Question

2. A resistor R and inductor L are connected in series with an AC voltage source with frequency fand maximum voltage Vo. a. Find the complex impedance (in the form Z = R + jX). If the impedance is written in polar form (Z = Zejo), find expressions for Z and Ø. Write your answers in terms of the variables R, L, and o(= 21f). b. If the voltage source is described by the phasor V = Voejut, and the current is described by I = 1, ej(wt-o), find lo in terms of R, L, 0, and Vo. [Use I = V/Z and the properties of complex numbers.] c. Find the amplitude of the voltage across the resistor, VR, in terms of R, L, 0, and Vo. d. If R = 11 12 and L = 17 mH, find the frequencies for which VR = 0.90 Vo and VR = 0.10 Vo. [Note that the voltage across the resistor is much larger for the lower frequency. If a load was placed across the resistor, a sizeable signal would be transferred to the load only for lower frequencies – therefore connecting across a resistor in an R-L circuit is another way to make a low-pass filter.]

Answer 1

part 1

complex impedance is

Z= R + MXL

$Z = R + j \omega L$

Since

$\textbf Z = Z e^{j\phi}$

So

$Z= \sqrt{R^2+(\omega L)^2}$

and

$tan(\phi)= \frac{X_L}{R}$

$\phi= tan^{-1}(\frac{\omega L}{R})$

Part b

V = Voejut

$\textbf I = I_0 e^{j(\omega t-\phi)}$

$\textbf I = \frac{\textbf V}{\textbf Z}$

$I_0 e^{j(\omega t-\phi)} = \frac{ V_0e^{j\omega t}}{Ze^{j\phi}}$

$I_0 = \frac{ V_0}{Z}$

Part c

voltage across R

$V_R = \frac{V_0}{\sqrt{R^2+(\omega L)^2}}R$

Part d

for

$0.9V_0= \frac{V_0}{\sqrt{R^2+(\omega L)^2}}R$

$\frac{\omega^2 L^2}{R} = 0.234$

$\omega = R\frac{\sqrt{0.234}}{L}$

$\omega = 11\frac{\sqrt{0.234}}{17 \times 10^{-3}}$

$\omega = 313.005 rad/s$

Now

$0.1V_0= \frac{V_0}{\sqrt{R^2+(\omega L)^2}}R$

$\omega = \sqrt{99}\frac{R}{L}$

$\omega = \sqrt{99}\frac{11}{17\times 10^{-3}}$

$\omega = 6438.15$

$\omega = 2049.32 Hz$