# 1.6: Obtaining Transfer Function Models

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The transfer function description of a dynamic system is obtained from the application of the Laplace transform to the ODE model assuming zero initial conditions. The transfer function describes the input-output relationship in the form of a rational function of a complex frequency variable ‘\(s\)’. The development of transfer function in the case of first and second-order systems is described below.

**First-Order ODE Model.** We consider a first-order ODE model with input \(u(t)\) and output \(y(t)\), described as: \(\tau \frac{{\rm d}y(t)}{{\rm d}t} +y(t)=u(t)\). We use the Laplace transform to describe it as an algebraic equation: \((\tau s+1)y(s)=u(s)\). The resulting input–output transfer function is given as: \[\frac{y(s)}{u(s)} =\frac{1}{\tau s+1}\]

**Second-Order ODE Model.** We consider a mass–spring–damper model (Example 1.8), described by a second-order ODE, **\(m\ddot{x}\; +\; b\dot{x}\; +\; kx=f\).** The model has a Laplace transform description: \[ms^{2} x(s)+bsx(s)+kx(s)=f(s).\] The input–output relation (transfer function) for the mass-spring-damper system with force input and displacement output is given as: \[\frac{x(s)}{f(s)} =\frac{1}{ms^{2} +bs+k} .\] The system transfer function is a ratio of two polynomials in \(s\). The transfer function of a physical system is a proper fraction, i.e., the degree of the denominator polynomial is greater than the degree of numerator polynomial.

**Example 1.10:** A bandpass RLC network

We consider a bandpass RLC network (Figure 11). By identyfying the capacitor voltage \(v_C\) and inductor curret \(i_L\) as natural variables, the two first-order ODE’s that describe the network are given as: \[C\frac{dv_C}{dt}=\frac{V_s-v_C}{R}-i_L,\ \ L\frac{di_L}{dt}=v_C\] Application of Laplace transform then gives the corresponding algebraic equations as: \[\left(sC+\frac{1}{R}\right)v_C+i_L=\frac{V_s}{R},\ \ \ sLi_L-v_C=0\] A relationship between input \(V_s\) and output \(v_C\) is obtained by eliminating \(i_L\) from the equaitons. The resulting transfer function is given as: \[\frac{v_C\left(s\right)}{V_s\left(s\right)}=\frac{sL/R}{s^2LC+sL/R+1}\]

Figure 11: A bandpass RLC network