Using the Transfer Function with Several Input Components
Suppose that the input signal for the filter of Figure 6.4 is given by

v_in(t) = 3 + 2 cos(2000πt) + cos(4000πt – 70°)

Find an expression for the output signal.

Question

Using the Transfer Function with Several Input Components
Suppose that the input signal for the filter of Figure 6.4 is given by

v_in(t) = 3 + 2 cos(2000πt) + cos(4000πt – 70°)

Find an expression for the output signal.

Accepted Answer

We start by breaking the input signal into its components. The first component is

[latex]v_{\mathrm{in1}}(t)=3[/latex]

and the second component is

[latex]v_{\mathrm{in2}}(t)=2\cos{(2000\pi t)}[/latex]

and the third component is

[latex]v_{\mathrm{in3}}(t)= \cos{(4000\pi t-70^\circ)}[/latex]

By inspection, we see that the frequencies of the components are 0, 1000, and 2000Hz, respectively. Referring to the transfer function shown in Figure 6.4, we find that

[latex]H(0) = 4[/latex]

[latex]H(1000)=3\angle 30^\circ[/latex]

and

[latex] H(2000)=2\angle 60^\circ [/latex]

The constant (dc) output term is simply H(0) times the dc input:

[latex]v_{\mathrm{out1}}=H(0)v_{\mathrm{in1}}=4 imes 3=12[/latex]

The phasor outputs for the two input sinusoids are

[latex]\mathrm{V_{out2}}=H(1000) imes \mathrm{V_{in2}}=3\angle 30^\circ imes 2\angle 0^\circ =6\angle 30^\circ[/latex]

[latex]\mathrm{V_{out3}}=H(2000) imes \mathrm{V_{in3}}=2\angle 60^\circ imes 1\angle -70^\circ =2\angle-10^\circ [/latex]

Next, we can write the output components as functions of time:

[latex]v_{\mathrm{out1}}(t)=12[/latex]

[latex] v_{\mathrm{out2}}(t)=6\cos{(2000\pi t +30^\circ)}[/latex]

and

[latex] v_{\mathrm{out3}}(t)=2\cos{(4000\pi t -10^\circ)} [/latex]

Finally, we add the output components to find the output voltage:

[latex] v_{\mathrm{out}}(t)= v_{\mathrm{out1}}(t)+ v_{\mathrm{out2}}(t)+ v_{\mathrm{out3}}(t) [/latex]

and

[latex] v_{\mathrm{out}}(t)=12+6\cos{(2000\pi t +30^\circ)}+2\cos{(4000\pi t-10^\circ)} [/latex]

Question 6.2: Using the Transfer Function with Several Input Components Su...

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