Impedance and Admittance

Once phasors are established, the next step in AC circuit analysis is to characterize how each circuit element responds to sinusoidal excitation. Impedance is the generalization of resistance to AC circuits, expressed as a complex number that accounts for both the magnitude and phase of the voltage-to-current relationship. Its reciprocal, admittance, is equally important for parallel circuit analysis and is central to nodal methods in the sinusoidal steady state.

Core Concept Explanation

In DC circuits, Ohm's law states V = IR, where resistance R is a real number. In AC circuits operating at a sinusoidal frequency w, the same Ohm's law generalizes to V = ZI in phasor form, where Z is a complex number called impedance. Impedance accounts for both the amplitude scaling and the phase shift introduced by energy-storing elements. Without this generalization, every element would need separate differential equations.

Impedance Z is written in rectangular form as Z = R + jX. Here, R is the resistance (real part, always non-negative for passive elements) and X is the reactance (imaginary part). Positive X indicates inductive behavior and negative X indicates capacitive behavior. The magnitude |Z| gives the ratio of voltage amplitude to current amplitude, and the angle of Z gives the phase shift between voltage and current.

The reactance of an inductor is X_L = wL, so its impedance is Z_L = jwL. The reactance of a capacitor is X_C = -1/(wC), so its impedance is Z_C = 1/(jwC) = -j/(wC). The resistor has no reactance: Z_R = R. Notice that inductive reactance increases with frequency while capacitive reactance decreases with frequency, which explains frequency-selective behavior in filters.

Admittance Y is defined as the reciprocal of impedance: Y = 1/Z = G + jB. The real part G is the conductance and the imaginary part B is the susceptance. For a resistor, Y = 1/R = G. For an inductor, Y = 1/(jwL) = -j/(wL), giving susceptance B_L = -1/(wL). For a capacitor, Y = jwC, giving susceptance B_C = wC. Admittance is particularly convenient when analyzing parallel circuits, where admittances of parallel branches add directly just as conductances do in DC circuits.

Mathematical Expression

For a series RLC circuit, the total impedance is the sum of individual impedances: Z_total = R + jwL + 1/(jwC) = R + j(wL - 1/(wC)). The magnitude is |Z| = sqrt(R² + (wL - 1/(wC))²) and the phase angle is theta = arctan((wL - 1/(wC))/R). At resonance, the imaginary part is zero: wL = 1/(wC), giving minimum impedance equal to R.

For a parallel RLC circuit, the total admittance is the sum of branch admittances: Y_total = G + j(wC - 1/(wL)). Series combination uses impedance addition; parallel combination uses admittance addition. Converting between Z and Y: if Z = R + jX, then Y = (R - jX)/(R² + X²) = R/(R²+X²) - jX/(R²+X²). This gives G = R/(R²+X²) and B = -X/(R²+X²).

Practical Understanding

Impedance is frequency-dependent. At low frequencies, inductive reactance is small (inductors appear as short circuits) and capacitive reactance is large (capacitors appear as open circuits). At high frequencies, the reverse is true. This frequency dependence is the basis for all filter design: low-pass, high-pass, band-pass, and band-stop filters all exploit the way impedance varies with frequency.

In practical circuit analysis, impedance matching is critical for maximum power transfer in RF and communication systems. The condition for maximum power transfer to a load is that the load impedance must be the complex conjugate of the source impedance: Z_load = Z_source*. This is the AC generalization of the maximum power transfer theorem studied in DC circuits.

Solved Numerical Example

A series RLC circuit has R = 50 ohm, L = 10 mH, and C = 100 uF. It is driven by a 50 Hz source. The impedance is calculated by finding the inductive and capacitive reactances at 50 Hz and combining them with the resistance. The admittance is then found as the reciprocal of the total impedance expressed in rectangular form.

Given:
R = 50 ohm, L = 10 mH = 0.01 H, C = 100 uF = 100×10⁻⁶ F
Frequency f = 50 Hz, so w = 2*pi*50 = 314.16 rad/s

Why this formula applies:
Series RLC: Z = R + j(X_L - X_C) where X_L = wL, X_C = 1/(wC)

Formula:
Z = R + j(wL - 1/(wC))

Substitution:
X_L = 314.16 × 0.01 = 3.14 ohm
X_C = 1 / (314.16 × 100×10⁻⁶) = 1 / 0.03142 = 31.83 ohm
Z = 50 + j(3.14 - 31.83) = 50 - j28.69 ohm

Calculation:
|Z| = sqrt(50² + 28.69²) = sqrt(2500 + 823.12) = sqrt(3323.12) = 57.67 ohm
angle(Z) = arctan(-28.69 / 50) = arctan(-0.5738) = -29.84°
Y = 1/Z = (50 + j28.69) / (50² + 28.69²) = (50 + j28.69) / 3323.12
G = 50/3323.12 = 0.01505 S, B = 28.69/3323.12 = 0.00864 S

Final Answer with units:
Z = 50 - j28.69 ohm = 57.67 angle(-29.84°) ohm
Y = 0.01505 + j0.00864 S

Exam Tip: In GATE, when converting Z = R + jX to admittance Y = G + jB, remember G is NOT 1/R unless X = 0. The correct formula is G = R/(R²+X²) and B = -X/(R²+X²). Confusing G = 1/R is a very common error in mixed impedance-admittance problems.

• Impedance Z = R + jX generalizes Ohm's law to AC: V = ZI in phasor domain. R is resistance, X is reactance.
• Inductive reactance X_L = wL (positive, increases with frequency). Capacitive reactance X_C = -1/(wC) (negative, decreases with frequency).
• Admittance Y = 1/Z = G + jB. G is conductance, B is susceptance. Admittance adds for parallel circuits.
• For series RLC: Z = R + j(wL - 1/(wC)). At resonance wL = 1/(wC) and Z = R (minimum impedance).
• Converting: G = R/(R²+X²) and B = -X/(R²+X²). These are NOT simply 1/R and -1/X.

Quick Revision

• Z = R + jX (ohms). R = resistance, X = reactance. |Z| = sqrt(R²+X²), angle = arctan(X/R).
• Z_R = R, Z_L = jwL, Z_C = 1/(jwC) = -j/(wC).
• Y = G + jB (Siemens). Y_R = 1/R, Y_L = 1/(jwL) = -j/(wL), Y_C = jwC.
• Series: Z_total = Z1 + Z2 + ... Parallel: Y_total = Y1 + Y2 + ...
• Resonance in series RLC: w_r = 1/sqrt(LC), Z = R (minimum), current maximum.
• GATE trap: G = 1/R is valid ONLY when X = 0. In general, G = R/(R²+X²).
• X > 0 means inductive (lagging current). X < 0 means capacitive (leading current).