A student says: 'Inductors combine just like resistors — series adds, parallel uses product-over-sum — so I can analyze an inductor network exactly the same way I analyze a resistor network.' What important distinction does this overlook?
AThe formulas are different — series inductors multiply rather than add
BInductors store energy in a magnetic field and return it to the circuit, while resistors dissipate energy permanently — so while the equivalent inductance formula mirrors resistors, the energy behavior and transient dynamics are fundamentally different
CThe parallel combination rule for inductors adds inductances, while resistors use reciprocals
DInductors and resistors cannot be in the same circuit, making the analogy invalid
The combination formulas are identical in form (series: add; parallel: reciprocal sum), but inductors are reactive elements that store energy in a magnetic field during one part of the cycle and return it during another. Resistors convert electrical energy to heat irreversibly. This means inductor networks have time-dependent (transient) behavior — current through an inductor cannot change instantaneously — that has no parallel in resistor networks. The formula analogy is useful for finding L_eq, but the circuit behavior is governed by differential equations, not Ohm's law.
Question 2 Multiple Choice
Two inductors L₁ = 6 H and L₂ = 3 H are connected in parallel. What is the equivalent inductance?
A9 H — they add in parallel
B2 H — product over sum: (6 × 3)/(6 + 3)
C18 H — they multiply in parallel
D4.5 H — the average of the two values
For two inductors in parallel: L_eq = (L₁ × L₂)/(L₁ + L₂) = (6 × 3)/(6 + 3) = 18/9 = 2 H. The equivalent inductance is always less than either individual value — adding parallel paths makes it easier to change the total current, reducing the effective inductance. This mirrors the parallel resistor formula exactly. Option A (adding) is the series formula applied incorrectly to parallel; option C confuses inductors with some other element.
Question 3 True / False
Adding a second inductor in parallel with an existing one always decreases the equivalent inductance below either individual inductor's value.
TTrue
FFalse
Answer: True
1/L_eq = 1/L₁ + 1/L₂, so L_eq = (L₁L₂)/(L₁+L₂). Since L₁L₂ < L₁(L₁+L₂) (as L₂ < L₁+L₂), we have L_eq < L₁, and by symmetry L_eq < L₂. The parallel combination is always smaller than the smallest individual element. Physically, adding a parallel inductor provides an additional current path, making the combined element easier to drive — requiring less voltage per unit of di/dt, which is equivalent to a smaller inductance.
Question 4 True / False
Because inductors store energy rather than dissipate it, the formulas for combining series and parallel inductor networks differ from those used for resistors.
TTrue
FFalse
Answer: False
The combination formulas are mathematically identical: series inductors add (L_eq = L₁ + L₂ + ...) just like series resistors, and parallel inductors use the reciprocal sum (1/L_eq = 1/L₁ + 1/L₂ + ...) just like parallel resistors. The formulas follow from applying KVL/KCL to v = L·(di/dt), which has the same mathematical structure as V = IR with respect to how elements combine. The distinction between energy storage and dissipation changes the circuit's transient behavior but not the combining rules for equivalent inductance.
Question 5 Short Answer
Starting from v = L·(di/dt) and Kirchhoff's current law, explain why inductors in parallel combine with the formula 1/L_eq = 1/L₁ + 1/L₂.
Think about your answer, then reveal below.
Model answer: In parallel, both inductors share the same voltage v across their terminals. By KCL, the total current derivative is di_total/dt = di₁/dt + di₂/dt. Since each inductor sees voltage v, its current derivative is di/dt = v/L, giving di_total/dt = v/L₁ + v/L₂ = v·(1/L₁ + 1/L₂). For the equivalent inductance, di_total/dt = v/L_eq, so 1/L_eq = 1/L₁ + 1/L₂. This mirrors the parallel resistor derivation (I_total = V/R₁ + V/R₂) because v = L·di/dt and V = IR have the same algebraic structure with respect to how branch quantities add.
The derivation should make clear that the 'reciprocal sum' rule is not a fact to memorize but a consequence of KCL plus the constitutive relation v = L·di/dt. The same logical structure (shared voltage → additive current derivatives → reciprocal inductances) applies because KCL doesn't care whether elements are resistors or inductors. Understanding the derivation also clarifies the analogy's limits: the formula gives L_eq for DC and AC analysis, but in transient circuits you must still write the differential equation, not just replace the network with L_eq and apply Ohm's law.