Impedance and Admittance

Every transmission line, transformer, and cable in the power grid is characterized by its impedance (how much it resists current) or equivalently its admittance (how easily current flows). These parameters determine how power flows through the network.

Starting Simple: Resistance

In DC circuits, resistance $R$ relates voltage and current via Ohm's Law:

$$V = I \cdot R$$

Resistance dissipates energy as heat. A conductor with resistance $R$ carrying current $I$ loses power:

$$P_{\text{loss}} = I^2 R$$

This is why high-voltage transmission exists — raising voltage lets us reduce current, cutting $I^2R$ losses.

Units: Ohms ($\Omega$)

AC Circuits: Enter Reactance

AC circuits have inductors and capacitors that store energy in fields rather than dissipating it. These elements have reactance $X$:

Inductive Reactance

Inductors (coils of wire) store energy in magnetic fields. Current through an inductor creates a magnetic field that opposes changes in current.

$$X_L = \omega L = 2\pi f L$$

where:

• $L$ is inductance in Henries (H)
• $f$ is frequency (60 Hz in North America, 50 Hz in Europe)
• $\omega = 2\pi f$ is angular frequency

Physical intuition: Inductors "resist" current changes. At higher frequencies, this resistance increases.

Capacitive Reactance

Capacitors store energy in electric fields between plates. Voltage across a capacitor opposes changes in voltage.

$$X_C = \frac{1}{\omega C} = \frac{1}{2\pi f C}$$

where $C$ is capacitance in Farads (F).

Physical intuition: Capacitors "resist" voltage changes. At higher frequencies, this resistance decreases.

Units: Ohms ($\Omega$), same as resistance

Impedance: R and X Together

Impedance $\mathbf{Z}$ combines resistance and reactance into a single complex number:

$$\mathbf{Z} = R + jX$$

where:

• $R$ is resistance (real part) — dissipates energy
• $X$ is reactance (imaginary part) — stores energy
• $j = \sqrt{-1}$ (engineers use $j$ instead of $i$ to avoid confusion with current)

Ohm's Law for AC

$$\mathbf{V} = \mathbf{I} \cdot \mathbf{Z}$$

Now voltage and current are complex phasors, and impedance is complex.

Magnitude and Angle

$$|\mathbf{Z}| = \sqrt{R^2 + X^2}$$

$$\angle \mathbf{Z} = \arctan\left(\frac{X}{R}\right)$$

A transmission line with $\mathbf{Z} = 5 + j20$ $\Omega$ has:

• $|\mathbf{Z}| = \sqrt{25 + 400} = 20.6$ $\Omega$
• $\angle \mathbf{Z} = \arctan(20/5) = 76°$ (mostly reactive)

Transmission Line Parameters

A transmission line has both series impedance and shunt admittance. The standard $\pi$-model looks like:

      R + jX
 ───┬──/\/\/──┬───
    │         │
   jB/2     jB/2
    │         │
   ─┴─       ─┴─

Series Elements: R and X

Resistance $R$: Comes from conductor material (aluminum, copper). Causes $I^2R$ losses.

$$R = \rho \frac{\ell}{A}$$

where $\rho$ is resistivity, $\ell$ is length, $A$ is cross-sectional area.

Reactance $X$: Dominated by the magnetic field around and between conductors (inductance). Transmission lines are almost purely inductive.

$$X = \omega L \approx 0.3 \text{ to } 0.5 \text{ } \Omega/\text{km (typical)}$$

Key insight: For transmission lines, $X >> R$ (typically 3-10× larger). This is why DC power flow ignores resistance — reactance dominates.

Shunt Elements: B (Line Charging)

Long transmission lines have capacitance between conductors and to ground. This creates line charging — the line generates reactive power even with no load.

$$B = \omega C$$

The $\pi$-model splits this capacitance equally at both ends: $B/2$ at each bus.

Physical intuition: Line charging is why lightly-loaded transmission lines can cause overvoltage — they're pumping reactive power into the system.

Admittance: The Inverse

Admittance $\mathbf{Y}$ is the reciprocal of impedance:

$$\mathbf{Y} = \frac{1}{\mathbf{Z}} = G + jB$$

where:

• $G$ is conductance (real part) — how easily real current flows
• $B$ is susceptance (imaginary part) — how easily reactive current flows

Why Use Admittance?

For network analysis, admittances are easier to work with:

Impedances in series add: $\mathbf{Z}_{\text{total}} = \mathbf{Z}_1 + \mathbf{Z}_2$

Admittances in parallel add: $\mathbf{Y}_{\text{total}} = \mathbf{Y}_1 + \mathbf{Y}_2$

Since buses connect multiple branches in parallel, building the network matrix (Y-bus) is simpler with admittances.

Converting Z to Y

$$\mathbf{Y} = \frac{1}{\mathbf{Z}} = \frac{1}{R + jX} = \frac{R - jX}{R^2 + X^2}$$

So:

$$G = \frac{R}{R^2 + X^2}$$

$$B = \frac{-X}{R^2 + X^2}$$

Note the sign: Positive $X$ (inductive) gives negative $B$.

Units: Siemens (S), the reciprocal of ohms. $1 \text{ S} = 1/\Omega$

Transformer Impedance

Transformers also have impedance, primarily reactive (leakage inductance). The per-unit impedance is typically 5-15% for power transformers:

$$Z_{\text{p.u.}} = 0.05 \text{ to } 0.15$$

This means if you short-circuit the secondary, the fault current is limited to:

$$I_{\text{fault}} = \frac{1}{Z_{\text{p.u.}}} = 6.7 \text{ to } 20 \times I_{\text{rated}}$$

Tap Ratio

Transformers also have a tap ratio $t$ (turns ratio):

$$\frac{V_1}{V_2} = t$$

Tap changers adjust voltage ±10% in discrete steps. Off-nominal taps affect the admittance matrix — see Y-Bus Matrix.

Per-Unit Impedance

Transmission studies use per-unit values to normalize across voltage levels:

$$Z_{\text{p.u.}} = \frac{Z_{\text{actual}}}{Z_{\text{base}}}$$

where:

$$Z_{\text{base}} = \frac{V_{\text{base}}^2}{S_{\text{base}}}$$

Example: 345 kV system, 100 MVA base:

$$Z_{\text{base}} = \frac{(345 \times 10^3)^2}{100 \times 10^6} = 1190 \text{ } \Omega$$

A line with $Z = 11.9 + j119$ $\Omega$ becomes $Z_{\text{p.u.}} = 0.01 + j0.1$ p.u.

Advantage: Per-unit values are similar across voltage levels, making it easy to spot unusual values.

See Units & Conventions for details.

Summary Table

Key Takeaways

1. Impedance $\mathbf{Z} = R + jX$ combines resistance (losses) and reactance (storage)
2. Transmission lines have $X >> R$ — reactance dominates
3. Admittance $\mathbf{Y} = 1/\mathbf{Z} = G + jB$ is used for network analysis
4. Line charging $B$ causes reactive power generation on lightly-loaded lines
5. Per-unit values normalize across voltage levels

See Also

• Y-Bus Matrix — Building the network admittance matrix
• Power Flow Theory — How impedance affects power flow
• Complex Power — Power flowing through impedances
• Units & Conventions — Per-unit system