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Frequency Response

Frequency Response - 1 Frequency Response and Bode Plots...

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1 © Bob York 2009 1 Frequency Response and Bode Plots 1.1 Preliminaries The steady-state sinusoidal frequency-response of a circuit is described by the phasor transfer function ( ) H j . A Bode plot is a graph of the magnitude (in dB) or phase of the transfer function versus frequency. Of course we can easily program the transfer function into a computer to make such plots, and for very complicated transfer functions this may be our only recourse. But in many cases the key features of the plot can be quickly sketched by hand using some simple rules that identify the impact of the poles and zeroes in shaping the frequency response. The advantage of this approach is the insight it provides on how the circuit elements influence the frequency response. This is especially important in the design of frequency-selective circuits. We will first consider how to generate Bode plots for simple poles, and then discuss how to handle the general second-order response. Before doing this, however, it may be helpful to review some properties of transfer functions, the decibel scale, and properties of the log function. Poles, Zeroes, and Stability The s -domain transfer function is always a rational polynomial function of the form 1 2 1 2 1 0 1 2 1 2 1 0 ( ) ( ) ( ) m m m m m n n n n n s a s a s a s a N s H s K K D s s b s b s b s b (1.1) As we have seen already, the polynomials in the numerator and denominator are factored to find the poles and zeroes; these are the values of s that make the numerator or denominator zero. If we write the zeroes as 1 2 3 , , z z z etc., and similarly write the poles as 1 2 3 , , p p p , then ( ) H s can be written in factored form as 1 2 1 2 ( )( ) ( ) ( ) ( )( ) ( ) m n s z s z s z H s K s p s p s p (1.2)
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2 Frequency Response and Bode Plots © Bob York 2009 The pole and zero locations can be real or complex. When the roots are real they are called simple poles or simple zeros . When the roots are complex they always occur in pairs that are complex conjugates of each other. Another important observation is that stable networks must always have poles and zeroes in the left-half of the complex s-plane, such that the real parts of the poles/zeroes will be negative . As an example, lets assume a stable network with simple poles at 1 1 p   and 2 10 p   . The transfer function would then be 1 2 1 1 ( ) ( )( ) ( 1)( 10) H s s p s p s s (1.3) Thus for stable networks we always will find terms of the form ( ) s a in the denominator, where a is a positive number. Students sometimes get confused by the use of ( ) s p or ( ) s a to represent the same pole location; just remember that the poles are the values of s that make the denominator zero, i.e. s p or s a   in this example; clearly these will represent the same pole if p a   , and will represent a stable pole if Re{ } 0 a or Re{ } 0 p .
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