Question

(a) Show that for an L-R-C series circuit the power factor is equal to$\frac{\mathbf{R}}{\mathbf{Z}}$(b) An L-R-C series circuit has phase angle$\mathbf{-}\mathbf{31}\mathbf{.}\mathbf{5}\mathbf{°}$. The voltage amplitude of the source is 90.0 V. What is the voltage amplitude across the resistor?

Short answer

The power factor is equal to$\frac{\mathrm{R}}{\mathrm{Z}}$ and the voltage amplitude across the resistor is 76.7V.

Step-by-step solution

Step-1: Power factor definition

The power factor is the cos of the phase angle of the voltage with respect to the current.

Step-2: Calculations for power factor formula

$$\begin{gathered} \mathrm{From}\mathrm{figure},\mathrm{cos\varphi }=\frac{\mathrm{IR}}{\mathrm{IZ}} \\ =\frac{\mathrm{R}}{\mathrm{Z}} \end{gathered}$$

Therefore, the formula for power factor is$\frac{\mathrm{R}}{\mathrm{Z}}$

Step-3: Calculations for voltage amplitude of the resistor

Since the phase angle given is negative, therefore the phase angle is negative, which means that the voltage lags the current. In part(a),the term IZ represents the voltage amplitude in the circuit and it is given by 90V. So, we can get the voltage amplitude of the resistor$V_R$ as next

$$\begin{gathered} V_R=IR \\ =\frac{V}{Z}\left(R\right) \\ =\left(V\right)\frac{R}{Z} \\ =\left(V\right)\cos \varphi \\ =\left(90V\right)\cos \left(-31.5°\right) \\ =76.7V \end{gathered}$$

Hence, the power factor is$\frac{\mathrm{R}}{\mathrm{Z}}$ and the resistor voltage amplitude is 76.7V.