Problem

A rectangular voltage pulse v_i = [u(t) - u(t - 1)] V is applied to the circuit in Fig. P13.59. Use the convolution integral to find v_o.

Step-by-step solution

1. Step 1 of 8

   Refer to Figure P13.59 in the textbook for the circuit diagram.

   The s-domain representation of the circuit is shown in Figure 1.

   

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2. Step 2 of 8

   Calculate the output voltage using voltage division rule.

   

   The transfer function is,

   

   Apply inverse Laplace transform on both sides.

   

   Replace  with .

   

   Here, the impulse response is a decaying exponential. So, plot the impulse function.

   

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3. Step 3 of 8

   Consider the input voltage, . Replace  with .

   

   The mathematical representation is,

   

   Plot the input signal.

   

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4. Step 4 of 8

   The folded excitation function  is shown in Figure 3.

   

   The convolution integral is,

   

   Plot the signal, .

   

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5. Step 5 of 8

   Apply the convolution to both Figure 4 and Figure 1.

   

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6. Step 6 of 8

   From Figure 5, the output voltage in the interval of  is,

   

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7. Step 7 of 8

   Now apply the convolution to both Figure 4 (when) and Figure 1.

   

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8. Step 8 of 8

   From Figure 5, the output voltage in the interval of  is,

   

   The output waveform is shown in Figure 7.

   

   Hence, the output has been obtained.

   Comments (1)