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4.6: Transfer Function - General Definition

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  • For any LTI, single-input-single-output (SISO) physical system, we denote the input as \(u(t)\) and the output as \(x(t)\). For an \(n\)th order system, in general, the input and output are related by an ODE of the form

    \[a_{1} \frac{d^{n} x}{d t^{n}}+a_{2} \frac{d^{n-1} x}{d t^{n-1}}+\ldots+a_{n+1} x=b_{1} \frac{d^{m} u}{d t^{m}}+b_{2} \frac{d^{m-1} u}{d t^{m-1}}+\ldots+b_{m+1} u\label{eqn:4.21}\]

    \(a_{1}, \ldots, a_{n+1}\) and \(b_{1}, \ldots, b_{m+1}\) are constants (with the numbering system keyed to MATLAB notation), and \(m\) \(\leq\) \(n\). Also, we assume that the system is stable, which is defined more precisely in Section 4.7. Taking the Laplace transform of the ODE, with all ICs equal to zero, gives

    \[\left(a_{1} s^{n}+a_{2} s^{n-1}+\ldots+a_{n+1}\right) L[x(t)]_{I C_s=0}=\left(b_{1} s^{m}+b_{2} s^{m-1}+\ldots+b_{m+1}\right) L[u(t)]\label{eqn:4.22}\]

    Then, from Equation \(\ref{eqn:4.22}\), the system transfer function, defined to be the ratio of the output transform to the input transform, with zero ICs, is the ratio of two polynomials,

    \[T F(s) \equiv \frac{L[x(t)]_{I C_s=0}}{L[u(t)]}=\frac{b_{1} s^{m}+b_{2} s^{m-1}+\ldots+b_{m+1}}{a_{1} s^{n}+a_{2} s^{n-1}+\ldots+a_{n+1}}\label{eqn:4.23}\]

    It is appropriate to state here (without proof) that the transfer function of any physically realizable system has \(m\) \(\leq\) \(n\), i.e., the degree of the numerator polynomial is less than or equal to the degree of the denominator polynomial. The condition \(m\) \(\leq\) \(n\) makes the transfer function causal, which means that the current (in time) output of the system is dependent only upon past and present (not future) values of the input. In general, the future values of an input cannot be predicted, so it is logical that a real system and its transfer function must be causal. See Bélanger, 1995, page 440.

    Note also from Equation \(\ref{eqn:4.23}\) that, if given \(TF(s)\) and input \(u(t)\), we can express the transform of the output with zero initial conditions as

    \[\left.L[x(t)]\right|_{I C s=0}=T F(s) \times L[u(t)]\label{eqn:4.24}\]