Power Factor Correction (Multi-Valued)

Electricity and Magnetism Level 3

An ideal AC voltage source supplies a composite RLC load, as shown above. There are two possible values of capacitance, C 1 C_1 and C 2 C_2 , which give the load unity power factor.

What is ( C 1 + C 2 ) (C_1 + C_2) (in microfarads)?

Note: All RLC components in the diagram above count as being part of the load.


The answer is 114.4318.

Steven Chase by Steven Chase , 37, USA

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1 solution

Steven Chase
Jan 27, 2018

A parameter sweep is the easiest way to solve this problem, but just for fun, let's derive a formal solution. Based on the notation in the picture above, we expect the capacitor reactance to turn out negative.

Intermediate parallel impedance:

Z P = j X ( R 2 + j X 2 ) R 2 + j ( X + X 2 ) = j X ( R 2 + j X 2 ) R 2 + j ( X + X 2 ) ⋅ R 2 − j ( X + X 2 ) R 2 − j ( X + X 2 ) Z_P = \frac{jX \, (R_2 + jX_2)}{R_2 + j(X + X_2)} \\ = \frac{jX \, (R_2 + jX_2)}{R_2 + j(X + X_2)} \cdot \frac{R_2 - j(X + X_2)}{R_2 - j(X + X_2)}

Reactive part of parallel impedance:

X P = X R 2 2 + X X 2 ( X + X 2 ) R 2 2 + ( X + X 2 ) 2 X_P = \frac{X R_2^2 + X X_2 (X + X_2)}{R_2^2 + (X + X_2)^2}

Equating this to the negative of the series inductive reactance:

X 1 ( R 2 2 + ( X + X 2 ) 2 ) = − X R 2 2 − X X 2 ( X + X 2 ) X 1 ( R 2 2 + X 2 + 2 X X 2 + X 2 2 ) = − X R 2 2 − X 2 X 2 − X 2 2 X X 1 R 2 2 + X 1 X 2 + 2 X X 1 X 2 + X 1 X 2 2 = − X R 2 2 − X 2 X 2 − X 2 2 X X_1 (R_2^2 + (X + X_2)^2) = -X R_2^2 - X X_2 (X + X_2) \\ X_1 (R_2^2 + X^2 + 2 X X_2 + X_2^2) = -X R_2^2 - X_2 X^2 - X_2^2 X \\ X_1 R_2^2 + X_1 X^2 + 2 X X_1 X_2 + X_1 X_2^2 = -X R_2^2 - X_2 X^2 - X_2^2 X

Getting the equivalence expression into a quadratic form in X X :

X 2 [ X 1 + X 2 ] + X [ R 2 2 + X 2 2 + 2 X 1 X 2 ] + X 1 ( R 2 2 + X 2 2 ) = 0 X^2 \Big[ X1 + X_2 \Big] + X \Big[ R_2^2 + X_2^2 + 2 X_1 X_2 \Big] + X_1 (R_2^2 + X_2^2) = 0

Results of quadratic solution:

X = − 26.8241 Ω X = − 170.6529 Ω C 1 = 98.8880 μ F C 2 = 15.5437 μ F X = -26.8241 \, \Omega \\ X = -170.6529 \, \Omega \\ C_1 = 98.8880 \, \mu F \\ C_2 = 15.5437 \, \mu F

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