Questions: Impedance and Admittance in AC Networks

Question 1

A circuit has three identical admittances Y₁ = Y₂ = Y₃ = 0.1 + j0.2 S connected in parallel. What is the total admittance?

• A. You must convert each to impedance, combine using the reciprocal formula, then convert back
• B. Y_total = 0.3 + j0.6 S — parallel admittances add directly
• C. Y_total = 0.033 + j0.067 S — parallel admittances average
• D. Y_total = 10 − j5 Ω — parallel elements require the impedance domain

Answer: (B)

One of the main motivations for admittance is exactly this: parallel admittances add directly, just as series impedances add directly. Y_total = Y₁ + Y₂ + Y₃ = 0.3 + j0.6 S. Option A describes the impedance approach, which works but requires extra steps. The admittance approach is cleaner for parallel circuits.

Question 2

An inductor L has reactance X_L = +jωL. What is its susceptance?

• A. B_L = +jωL — susceptance equals reactance for inductive elements
• B. B_L = +1/(ωL) — susceptance is the magnitude of the reciprocal
• C. B_L = −1/(ωL) — susceptance and reactance have opposite signs
• D. B_L = −jωL — susceptance is the negative imaginary part of impedance

Answer: (C)

For an inductor, Z = jωL, so Y = 1/Z = 1/(jωL) = −j/(ωL). The susceptance (imaginary part of admittance) is B_L = −1/(ωL), which is negative — opposite in sign to the positive reactance X_L = +ωL. A capacitor has negative reactance but positive susceptance. Always derive susceptance from Y = 1/Z rather than assuming it matches the sign of reactance.

Question 3

For parallel circuit elements, using admittances instead of impedances simplifies the calculation because parallel admittances add directly.

• T. True
• F. False

Answer: True

True. This is the primary motivation for introducing admittance. In the impedance domain, parallel elements require 1/Z_total = 1/Z₁ + 1/Z₂ + ... — a cumbersome sum of reciprocals. In the admittance domain, Y_total = Y₁ + Y₂ + ... — a simple sum. The symmetry is complete: series circuits are natural in the impedance domain; parallel circuits are natural in the admittance domain.

Question 4

An element's susceptance and reactance usually have the same algebraic sign.

• T. True
• F. False

Answer: False

False. Susceptance and reactance have opposite signs for the same element. An inductor has positive reactance (+ωL) but negative susceptance (−1/(ωL)). A capacitor has negative reactance (−1/(ωC)) but positive susceptance (+ωC). This follows from Y = 1/Z: taking the reciprocal of a purely imaginary number flips its sign. Forgetting this sign flip is one of the most common errors when switching between impedance and admittance representations.

Question 5

Why might an engineer choose to analyze a mixed series-parallel AC network partly in the impedance domain and partly in the admittance domain, rather than committing to one representation throughout?

Model answer: Series sub-circuits are most efficiently analyzed in the impedance domain (series impedances add), while parallel sub-circuits are most efficiently analyzed in the admittance domain (parallel admittances add). In a mixed network, switching representations at the boundary between series and parallel sections keeps the algebra simpler and reduces errors from repeatedly computing reciprocals.

The power of the dual representation is this flexibility. Neither domain is universally superior — the choice depends on circuit topology at each stage. Skilled analysis involves recognizing which domain makes each step easier and converting between them as needed, rather than forcing a complex mixed network into a single representation.