Second Order Frequency Response

How second order systems respond to sinusoidal/oscillating input. Similar to first order.

Gain and Phase for Second Order Systems

For a 2nd order system in standard input-output form:

$$\frac{1}{{\omega }_n^2}\frac{d^2}{d{t}^2}y(t)+\frac{2\zeta }{{\omega }_n}\frac{d}{dt}y(t)+y(t)=u(t),y(0)=0,\frac{d}{dt}y(0)=0$$ $$G(s)=\frac{{\omega }_n^2}{s^2+2\zeta {\omega }_{n}s+{\omega }_n^2}$$ $$G(j\omega )=\frac{{\omega }_n^2}{({\omega }_n^2-{\omega }^2)+2j\zeta {\omega }_n\omega }$$

The gain and phase of the frequency response are therefore:

$$|G(j\omega )|=\frac{{\omega }_n^2}{[({\omega }_n^2-{\omega }^2{)}^2+4{\zeta }^2{\omega }_n^2{\omega }^2{]}^{1/2}}$$

$$\varphi (\omega )=\angle G(j\omega )=-{\tan }^{-1}\frac{2\zeta {\omega }_n\omega }{{\omega }_n^2-{\omega }^2}$$

Bode Plots, from Data Book

The plots show gain and phase shift for varying values of $\zeta$

Example

For the electrical system shown below with the values $R=1k\Omega$, $C=0.1\mu F$, $L=0.1H$ find:

• The undamped natural frequency ${\omega }_n$
• The damping factor $\zeta$
• Sketch the magnitude of the frequency response $|G(j\omega )|$
  • At what frequency is this at it's maximum?
• Sketch a bode plot using matlab

The system equation is: $$LC\frac{d^2}{d{t}^2}{e}_o(t)+RC\frac{d}{dt}{e}_o(t)+e_o(t)=e_i(t)$$

Undamped natural frequency: $$\frac{1}{{\omega }_n^2}=LC=10^{-8}⟹{\omega }_n=10^{4}rad{s}^{-1}$$

Damping factor: $$\frac{2\zeta }{{\omega }_n}=RC=10^4⟹\zeta =0.5$$

Using the graph from the data book

The graph peaks at approx $|G(j\omega )|=1.15$, so: $$\omega \approxeq 0.71{\omega }_n=0.71\times 10^{4}rad{s}^{-1}$$ $$f=\frac{0.71\times 10^4}{2\pi }=1129Hz$$

Matlab plot:

R = 1000
C = 10e-7
L = 0.1
sys = tf([1],[L*C R*C 1]); figure; step(sys);
bode(sys);