Concept explainers
Refer to the circuit of Fig. 8.95, which contains a voltage-controlled dependent voltage source in addition to two resistors. (a) Compute the circuit time constant. (b) Obtain an expression for vx valid for all t. (c) Plot the power dissipated in the 4 Ω resistor over the range of six time constants. (d) Repeat parts (a) to (c) if the dependent source is installed in the circuit upside down. (e) Are both circuit configurations “stable”? Explain.
Figures 8.95
(a)
Find the circuit time constant.
Answer to Problem 76E
The time constant of the circuit is
Explanation of Solution
Formula used:
The expression for the resistance of the circuit is as follows:
Here,
The expression for the time constant of circuit is as follows:
Here,
Calculation:
To find equivalent resistance of a circuit the independent current source is replaced by open circuit and
The circuit diagram is redrawn as shown in Figure 1.
Refer to the redrawn Figure 1:
Apply KVL in mesh 1:
Here,
Substitute
The expression for voltage across the
Here,
Substitute
Substitute
Rearrange for
Substitute
So, the equivalent resistance across inductor is
Substitute
So the time constant of the circuit is
Conclusion:
Thus, the time constant of the circuit is
(b)
Obtain an expression for
Answer to Problem 76E
The expression for the voltage
Explanation of Solution
Formula used:
The expression for the final response of the circuit valid for all
Here,
Calculation:
The unit-step forcing function as a function of time which is zero for all values of its argument less than zero and which is unity for all positive values of its argument.
Here,
The independent current source is:
Substitute
The current through
The
So, the value of the current flowing through the inductor for
The inductor does not allow sudden change in the current.
So,
Therefore, the current flowing in the circuit for
Substitute
So, the current flowing through the
The circuit diagram is redrawn as shown in Figure 2 for
Refer to the redrawn Figure 2:
Apply KCL in the circuit:
Substitute
Rearrange for
The expression for the current flowing through
Here,
Substitute
So, the current flowing through
Substitute
The expression for the voltage across the
Substitute
Conclusion:
Thus, the expression for the voltage
(c)
Plot the power dissipated in the
Explanation of Solution
Given data:
The range of the time is six time constant.
Formula used:
The expression for the power dissipated in the
Here,
Calculation:
Substitute
The time constant of the circuit is
The different value for the power dissipated in the
The graph for power dissipated in the
Calculation:
Thus, the graph for power dissipated in the
Conclusion:
(d)
Repeat parts (a) to (c) if the dependent source is installed in the circuit upside down.
Explanation of Solution
Calculation:
To find equivalent resistance of a circuit the independent current source is replaced by open circuit and
The circuit diagram is redrawn as shown in Figure 4:
Refer to the redrawn Figure 4:
Apply KVL in mesh 1:
Here,
Substitute
The expression for voltage across the
Here,
Substitute
Substitute
Rearrange for
Substitute
So, the equivalent resistance across inductor is
Substitute
So, the time constant of the circuit is
The unit-step forcing function as a function of time which is zero for all values of its argument less than zero and which is unity for all positive values of its argument.
Here,
The independent voltage source is:
Substitute
The current through
The
So, the value of the current flowing through the inductor for
The inductor does not allow sudden change in the current.
So,
Therefore, the current flowing in the circuit for
Substitute
So, the current flowing through the
The circuit diagram is redrawn as shown in Figure 5 for
Refer to the redrawn Figure 5:
Apply KCL in the circuit:
Substitute
Rearrange for
The expression for the current flowing through
Here,
Substitute
So, the current flowing through
Substitute
The expression for the voltage across the
Substitute
So, the expression for the voltage
Substitute
The time constant of the circuit is
The different value for the power dissipated in the
The graph for power dissipated in the
Conclusion:
Thus, the time constant of the circuit is
(e)
Are both circuit configurations “stable”? Explain.
Answer to Problem 76E
Both circuit configurations are “stable”.
Explanation of Solution
Refer to Figure 3 and Figure 6:
The response (output power dissipated in the
So, both circuit configurations “stable”.
Conclusion:
Thus, both circuit configurations are “stable”.
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