We are presented with a familiar first-order circuit: a capacitor driven by a Thevenin source. Now we have a sinusoidal drive that starts at t=0. so the differential equation is
RCdv(t)dt+v(t)=vI(t)=Vicos(ωt)u(t)
and we are given the initial state v(0−)=0.
We know how to solve this problem in the abstract.
To solve the given differential equation, RC dv(t)/dt + v(t) = Vi cos(ωt) u(t), with the initial condition v(0-) = 0, we can follow these steps:
1. Write the differential equation in standard form: dv(t)/dt + (1/RC) v(t) = (1/RC) Vi cos(ωt) u(t).
2. Find the integrating factor (IF): Multiply both sides of the equation by e^((1/RC) t). This will transform the left-hand side into the result of applying the product rule for differentiation and allow us to simplify the equation.
3. Multiply the equation by the integrating factor e^((1/RC) t): e^((1/RC) t) dv(t)/dt + (1/RC) e^((1/RC) t) v(t) = (1/RC) Vi e^((1/RC) t) cos(ωt) u(t).
4. Simplify the left-hand side by recognizing that d(e^((1/RC) t) v(t))/dt = e^((1/RC) t) dv(t)/dt + (1/RC) e^((1/RC) t) v(t) due to the product rule.
5. Rewrite the equation: d(e^((1/RC) t) v(t))/dt = (1/RC) Vi e^((1/RC) t) cos(ωt) u(t).
6. Integrate both sides with respect to t: ∫d(e^((1/RC) t) v(t))/dt dt = ∫(1/RC) Vi e^((1/RC) t) cos(ωt) u(t) dt.
7. Integrate the left-hand side and simplify using the fundamental theorem of calculus: e^((1/RC) t) v(t) - v(0-) = (1/RC) ∫Vi e^((1/RC) t) cos(ωt) u(t) dt.
8. Apply the initial condition v(0-) = 0: e^((1/RC) t) v(t) - 0 = (1/RC) ∫Vi e^((1/RC) t) cos(ωt) u(t) dt.
9. Simplify further: e^((1/RC) t) v(t) = (1/RC) ∫Vi e^((1/RC) t) cos(ωt) u(t) dt.
10. Solve the integral on the right-hand side: Determine the integral of cos(ωt) u(t) with respect to t to get an expression in terms of t.
11. Substitute the result of the integral back into the equation: e^((1/RC) t) v(t) = (1/RC) [integral_result].
12. Solve for v(t) by dividing both sides of the equation by e^((1/RC) t): v(t) = (1/RC) [integral_result] / e^((1/RC) t).
13. Simplify further if needed.
These steps will give you the solution to the given differential equation for the capacitor driven by a Thevenin source with a sinusoidal input.