2.4 Resonance and the Transfer Function

Chapter 2.4 Resonance and the Transfer Function

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

  • CHAPTER 2. ALTERNATING CURRENT CIRCUITS33Taking the real and imaginary components givesRe[Z]=RL/C− R/(ωC)[ωL− 1/(ωC)]R2 +[ωL− 1/(ωC)]2(2.65)=R/(C2ω2)R2 +[ωL− 1/(ωC)]2,(2.66)Im[Z]=−R2/(ωC) − L/C[ωL− 1/(ωC)]R2 +[ωL− 1/(ωC)]2.(2.67)The inverse tangent of the ratio of the imaginary to real parts isφ=tan−1−R2/(ωC) − ωL2/C + L/(ωC2)R/(C2ω2)(2.68)=tan−1−R2Cω − CL2ω3 + LωR(2.69)=tan−1ωR(L− CR2− CL2ω2) .(2.70)There is a resonance at ωL− 1/(ωC)=0⇒ ω =1/√LCand henceφres =tan−11R√LC(L− CR2− L)(2.71)=tan−1−RCL(2.72)=− tan−1RCL(2.73)=− tan−1(10−2)(2.74)≈ 0.(2.75)At ω =105 rad/s.φ=tan−11051021− 10−8(102)2− 10−8(1)2(105)2(2.76)=tan−1[103(1− 10−4− 102)](2.77)≈ tan−1(−105)(2.78)≈−π/2.(2.79)2.4Resonance and the Transfer FunctionLets now consider putting a sinusoidal source in our series RCL circuit and consider thevoltage across one of the circuit elements. The resistor for example in figure 2.7