in figure 2. The external source is an AC source with VAC(t) R, L, Vo, , o are arbitrary positive constants. R C 0000 = Vo cos(nt + p). Figure 2: 'R+LC' circuit with an external source. (a) Find the differential equation that describes the current in the inductor. You can use all what you learned about this circuit in PS2 but show again your work, including Kirchhoff equations. Remember to have sum of potentials to set the differential equation equal to V(t). Hint: just the differential equation. Do not solve it yet. (b) Find IL(t) ignoring the transient behavior. That is, get the steady response to the external source. Hint: one way is to set IL(t) = A cos Ωt + B sin Qt, with A, B some constants. Then, find A, B by solving the differential equation of IL(t). Some suggestions from class: i/ do not use a coefficient of 1 in the i term. Use instead (RLC)Ï. The algebra will have less fractions. ii/ Beware that V(t) has an arbitrary phase: . Use cos(Nt+) = cos Nt cos d- sin Nt sin p. iii/ at the end, check that A, B have units of current! (c) Given IL(t), find the voltage across the coil inductor VL(t)? Leave the answer in terms of A, B, or whichever labels you used in the previous questions. That is, you can, but it is not required to substitute the explicit expressions of A, B found in the previous question.

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Thevenin Theorem

Any linear circuit, no matter how complex, maybe simplified to an equivalent circuit utilizing only a single voltage source and series resistance coupled to a load, according to Thevenin's Theorem. The superposition principle says that all foundational equations must be linear, and the notion of "linear" is still the same (i.e., there should not be any exponentials or roots). This is particularly the case when operating with passive elements like resistors, inductors, and capacitors. However, some components (particularly some gaseous and semiconductor devices like tunnel diode, Schottky diode, transistors like BJT, MOSFET) are nonlinear, meaning their current resistance fluctuates with voltage and current. As a result, circuits made up of these parts are known as nonlinear circuits.