Question

Suppose the voltage source for a series RL-circuit were given as V0sin(ωt) instead of V0cos(ωt). Write an expression for the current amplitude in terms of V0, ω, R, and L.
Required:
a. Calculate the current amplitude, in milliamperes, when V0= 1.9 V, ω = 51 rad/s, R = 21 Ω, and L = 0.52 H.
b. Calculate the phase constant, in radians in the range -π/2 ≤ φ ≤ π/2, for the circuit parameter values given in part (b).

Answer 1

Answer:

Explanation:

This is an RL circuit, therefore:

Impedance; z = $\mathbf{\sqrt{R^2+L^2}}$

$\mathbf{z = \sqrt{R^2+(Lw)^2}}$

Current amplitude

$\mathbf{I_o = \dfrac{V_o}{z}} \\ \\ \mathbf{I_o = \dfrac{V_o}{\sqrt{R^2+L^2\omega ^2}}}$

a)

Given that:

$V_o = 1.9 \ V \\ \\ \omega= 51 \ rad/s\\\\ R = 21 \Omega \\ \\ L = 0.52 H$

∴

$I_o= \dfrac{1.9}{\sqrt{21^2+(0.52\times 51)^2}}$

$\mathbf{I_o= 0.0562} \\ \\ \mathbf{I_o = 56.2 \ mA}$

b)

Phase constant :

$tan \ \phi = \dfrac{L \omega}{R } \\ \\ tan \ \phi = \dfrac{0.52 \times 51}{21} \\ \\ tan \phi = 1.263$

$\text{Phase constant : }\phi = tan^{-1} (1.263) \\ \\ \phi = 51.6^0\\ \\\text{To radians} \phi = 51.6 \times \dfrac{\pi}{180} \\ \\ \phi = 0.287 \pi \\ \\ \mathbf{\phi = 0.9 \ rad}$