Admittance

Admittance $Y$ is the reciprocal of impedance:

$Y=\frac{1}{Z}=G+jB\text{(S, siemens)}.$

Current per volt applied. Large $Y$ means current flows easily. Like impedance, $Y$ is complex in general:

• $G=\mathrm{Re}(Y)$, conductance (S). The dissipative part. For a pure resistor, $Y=1/R=G$.
• $B=\mathrm{Im}(Y)$, susceptance (S). The reactive part. For a pure capacitor, $Y=j\omega C$, so $B=\omega C$ (capacitive susceptance, positive). For a pure inductor, $Y=1/(j\omega L)=-j/(\omega L)$, so $B=-1/(\omega L)$ (inductive susceptance, negative).

Watch the sign convention for $B$: it’s opposite to the sign of reactance $X$. A capacitor has $X<0$ (capacitive reactance) but $B>0$ (capacitive susceptance); an inductor has $X>0$ but $B<0$. The flip falls straight out of $Y=1/Z$: the reciprocal of $jX$ is $-j/X$, which inverts the imaginary sign.

Why admittance and not just impedance

Impedance is natural for series combinations: $Z_{\text{series}}=Z_1+Z_2+\dots$. Each branch’s impedance adds.

Admittance is natural for parallel combinations: $Y_{\text{parallel}}=Y_1+Y_2+\dots$. Each branch’s admittance adds.

When a circuit has many parallel branches (a noisy power line with many loads, or the input network of an amplifier), working in admittance is cleaner than computing each $Z_i$, inverting, summing, and inverting again. For mixed series-parallel networks, you switch between $Z$ and $Y$ at each level.

Computation from $Z$

To go from $Z=R+jX$ to $Y=G+jB$:

$Y=\frac{1}{R+jX}=\frac{R-jX}{R^2+X^2}=\frac{R}{R^2+X^2}-j\frac{X}{R^2+X^2}.$

So: $G=\frac{R}{R^2+X^2},B=-\frac{X}{R^2+X^2}.$

Watch out: $G\ne 1/R$ unless $X=0$. A resistor in series with a reactance has $G<1/R$; the reactance “shields” the resistance from the source.

Characteristic admittance

For a Transmission line with Characteristic impedance $Z_0$, the characteristic admittance is

$Y_0=\frac{1}{Z_0}.$

For $Z_0=50$ Ω: $Y_0=0.02$ S = 20 mS. Used in stub-matching and parallel-branch analysis of TL networks.

The normalized admittance is $y=Y/Y_0=Y{Z}_0$, same notation as normalized impedance.

Admittance on the Smith chart

A normalized impedance $z=z_L$ has reflection coefficient $\Gamma =(z-1)/(z+1)$. The corresponding admittance $y=1/z$ has reflection coefficient

${\Gamma }_y=\frac{y-1}{y+1}=\frac{1/z-1}{1/z+1}=\frac{1-z}{1+z}=-\Gamma .$

A 180° rotation of $\Gamma$ around the chart center sends $z$ to $1/z=y$. That’s the admittance trick on the Smith chart: to convert an impedance point to its admittance, draw a diameter through the chart center and reflect.

So the same Smith chart works as an “admittance chart” by rotating the labels 180°. Constant-$r$ circles become constant-$g$ circles; constant-$x$ arcs become constant-$b$ arcs (with sign flip). Most engineering charts have both grids overlaid in different colors.

This matters for stub matching: stubs are parallel elements, so you work in admittance to add them to the line admittance. Converting load impedance to admittance via the 180° rotation is step 1 of single-stub matching.

Worked example

A load $Z_L=50+j75$ Ω on a 50 Ω line. Normalized impedance $z_L=1+j1.5$.

$|z_L{|}^2=1^2+1.5^2=\mathrm{3.25.}$

Admittance:

$y_L=\frac{1}{z_L}=\frac{1-j1.5}{1^2+1.5^2}=\frac{1-j1.5}{3.25}\approx 0.308-j\mathrm{0.462.}$

So $g_L\approx 0.31$, $b_L\approx -0.46$. The original impedance was inductive ($x_L>0$); the admittance is correspondingly capacitive in susceptance sign ($b_L<0$). Denormalized: $G_L=g_L/Z_0=6.15$ mS, $B_L=b_L/Z_0=-9.23$ mS.

In AC circuit analysis broadly

Admittance is the natural quantity wherever:

• Parallel branches appear (loads on a power bus, frequency-selective filters, antenna feed networks).
• Stub matching is done on a transmission line.
• Nodal analysis is the preferred method. KCL at a node gives $\sum I_i=0$, which becomes $\sum Y_i{V}_i=0$ in admittance form, the basis of SPICE-style nodal matrices.
• Op-amp feedback networks with parallel-RC elements simplify by working in $Y(\omega )$.

R, C, and L are the things you build with; impedance is the most-cited derived quantity; admittance is its companion, used where parallel structure or nodal methods make it the natural choice.