Reactive Power Compensation

Electricity and Magnetism Level 5

An ideal $10 V_\text{rms}$ sinusoidal voltage source is connected across an RLC circuit, as shown above.

The series RL portion represents a load, such as a combination of heating elements and induction motors. The resistance is $3 \Omega$ and the inductive reactance is $4 \Omega$ .

A capacitor is placed in parallel with the load so that the net reactive power consumed by the total RLC combination is zero. Power utilities do this at their substations because they don't want to use up too much of their generation and transmission capability sourcing reactive power that brings in zero revenue.

To 2 decimal places, how many ohms of capacitive reactance are required?

Give your answer as a positive number.

The answer is 6.25.

1 solution

Chew-Seong Cheong
Jan 3, 2017

Let the voltage source to have a phase angle of 0, that is $V_s\angle 0 = 10\angle 0 = 10 \text{ V}$ .

Then the current through the load is $I_L = \dfrac {V_s}{Z_L} = \dfrac {10}{3+4i} = \dfrac {10(3-4i)}{25} = 1.2-1.6i \text{ A}$ .

Let the current through the capacitor be $I_C$ then the current from the source $I_s = I_L + I_C$ . For the reactive power consumed by the RLC combination to be zero, $I_s$ must be in phase with $|V_s| \angle 0$ or $I_s =| I_s| \angle 0$ . Therefore,

$\begin{aligned} I_s & = I_L + I_C \\ | I_s| + 0i & = 1.2 - 1.6i + I_C \\ \implies I_C & = 1.6i \\ \frac {V_s}{\color{#3D99F6}Z_C} & = 1.6i & \small \color{#3D99F6} \text{where }Z_C \text{ is the capacitive reactance.} \\ \frac {10}{Z_C} & = 1.6i \\ \implies Z_C & = \frac {10}{1.6i} \\ & = -\boxed{6.25}i \ \Omega \end{aligned}$

Reactive Power Compensation

The answer is 6.25.

1 solution

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