3.4 Complex Frequencies and the s-Plane

Chapter 3.4 Complex Frequencies and the s-Plane

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

  • CHAPTER 3. FILTER CIRCUITS49Example: Write the transfer function H(jω) for the network in figure 3.4 andfrom it find:1. the corner frequency,Treating the circuit like a voltage divider, the transfer function isH(jω)=1/(jωC)1/(jωC)+ jωL=11− ω2LC.(3.32)For ω≈ 0⇒ H(jω)→ 1.For large ω⇒ H(jω)→ −1ω2LC.For the corner frequency 1=1ω2C LC ⇒ ω2C =1LC.ThereforeωC =1√LC=1(1× 10−6)1/2=1× 103rad/s.(3.33)1 microF1 HFigure 3.4: Four-terminal network without resistance.2. the value of|H| at the corner frequency.At the corner frequencyH(jωC)=11− 1→∞.(3.34)3. How many degrees of phase shift are introduced by this network just belowand just above the corner frequency?Since H(jω) is always real there is no phase shift.3.4Complex Frequencies and the s-PlaneWe will now consider s-plane techniques. Not because we will use them, but more to under-stand some of the common electronics terminology.We can enhance the usefulness of the transfer function H(jω) by transforming to acomplex frequency. Define the complex variable s such thats = σ + jω,(3.35)where σ is an inverse time constant.