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Lagging and Leading

In an AC circuit, current is ideally in phase with voltage. However, inductive or capacitive devices produce a phase shift called phi ([-90° ... +90°]), described by "\(\cos(\phi)\)" or "Power Factor".

Many confusions can arise from the sign of this phase shift. For a consumer device, there are 2 situations:

Lagging

  • \(\phi\) > 0
  • The current lags (follows) the voltage
  • This is produced by inductive devices (motors, transformers, etc)
  • This produces what is named "Lagging reactive power", expressed in [kVAr]
  • We can say that the plant produces Reactive energy.

Leading

  • \(\phi\) < 0
  • The current is ahead of the voltage
  • This is produced by capacitive devices,
  • This produces what is named "Leading reactive power", expressed in [kVAr]
  • We can say that the plant consumes Reactive energy.

Inverter output

Inverter output circuits can electronically create a phase shift (consuming/generating reactive power) at "no energy cost"—that is, without consuming additional active power. This means the inverter produces a sinusoidal current that is not in phase with the grid voltage.

Therefore, grid managers may require PV plants to "consume" reactive power to compensate for the lagging reactive power produced by the numerous motors on the grid.

In this case, in the loss diagram, reactive energy is labeled "Reactive energy absorbed from the grid" when the required phi is negative (leading).

Impact of resistive, capacitive and inductive element

We explain the impact of resistive, capacitive and inductive elements with the examples of the \(R\!-\!L\!-\!C\) circuits and its equations. Linear AC circuit are commonly solved using the framework of the complex impedance \(Z\) which related complex current and tension \(V=ZI\)

A short- or medium-length overhead line can be represented by a single series impedance

\[ Z \;=\; R \;+\; j\!\bigl(X_L - X_C\bigr), \qquad X_L = \omega L, \qquad X_C = \frac{1}{\omega C}, \]

where

  • \(R\) — series resistance of the conductors \([\Omega]\),
  • \(L\) — series inductance \([\mathrm{H}]\),
  • \(C\) — shunt capacitance “reflected” into the series path \([\mathrm{F}]\),
  • \(\omega = 2\pi f\) — angular frequency \([\mathrm{rad}\,\mathrm{s}^{-1}]\).

If the sending-end rms voltage is \(V_{RMS}\) and the line carries rms current \(I_{RMS}\),

\[ I_{RMS} \;=\; \frac{V_{RMS}}{Z} \;=\; \frac{V_{RMS}}{R + j\bigl(X_L - X_C\bigr)}. \]

The complex power at that cross-section is

\[ S \;=\; P + jQ \;=\; V\,I^{*} \;=\; V\,V^{*} \frac{1}{Z} \;=\; \frac{V_{RMS}^{2}\bigl(R - j\,(X_L - X_C)\bigr)} {R^{2} + (X_L - X_C)^{2}}. \]

recalling that \(VV^*=V_{RMS}\). Thus

\[ P \;=\; \frac{V_{RMS}^{2}R}{R^{2} + \bigl(X_L - X_C\bigr)^{2}}, \qquad Q \;=\; \frac{V_{RMS}^{2}\bigl(X_L - X_C\bigr)} {R^{2} + \bigl(X_L - X_C\bigr)^{2}}. \]
  • \(P\) is always positive — real power dissipated as heat in \(R\).
  • \(Q\) is positive (lagging) if the branch is net-inductive \((X_L > X_C)\) and negative (leading) if it is net-capacitive \((X_C > X_L)\) or zero if those elements balance each others.