CHAPTER 2. ALTERNATING CURRENT CIRCUITS281. under damped (R2 < 4L/C): Ae−t/τ cos(ωt + φ),2. over damped (R2 > 4L/C): A1e−t/τ1 + A2e−t/τ2 , and3. critically damped (R2 =4L/C): (A1 + A2t)e−t/τ .RCL circuits have a variety of properties, especially when driven by sinusoidal sources,which will not be investigated here. My aim is simply to expose you to the area and get onto more interesting topics. Driven oscillating systems also appear in other areas of physicsand hopefully you will encounter them there. The detailed considerations lead to discussionson resonance and quality-factor Q.2.3Sinusoidal Sources and Complex ImpedanceWe now consider current and voltage sources with time average values of zero. We will useperiodic signals but the observation time could well be less than one period. Periodic signalsare also useful in the sense that arbitrary signals can usually be expanded in terms of aFourier series of periodic signals. Lets start withv(t)=V0 cos(ωt + φV )and(2.27)i(t)=I0 cos(ωt + φI ).(2.28)Notice that I have now switched to lowercase symbols. Lowercase is generally used for ACquantities while uppercase is reserved for DC values.Now is the time to get into complex notation since it will make our discussion easier andis encountered often in electronics. The above voltage and current signals can be writtenv(t)=V0ej(ωt+φV )and(2.29)i(t)=I0ej(ωt+φI ).(2.30)To be cleaver we will deﬁne one EMF in the circuit to have φ = 0. In other words, wewill pick t = 0 to be at the peak of one signal. The vector notation is used to remind usthat complex numbers can be considered as vectors in the complex plane. Although not socommon in physics, in electronics we refer to these vectors as phasors. Hence you shouldnow review complex notation.The presence of sinusoidal v(t)or i(t) in circuits will result in an inhomogeneous diﬀeren-tial equation with a time-dependent source term. The solution will contain sinusoidal termswith the source frequency.The extension of Ohm’s law to AC circuits can be written asv(ω, t)= Z(ω)i(ω, t),(2.31)where ω is the source frequency. Z is a generalized resistance referred to as the impedance.We can cancel out the common time dependent factors to obtain