Second order systems

Consider the following circuits composed of R, C, L connected in series:

By Kirchhoff's Voltage law, we have

$$v_{in}(t)=V_R + v_L + v_C = L \; \frac{di(t)}{dt} + R \; i(t) + \frac{1}{C} \int_0^t i(\tau) d\tau$$

where $v_{in}(t)$ is the input voltage across all three elements. Taking a time derivative of the equation, we get:

$$L \; \frac{d^2i(t)}{dt^2} + R \; \frac{di(t)}{dt} + \frac{1}{C} i(t) =\frac{dv_{in}(t)}{dt}$$

If the current is treated as the output $y(t)=i(t)$, this LTI system's response to a complex exponential input $v_{in}(t)=e^{st}$ is $i(t)=H(s)e^{st}$, and the equation becomes:

$$L \frac{d^2}{dt^2} e^{st}H(s)+R \frac{d}{dt} e^{st}H(s) +\frac{1}{C} e^{st}H(s)=\frac{d}{dt} e^{st}$$

i.e.,

$$H(s)\;(s L e^{st}+R e^{st} +\frac{1}{sC} e^{st})=e^{st}$$

The transfer function can be found to be

$$H(s)=\frac{I(s)}{V_{in}(s)}=\frac{1}{sL+R+(sC)^{-1}}$$

Alternatively, the impedances of R, C and L can be expressed in Laplace domain as:

$$Z_R=R,\;\;\;Z_C=1/sC,\;\;\;\;Z_L=sL$$

and the total impedance of the RCL series circuit is

$$Z_{total}(s)=Z_R(s)+Z_C(s)+Z_L(s)=R+\frac{1}{sC}+sL$$

we have

$$V_{in}(s)=Z_{total}(s) I(s)=[Z_L(s)+Z_R(s)+Z_C(s)]I(s) =(sL+R+\frac{1}{sC})I(s)$$

and we get the same transfer function:

$$H(s)=\frac{I(s)}{V_{in}(s)}=\frac{1}{sL+R+(sC)^{-1}}$$

If we consider the voltage across one of the three element (R,C,L) as the output, the circuit can be considered as a voltage divider with transfer function

$$H(s)=\frac{Z_{out}(s)}{Z_{total}(s)}$$

where $Z_{out}(s)$ depends on how the output is defined. If the output is the voltage across L,

$$H_L(s)=\frac{V_L(s)}{V_{in}(s)}=\frac{sL}{sL+R+(sC)^{-1}} =\frac{s^2}{s^2+2\zeta \omega_n s + \omega_n^2}$$

If the output is the voltage across R,

$$H_R(s)=\frac{V_R(s)}{V_{in}(s)}=\frac{R}{sL+R+(sC)^{-1}} =\frac{2\zeta \omega_n s}{s^2+2\zeta \omega_n s + \omega_n^2}$$

If the output is the voltage across C,

$$H_C(s)=\frac{V_C(s)}{V_{in}(s)}=\frac{(sC)^{-1}}{sL+R+(sC)^{-1}} =\frac{\omega_n^2}{s^2+2\zeta \omega_n s + \omega_n^2}$$

where

$$\left\{ \begin{array}{ll} \omega_n \stackrel{\triangle}{=} ... ...{L}} & \mbox{is the damping coefficient} \end{array} \right.$$

In general, the transfer function of a second order system can be written as

$$H(s)=\frac{N(s)}{D(s)}=\frac{N(s)}{s^2+2\zeta \omega_n s + \omega_n^2} =\frac{C_1}{s-s_1}+\frac{C_2}{s-s_2}$$

where $s_1$ and $s_2$ are the two roots of the denominator $D(s)$:

$$s_{1,2}=(-\zeta \pm j \sqrt{1-\zeta^2}) \omega_n = -\zeta \omega_n \pm j\omega_d$$

with

$$\omega_d \stackrel{\triangle}{=}\omega_n \sqrt{1-\zeta^2} < \omega_n$$

is called the damped natural frequency. The coefficients $C_1$ and $C_2$ can be found by partial fraction extension.