Concept explainers
The inductor current IL, voltage 'v' across the 2Ω resister and voltage
Answer to Problem 5.72HP
Explanation of Solution
Given:
The given circuit is shown below.
The switch is closed at t = 0 s and reopened at t = 5 s.
Calculation:
The capacitor does not allow the sudden change in voltage and the inductor does not allow the sudden change in current.
At t = 0, the capacitor behaves like short circuit and inductor behaves like open circuit.
The modified circuit diagram is:
From the circuit,
Relation between inductor current and voltage is:
Considering the following circuit, at
Applying Kirchhoff's voltage law in loop1,
Applying Kirchhoff's voltage law in top loop,
Differentiating the equation with respect to t,
Applying Kirchhoff's voltage law in bottom loop,
From equation 1, putting the value of
From equation 2, putting the value of
Differentiation of any constant value is zero.
Writing the equation in standard second order differential equation:
Dividing by 0.75,
Comparing the equation with standard second order differential equation:
The natural frequency is determined as:
The damping ratio is determined as follows:
The value of damping ratio is less than 1.
Hence, it is an underdamped second order circuit.
The following expression is used to solve the complete solution.
The roots
At t = 0, the inductor's natural response current is zero.
Differentiating the above equation and substituting t = 0,
Substituting,
Thus, expression for inductor current is:
The voltage across the inductor is:
The voltage across the resistor is:
The voltage across the capacitor is:
Considering that at t = 5 s, the switch moved to open position.
The initial value of the inductor current is:
The initial value of the capacitor voltage is:
Considering the following circuit to determine the initial values:
The inductor voltage:
Considering the following circuit for t > 5 s.
Applying Kirchhoff's voltage law in top loop,
Differentiating the equation with respect to t,
Applying Kirchhoff's voltage law in bottom loop,
Comparing with standard second order equation:
The natural frequency is determined as:
The damping ratio is determined as follows:
The value of damping ratio is less than 1.
Hence, it is an underdamped second order circuit.
The following expression is used to determine the complete solution:
The roots
Substituting
Hence, the expression of inductor current is:
The voltage of inductor is:
The voltage of 2O resistor:
The voltage across capacitor is:
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Chapter 5 Solutions
Principles and Applications of Electrical Engineering
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