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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).

Respuesta :

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Answer:

Explanation:

This is an RL circuit, therefore:

Impedance; z = [tex]\mathbf{\sqrt{R^2+L^2}}[/tex]

[tex]\mathbf{z = \sqrt{R^2+(Lw)^2}}[/tex]

Current amplitude

[tex]\mathbf{I_o = \dfrac{V_o}{z}} \\ \\ \mathbf{I_o = \dfrac{V_o}{\sqrt{R^2+L^2\omega ^2}}}[/tex]

a)

Given that:

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

∴

[tex]I_o= \dfrac{1.9}{\sqrt{21^2+(0.52\times 51)^2}}[/tex]

[tex]\mathbf{I_o= 0.0562} \\ \\ \mathbf{I_o = 56.2 \ mA}[/tex]

b)

Phase constant :

[tex]tan \ \phi = \dfrac{L \omega}{R } \\ \\ tan \ \phi = \dfrac{0.52 \times 51}{21} \\ \\ tan \phi = 1.263[/tex]

[tex]\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}[/tex]

(a)The value of current amplitude will be 56.2 milliamperes.

(b)The value of the phase constant will be 0.9 rad.

What is an RL circuit?

RL Circuits often known as RL networks or RL filters are a form of a circuit that uses a mix of inductors and resistors and is powered by a source of electricity.

The given data in the problem is;

I₀ is the current amplitude =?

V₀= 1.9 V

ω is the angular velocity = 51 rad/s

R is the resistance in the circuit = 21 Ω

Ф is the phase constant=?

(a) The value of current amplitude will be 56.2 milliamperes.

The formula for the current amplitude is given as;

[tex]\rm I_0= \frac{V_0}{\sqrt{R^2+L^2 \omega^2} } \\\\[/tex]

[tex]\rm I_0= \frac{1.9}{\sqrt{(21)^2+(0.52\times 51)^2} } \\\\[/tex]

[tex]\rm I_0 = 0.0562 \ A \\\\ \rm I_0 =56.2 mA[/tex]

Hence the value of the current amplitude will be 56.2 milliamperes.

(b)The value of the phase constant will be 0.9 rad.

The formula for phase constant is given by;

[tex]\rm tan\phi=\frac{L\omega}{R} \\\\ \rm tan\phi=\frac{0.52\times 51}{21} \\\\ \rm tan\phi=1.263 \\\\ \rm \phi = tan^{-1}(1.263) \\\\ \phi= 51.6 ^0[/tex]

To convert degree into radian the following formula is used;

[tex]1 \pi =180^0 \\\\ 1^0=\frac{\pi}{180} \\\\ 51.6^0=51.6 \times \frac{\pi}{180} \\\\ \phi=0.9 \ rad[/tex]

Hence the value of the phase constant will be 0.9 rad.

To learn more about the RL circuit refer to the link;

https://brainly.com/question/16795672

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