Frequency Domain Analysis

We analyze these using transfer function models where our docus will be on LTI systems

Frequency

In systems, this is a man made concept whereas the time domain is a physical concept.

The output of a linear Dynamic Systems is the transient response + the steady state response $x\left(t\right)=H\left(s\right)e^{st}$

We can represent a time signal as a sum of sinusoids and for any single sinusoid, we can represent that as a combination of sinusoids

If $s=j\omega$ (on the imaginary axis), then $H\left(j\omega \right)$ is the system’s frequency response.

Where $e^{st}$ can be represented using Euler’s Identity

Frequency response is the steady-state response to a sinusoid (after the transient response disappears)

$x\left(t\right)=\left|H(j\omega t\right|e^{j\omega t+<H\left(j\omega \right)}$
Where the first term is the amplification by the system and the second term is the phase shift by the system.

If we say that $f\left(t\right)=A\sin {\omega }_t$ then the output $x\left(t_{}\right)=A\left|H\left(j{\omega }_{}\right)\right|\sin \left(\omega t+\varphi \right)$ where $\varphi =<J\left(j\omega \right)$

We can apply frequency domain analysis to:

• 1st Order Dynamic Systems
• 2nd Order Dynamic Systems

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Frequency Domain Specifications

1. Resonance peak $M_r$: maximum value of $M_G:=\Vert G\left(j\omega \right)\Vert$ this is not desirable and we should suppress this.
2. Resonance frequency ${\omega }_r$ which is the frequency required to achieve $M_r$
3. Bandwidth, ${\omega }_{BW}$: range of frequency that a system can normally operate, this is where the $M_G$ drops to $\frac{1}{\sqrt{2}}$ of the DC gain
4. Cutoff rate: the rate at which the high frequency signals are attenuated and is the slope of $M_G$ (in dB) at high frequencies which depend on relative the relative degree of the system (n - m (denominator degree - numerator degree))

See 1st Order Frequency Domain Specs and 2nd Order Frequency Domain Specs.