Question

In a semi-conductor factory, an engineer is required to analyse the functionality of electric circuit boards. The circuit consists of a switch, an electromotive force E (usually supplied by a battery or generator), a resistor R, an inductor L, and a capacitor C. If the charge (Q) on the capacitor at time t is Q(t), then the current (I) is the rate of change of charge with respect to t, i.e., I(t)= dt/dQ. The electric circuits can be represented as second-order linear differential equation with constant coefficients as follow : L d^2Q/dt^2 + R dq/dt + 1/c Q = E(t). A series circuit is given to the engineer to do the analysis. Given that the circuit contains a resistor with R=24Ohm(Ω), an inductor with L=2Henry(H), and a capacitor with C= 0.005 Farad (F). The engineer needs to determine a) the charge at time t,Q(t) when the switch if off and without battery supply. b) the charge at time t,Q(t) when the switch if on and with a 12-Volt battery supply i. using the method of Undetermined Coefficients; and ii. using the method of Variation of Parameters. c) the current at time t,I(t) based on the charge with battery supply in question (b) above. d) the current at time t,I(t) if given that when the electric circuit has initial charge with Q=0.001 Coulomb (C) and the initial current with I=0 Ampere (A). (5 marks

Answer 1

a) When the switch is off and there's no battery supply, the electric potential difference, or electromotive force E(t), is zero. That means the right-hand side of the given differential equation is zero. Therefore, the solution to the homogeneous differential equation represents the charge on the capacitor:

L d²Q/dt² + R dQ/dt + 1/C Q = 0

In this case, since there's no initial charge or current supplied, Q(t) = 0 for all t.

b) When the switch is turned on and a 12-Volt battery is connected:

i. The method of Undetermined Coefficients:

We can solve this by proposing a particular solution that has the same form as the non-homogeneous term, E(t). As E(t) = 12 volts is a constant, we propose Q(t) = A as a constant.

After substituting Q(t) = A into the equation, we would be able to find the value of A, which would give us the particular solution. The general solution would then be the sum of this particular solution and the solution to the homogeneous equation (obtained from part (a)).

ii. The method of Variation of Parameters:

In this method, we would make use of the solutions of the homogeneous differential equation. After finding these, we would propose a solution for the non-homogeneous differential equation in terms of these solutions, and a pair of functions (u and v) to be determined. We then substitute this proposal into the differential equation to obtain a set of two new first-order differential equations for u and v.

c) Once we've found the charge Q(t) in part (b), we can find the current I(t) by differentiating Q(t), as I(t) = dQ/dt.

d) With the given initial conditions (Q = 0.001 C, I = 0 A), we can substitute these into the general solution and its derivative obtained in part (b). We would then solve the resulting system of two equations to find the constants involved, allowing us to determine the specific solution for these initial conditions.

Step-by-step explanation:

Complex question. Answer depends on data provided and format of equations provided.