Fundamentals of Electric Circuits Chapter 14 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Overview 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. The concept of resonance as applied to LRC circuits . Frequency filters.

Frequency Response Frequency response is the variation in a circuit’s behavior with change in signal frequency. 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.

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).

Transfer Function There are four possible input/output combinations:

Example For the RC circuit, obtain the transfer function and its frequency response. Let

Example For the RL circuit, obtain the transfer function and its frequency response. Let

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. 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)

Example For the RL circuit, calculate the gain and its poles and zeros

Decibel Scale 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)

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 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.

Standard Form The transfer function may be written in terms of factors with real and imaginary parts. For example: 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+j22ω/ ωn+ (jω/ ωn)2] or zero 1/[1+j21ω/ ωn+ (jω/ ωk)2]

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°

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: This can be approximated as a flat line and sloped line that intersect at ω=z1. This is called the corner or break frequency

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

Bode Plots Quadratic pole/zero: The magnitude of the quadratic pole 1/[1+j22ω/ ωn+ (jω/ ωn)2] is -20log10 [1+j22ω/ ωn+ (jω/ ωn)2] This can be approximated as: 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

Bode Plots The phase can be expressed as: 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.

Bode Plots

Bode Plots

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 requires at least one capacitor and inductor.

Construct the Bode plots for the transfer function Example Construct the Bode plots for the transfer function

Draw the Bode plots for the transfer function Example Draw the Bode plots for the transfer function

Draw the Bode plots for the transfer function Example Draw the Bode plots for the transfer function

Draw the Bode plots for the transfer function Example Draw the Bode plots for the transfer function

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 is called the resonant frequency

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:

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. It is also a measure of the ratio of the resonant frequency to its bandwidth, B

Example In the circuit, Find . (2)Calculate Q and B (3) Determine the amplitude of the current at

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.

Parallel Resonance The relevant equations for the parallel resonant circuit are:

Determine the resonant frequency of the circuit. Example Determine the resonant frequency of the circuit.

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.

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

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: This is also referred to as the cutoff frequency. The filter is designed to pass from DC up to ωc

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.

Bandpass Filter The RLC series resonant circuit provides a bandpass filter when the output is taken off the resistor. The center frequency is: 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.

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.

Example Determine what type of filter is shown in the figure Example Determine what type of filter is shown in the figure. Calculate the corner or cutoff freqency.

Example Determine what type of filter is shown in the figure Example Determine what type of filter is shown in the figure. Calculate the corner or cutoff freqency.

Example If the bandpass filter in the figure is to reject a 200 Hz sinusoid while passing other freq. ,calculate the values of L, C and .

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.

First Order Lowpass The corner frequency will be:

First Order Highpass The corner frequency will be:

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.

Figure 14.45

The analysis of the bandpass filter

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.

Figure 14.47

The analysis of the bandreject filter

Example Design a lowpass active filter with a dc gain of 4 and a corner frequency of 500 Hz.

Example Design a highpass active filter with a high frequency gain of 5 and a corner frequency of 2kHz. Use a 0.1 capacitor in your design