not have been possible without the development of filters. 23 Passive Filters • There are four types of filters: – Lowpass passes only low frequencies and blocks high frequencies. – Highpass does the opposite of lowpass – Bandpass only allows a range of frequencies to pass through. – Bandstop does the opposite of bandpass 24 Lowpass Filter • A typical lowpass filter is formed when the output of a RC circuit is taken off the capacitor. • The half power frequency is: c 1 RC • This is also referred to as the cutoff frequency. • The filter is designed to pass from DC up to ωc 25 Highpass Filter • A highpass filter is also made of a RC circuit, with the output taken off the resistor. • The cutoff frequency will be the same as the lowpass filter. • The difference being that the frequencies passed go from ωc to infinity. 26 Bandpass Filter • The RLC series resonant circuit provides a bandpass filter when the output is taken off the resistor. • The center frequency is: 0 1 LC • The filter will pass frequencies from ω1 to ω2. • It can also be made by feeding the output from a lowpass to a highpass filter. 27 Bandstop Filter • A bandstop filter can be created from a RLC circuit by taking the output from the LC series combination. • The range of blocked frequencies will be the same as the range of passed frequencies for the bandpass filter. 28 Active Filters • Passive filters have a few drawbacks. – They cannot create gain greater than 1. – They do not work well for frequencies below the audio range. – They require inductors, which tend to be bulky and more expensive than other components. • It is possible, using op-amps, to create all the common filters. • Their ability to isolate input and output also makes them very desirable. 29 First Order Lowpass • If the input and feedback elements in an inverting amplifier are selectively replaced with capacitors, the amplifier can act as a filter. • If the feedback resistor is replaced with a parallel RL element, the amplifier becomes a lowpass filter. • The corner frequency will be: c 1 Rf C f 30 First Order Highpass • Placing a series RL combination in place of the input resistor yields a highpass filter. • The corner frequency of the filter will be: c 1 Ri Ci 31 Bandpass • To avoid the use of an inductor, it is possible to use a cascaded series of lowpass active filter into a highpass active filter. • To prevent unwanted signals passing, their gains are set to unity, with a final stage for amplification. 32 Bandreject • Creating a bandstop filter requires using a lowpass and highpass filter in parallel. • Both output are fed into a summing amplifier. • It will function by amplifying the desired signals compared to the signal to be rejected. 33

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