An AC circuit has an inductor and a resistor of resistance R in series, such that X L = 3 R . Now, a capacitor is added in series such that X C = 2 R . the ratio of the new power factor with the o

Question

An AC circuit has an inductor and a resistor of resistance R in series, such that X L ​ = 3 R . Now, a capacitor is added in series such that X C ​ = 2 R . the ratio of the new power factor with the old power factor of the circuit is 5 ​ : x . The value of x is

TestHub · Accepted Answer

First Recall the formula of power factor, $cos ϕ = \frac{R}{Z}$ $\Rightarrow cos ϕ_{ini t ia l} = \frac{R}{Z} = \frac{R}{( X _{L} ) ^{2} + R ^{2}} = \frac{R}{9 R ^{2} + R ^{2}} = \frac{1}{10}$ .

$\Rightarrow cos ϕ_{f ina l} = \frac{R}{Z} = \frac{R}{( X _{L} - X _{C} ) ^{2} + R ^{2}} = \frac{R}{R ^{2} + R ^{2}} = \frac{1}{2}$

$\Rightarrow \frac{c o s ϕ _{f ina l}}{c o s ϕ _{ini t ia l}} = \frac{5}{x} \Rightarrow x = 1$

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