Tr.Fu.II_V4

Tr.Fu.II_V4 - L 302-4.V4 Drexel University Electrical and...

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L302-4.V4 4-1 Drexel University Electrical and Computer Engr. Dept. Electrical Engineering Laboratory II, ECEL 302 E. L. Gerber TRANSFER FUNCTION ANALYSIS II Object The object of this experiment is to learn how to identify transfer functions with second-order factors. After completing this experiment, you should be able to identify the three types of systems: overdamped, critically damped, and under- damped. See Chapter - 7 in Irwin. You should be able to determine the exact solution to these functions using the software available in the Lab. Introduction Passive circuits consisting of inductors and capacitors will result in transfer functions with higher-order factors. Active circuits, with op-amps or transistors, will result in transfer functions with second-order factors with only two energy elements (L and/or C). An active filter is defined as an op-amp with resistors and capacitors only. These circuits can result in transfer functions with second-order factors or higher. Theory The transfer function of a circuit under study can be expressed in either the `polyno- mial’ or `factored’ form. In the factored form the poles and zeros are explicitly expressed. The transfer function in the expression below has first-order factors only. H(s) = K(s + z 1 )(s + z 2 ) ...(s + z m ) (s + p 1 )(s + p 2 ) ... (s + p n ) Factored Form Below we have two transfer functions with repeated factors. In both cases the break frequency is at 10 r/s; however, the asymptotic slope is –40 dB/dec for the second- order term and -(20r)dB/dec for the r- th order term, where r is an integer.
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L302-4.V4 4-2 H ( s ) = K ( s + 10) 2 H ( s ) = K ( s + 10) r 1) Second-Order Quadratic Factors The function below is a common second-order low pass filter transfer function with a quadratic factor in the denominator. We will investigate its characteristics. H(s) = K s 2 + as + b = K s 2 + 2 ω n ζ s + ω n 2 The parameters 'a' and 'b' are dependent on the circuit elements. The system parame- ters, ω n and ζ , are often used, and they are the break frequency and damping ratio respectively. The system parameters are dependent on the circuit elements as well, b = ω n 2 and a = 2 ω n ζ . The roots of this factor (poles) are found from the quadratic formula for each case: s 1 , s 2 = a 2 ± a 2 2 b = − ω n ζ ± ω n ζ ( ) 2 − ω n 2 a) Root Types: A second-order polynomial will have two roots that depend on the element values. There are three types of roots that can occur as can be seen from the above equation for s 1 and s 2 . 1) When ζ > 1, s 1 and s 2 are real and negative, distinct roots.
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Tr.Fu.II_V4 - L 302-4.V4 Drexel University Electrical and...

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