RCL Circuit analysis

Question

Figure 4: Representative circuit of an electric guitar's pickup.

Consider a familiar electric guitar or bass. Below each string, there is a permanent magnet (sometimes two) called pole pickups. The pole pickups magnetize the metal strings. The pickup has a circuit modeled in figure 4 When a string is plucked, the string vibrates, and its magnetic field fuctuates. These fluctuations act as an external AC source for the pickup circuit. Depending on the values of Ro, RI, C, L, the response of the circuit can be adjusted to pick up higher or lower frequencies. In the following, consider that the effect of one vibrating string on the circuit can be characterized by VAc(t) = Vo cos(St + ф).

TLTR: Find the total power dissipated by the circuit over one period of the AC source and prove that there's a value of 1 that maximizes it.

You should go through all the following steps. As you already know, it's always a good idea to check units.

(a) Find the differential equation that governs the charge in the capacitor. Hint:

It is not a second order differential equation. (3 pts)

(b) Find the differential equation that governs the current across the inductor.

Hint: It is not a second order differential equation. (3 pts)

(c) Solve the steady state of the charge in the capacitor: Qc(t) = A cos It +

B sin It, with A, B some constants that you need to find. (4 pts)

(d) Find the instantaneous power dissipated in the resistor of the conductor, .e., in Ro. (2pts)

(e) Find the average power dissipated in the resistor of the capacitor over one period of the AC source. Let's call it < PRe>. (3 pts)

(f) Sketch < PRo > for different values of S while keeping the other parameters constant. Does < PRo > increase or decrease with frequency? (2 pts)

(g) Solve the steady state of the current in the inductor: Is (t) = C cos ft +

D sin St, with C, D some constants that you need to find. (4 pts)

Answer 1

I have attachment file separately. You can ask