2.2.4 RCL Circuit

Chapter 2.2.4 RCL Circuit

Physics Lecture Notes – Phys 395 Electronics Book
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Physics Lecture Notes – Phys 395 Electronics Book

  • CHAPTER 2. ALTERNATING CURRENT CIRCUITS27where Q(t =0)= Q0 is the initial charge on the capacitor and φ is an arbitrary phaseconstant. Considering the cases of Q0 = Qmax,gives φ = 0. The angular frequency ω istotally determined by the other parameters of the circuitω2 =1/(LC)(2.20)and ωr≡ 1/√LC is the natural or resonance frequency of the circuit.We can also solve for the current and voltage across the capacitorQ(t)=Q0 cos(ωrt),(2.21)I(t)=dQdt=−Q0ωr sin(ωrt),(2.22)=−I0 sin(ωrt)= I0 cos(ωrt + π/2),and(2.23)V (t)=Q(t)C=Q0Ccos(ωrt)= V0 cos(ωrt).(2.24)Notice that unlike the transient current and voltage responses of the RC and RL circuits,the LC circuit oscillates. The energy in the circuit is shared back and forth between theinductor and capacitor.2.2.4RCL CircuitLets now consider the case of all three passive circuit elements in series, as in figure 2.4.RCLFigure 2.4: RCL circuit.Applying Kirchoff’s law around the loop and using I = dQ/dt givesLdIdt+ RI +QC=0and(2.25)Ld2Qdt2+ RdQdt+QC=0.(2.26)The solution will not only depend on the initial conditions but also the relative values ofR, C and L.There are three possible solutions: