Question

Find ) in the integro-differential equation dv/dt+5v(t) + 4 f vat = 24 sin (4t+10°) using the phasor approach. Please report your answer so the magnitude is positive and all angles are in the range of negative 180 degrees to positive 180 degrees. The value of v(t) is ______cos(4t-______ °) V.

Answer 1

The value of v(t) is 12 cos(4t - 10°) V.

Step-by-step explanation:

The given integro-differential equation is dv/dt + 5v(t) + 4 ∫ v(t) dt = 24 sin(4t + 10°). To solve this using the phasor approach, we convert the time-domain equation into the frequency domain. Let V and I be the phasor representations of v(t) and 24 sin(4t + 10°), respectively.

Taking Laplace transforms of both sides, we get sV + 5V + 4/s V = I, where s is the complex frequency. Solving for V yields V = 24 / (s(s + 4)(s + 5)) e^(-j10°).

Now, expressing V in polar form |V|e^(jφ), we find |V| = 12 and φ = -10°. Converting this back to the time domain, the final answer is v(t) = 12 cos(4t - 10°) V.

In summary, by applying the phasor approach and Laplace transforms, we transformed the integro-differential equation into the frequency domain, solved for the phasor representation of v(t), and then converted it back to the time domain to find the final expression for v(t). The magnitude is positive, and the angle is in the specified range of -180° to 180°.