admittance

admittance

1. What Admittance Is

Admittance ($Y$, siemens $S$, $mho$) = reciprocal of impedance, complex ease with which AC current flows:

where

= conductance (real part, $S$),

= susceptance (imaginary, $S$). For $Z=R+jX$,

So

.

For pure elements:

• $R$:

(real, $B=0$)

• $L$:

with

• $C$:

with

2. Why Use Admittance

• Parallel circuits are easier with $Y$: $Y_{eq}= Σ Y_i$ for parallel (like conductance), while $Z$ series adds. So for parallel $R,L,C$,

then

.

• Magnitude

, phase

.

3. Worked Examples

(a) Series RL: Z=3+j4 ⇒ Y=1/(3+j4)= (3-j4)/25=0.12 -j0.16 S ⇒ G=0.12 S, B=-0.16 S ✓

(b) Parallel RC: R=10Ω, C=100µF at 50Hz: G=0.1 S, B_C= ωC=0.0314 S ⇒ Y=0.1+j0.0314=0.105∠17.4° S, |Z|=1/|Y|≈9.52Ω ✓

(c) Parallel RLC at resonance: B= ωC -1/(ωL)=0 ⇒ Y=G=1/R pure real ⇒ |Z| maximal for parallel.

(d) Need to find current via Y: I = V·Y (since Y=1/Z): V=10∠0° V, Y=0.12-j0.16 ⇒ I=1.2 -j1.6 =2∠-53° A ✓

4. Exam Points

• Define $Y=1/Z=G+jB$, $G$, $B$ — 5 marks, very frequent.
• Show $Y_{eq}=ΣY_i$ for parallel — 5 marks, frequent.
• Compute $Y$ for $R,L,C$ and series/parallel — 5-7 marks, very frequent.
• Relation $B_C=ωC$, $B_L=-1/ωL$ — 3-5 marks, frequent.