Transfer Function of a System with Sinusoidal Input 2.5 2000/07/10 2004/08/10 13:32:37.397 GMT-5 Don Johnson dhj@rice.edu Liqun Wang liqun@rice.edu Don Johnson dhj@rice.edu transfer function sinusoidal input Discussing the special case of Transfer Function, that is when the input voltage is a sinusoid. One very important special case of transfer functions occurs when the input voltage is a sinusoid. Because a sinusoid is the sum of two complex exponentials, each having a frequency equal to the negative of the other, we can directly predict the output voltage by examining the transfer function. If the source is a sine wave, we know that v in t A 2 f t A 2 2 f t 2 f t Since the input is the sum of two complex exponentials, we know that the output is also a sum of two similar complex exponentials, the only difference being that the complex amplitude of each is multiplied by the transfer function evaluated at each exponential's frequency. v out t A 2 H f 2 f t A 2 H f 2 f t As noted earlier, the transfer function is most conveniently expressed in polar form: H f H f H f . Furthermore, H f H f (even symmetry of the magnitude) and H f H f (odd symmetry of the phase). The output voltage expression simplifies to v out t A 2 H f 2 f t H f A 2 H f 2 f t H f A H f 2 f t H f The circuit's output to a sinusoidal input is also a sinusoid, having a gain equal to the magnitude of the circuit's transfer function evaluated at the source frequency and a phase equal to the phase of the transfer function at the source frequency. It will turn out that this input-output relation description applies to any linear circuit having a sinusoidal source. This input-output property is a special case of a more general result. Show that if the source can be written as the imaginary part of a complex exponential— v in t V 2 f t —the output is given by v out t V H f 2 f t . Show that a similar result also holds for the real part. The key notion is writing the imaginary part as the difference between a complex exponential and its complex conjugate: V 2 f t V 2 f t V 2 f t 2 The response to V 2 f t is V H f 2 f t , which means the response to V 2 f t is V H f 2 f t . as H f H f , the Superposition Principle says that the output to the imaginary part is V H f 2 f t . The same argument holds for the real part: V 2 f t V H f 2 f t .