Two ideal inductors, L1 and L2, have zero internal resistance and are far apart, so their magnetic fields do not influence each other. (a) Assuming these inductors are connected in series, show that they are equivalent to a single ideal inductor having Leq = L1 + L2. (b) Assuming these same two inductors are connected in parallel, show that they are equivalent to a single ideal inductor having 1/Leq = 1/L1 + 1/L2. (c) What If? Now consider two inductors L1 and L2 that have nonzero internal resistances R1 and R2, respectively. Assume they are still far apart, so their mutual inductance is zero, and assume they are connected in series. Show that they are equivalent to a single inductor having Leq = L1 + L2 and Req = R1 + R2. (d) If these same inductors are now connected in parallel, is it necessarily true that they are equivalent to a single ideal inductor having 1/Leq = 1/L1 + 1/L2 and 1/Req = 1/R1 + 1/R2? Explain your answer.
(a)
To show: The equivalent inductance is
Answer to Problem 32.30P
Explanation of Solution
Given info: The inductance of the inductor are
For a series connection, both inductor carry equal currents at every instant. So the change in current
Write the expression to calculate the voltage across the pair.
Conclusion:
Therefore, the equivalent inductance is
(b)
To show: The equivalent inductance in parallel combination is
Answer to Problem 32.30P
Explanation of Solution
Given info: The inductance of the inductor are
For a parallel connection, the voltage across each inductor is same for both.
Write the expression to calculate the voltage across each inductor.
The current in the connection is,
Here,
The change in current in equivalent inductor is,
The change in current in first inductor is,
The change in current in second inductor is,
Substitute
Thus, the equivalent inductance in parallel combination is
Conclusion:
Therefore, the equivalent inductance in parallel combination is
(c)
To show: The equivalent inductance and resistance when their internal resistance is non zero in series combination is
Answer to Problem 32.30P
Explanation of Solution
Given info: The inductance of the inductor are
Write the expression to calculate the voltage across the connection.
Here,
Thus, the equivalent inductance and resistance when their internal resistance is non zero in series combination is
Conclusion:
Therefore, the equivalent inductance and resistance when their internal resistance is non zero in series combination is
(d)
Answer to Problem 32.30P
Explanation of Solution
Given info: The inductance of the inductor are
If the circuit elements are connected in parallel then two conditions always true.
The first condition is,
The second condition is,
Thus, it is necessarily true that they are equivalent to a single ideal inductor having
Conclusion:
Therefore, it is necessarily true that they are equivalent to a single ideal inductor having
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Chapter 32 Solutions
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