# 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

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