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Lagging and Leading

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

Many confusions may arise with the sign of this Phase shift. For a consumer device, we have 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

Now the output circuits of the inverters have the possibility of electronically creating a Phase shift (consuming/generating reactive power) at "no energy cost", i.e. without consuming any additional Active Power. This means that the inverter produces a sinusoïdal current which is not in phase with the grid voltage.

Therefore the Grid managers may require from the PV plants to "consume" Reactive power, for compensating the Lagging Reactive power produced by the numerous motors on the grid.

In this case on the loss diagram, the denomination of the reactive energy is labelled "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.