TR.I_V5 - L 302-3.V5 Drexel University Electrical and...

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L302-3.V5 3-1 Drexel University Electrical and Computer Engr. Dept. Electrical Engineering Laboratory II, ECEL 302 E. L. Gerber TRANSFER FUNCTION ANALYSIS I Objective The object of this experiment is to learn how to determine and characterize the transfer function of a circuit. And then identify the various types of transfer functions and graph their gain and phase response by inspection and with a computer. You will investigate pole-zero diagrams of transfer functions. The Bode Approximation method will be employed to quickly obtain transfer function response curves. Introduction Two and four terminal networks (amplifiers and filters) are often characterized by determining the ratio of two network signals. For example, the ratio of the output voltage to the input voltage is defined as the voltage gain transfer function. The current gain would be the ratio of the output current to the input current. Usually the network contains frequency dependent elements (L and C); hence, the transfer function is frequency dependent. In general, the transfer function is expressed in terms of the complex frequency variable s (= σ + j ω ). Where s is also the operator, s = d /dt.
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L302-3.V5 3-2 Theory 1) The Transfer Function : When we solve a linear circuit for its transfer function we can write it as T(s) or H(s). After simplifying the function we can express it in the polynomial form , T ( s ) = H ( s ) = N ( s ) D ( s ) = a m s m + a m 1 s m 1 + ••• + a o ( ) b n s n + b n 1 s n 1 + + b o ( ) where N(s) is the numerator polynomial and D(s) is the denominator of the transfer function. When expressed in this form it may be possible to solve for the roots of the two polynomials. The roots of the numerator are called the zeros, ‘z’, and the roots of the denominator are called the poles, ‘p’. If the polynomial N(s) is a m- th order polynomial then there are ‘m’ roots and ‘m’ zeros in the function. If the polynomial D(s) is an n- th order polynomial then there are ‘n’ roots and ‘n’ poles in the function. In the factored form , the poles and zeros are expressed explicitly. The transfer function can be expressed as follows, H(s) = K(s + z 1 )(s + z 2 ) ...(s + z m ) (s + p 1 )(s + p 2 ) ... (s + p n ) A zero is that value of s that makes the transfer function equal to zero; a pole is that value of s that makes the transfer function equal to infinity. K is a constant of the system. The poles and zeros of a transfer function are important characteristics of the function and it will be necessary to solve for their values. They are constants of the system and may be complex values. They have the same dimensions as frequency, rad/sec. We often plot their location on the complex frequency plane, the s-plane . We call the plot the ‘pole- zero diagram’ of the system. 2) Logarithmic Gain: We usually graph the magnitude of the transfer function and its phase against the frequency, ω . It is common to use logarithmic plots instead of linear plots because of the large range of values. The logarithmic gain is defined as the logarithm of the magnitude
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TR.I_V5 - L 302-3.V5 Drexel University Electrical and...

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