Bandwidth of a First-Order Model Consider the model [latex]τ \dot{v} + v = f (t)[/latex], for which [latex]M = \frac{V}{F} = \frac{1}{\sqrt{1 + ω² τ²}} [/latex] Obtain an expression for the bandwidth in terms of [latex]\tau[/latex] and interpret its significance

The peak in M occurs at ω = 0 and is [latex]M_{\text{peak}}[/latex] = 1. Thus [latex]ω_{1} = 0[/latex] and [latex]ω_{2}[/latex] is found from (8.4.1) as follows: [latex]M(ω_{1}) ≤ \frac{M_{peak}}{\sqrt{2}} ≥ M(ω_{2}) [/latex] (8.4.1) [latex]\frac{M_{peak}}{\sqrt{2}} = \frac{1}{\sqrt{2}} = M(ω_{2}) = \frac{1}{\sqrt{1 + ω^{2}_{2} \tau^{2}}} [/latex] This gives [latex]ω_{2} = 1/\tau[/latex] . Thus the bandwidth of the model [latex]\tau \dot{v} + v = f (t) [/latex] is the range of frequencies [latex](0, 1/τ).[/latex] Because a small time constant indicates a fast system, the bandwidth is another measure of the speed of response. So the faster the system, the larger the bandwidth, and some engineers describe a system’s response speed in terms of its bandwidth rather than in terms of its time constant. However, for other models, the relation between the time constant and the bandwidth is not always so simple, as we will see.

Question 8.4.1: Bandwidth of a First-Order Model Consider the model τv + v =...

System Dynamics

Bandwidth of a First-Order Model

Consider the model $τ \dot{v} + v = f (t)$ , for which

M = \frac{V}{F} = \frac{1}{\sqrt{1  +  ω² τ²}}

Obtain an expression for the bandwidth in terms of $\tau$ and interpret its significance

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The peak in M occurs at ω = 0 and is $M_{\text{peak}}$ = 1. Thus $ω_{1} = 0$ and $ω_{2}$ is found from (8.4.1) as follows:

$M(ω_{1}) ≤ \frac{M_{peak}}{\sqrt{2}} ≥ M(ω_{2})$ (8.4.1)
$\frac{M_{peak}}{\sqrt{2}} = \frac{1}{\sqrt{2}} = M(ω_{2}) = \frac{1}{\sqrt{1 + ω^{2}_{2} \tau^{2}}}$
This gives $ω_{2} = 1/\tau$ . Thus the bandwidth of the model $\tau \dot{v} + v = f (t)$ is the range of frequencies $(0, 1/τ).$
Because a small time constant indicates a fast system, the bandwidth is another measure of the speed of response. So the faster the system, the larger the bandwidth, and some engineers describe a system’s response speed in terms of its bandwidth rather than in terms of its time constant. However, for other models, the relation between the time constant and the bandwidth is not always so simple, as we will see.