Ph272- General Physics II (Electricity and Magnetsism)

Lecture 22: Ch. 31 - AC - Circuits; Complex Impedance

Key Topics

Overview

• In the previous lecture, you see that using cosine function for the LCR circuit involve some messy algebra.
• We will solve the DE using complex function.
• We will introduce complex impedance for capacitor and inductor.
• The rules for combining impedance are exactly like those for resistance, excpet these are complex numbers.
• The magnitude of the impedance give you the "resistance" and the angle tells you the phase shift of the current relative to the Emf.

Review Complex number

• Click here for a brief review of complex numbers (Read this before lecture).

Complex impedance

Note: The impedance is minimum at resonance, i.e. w = natural frequency and the |Z| = R.

• I(max)=EMF/|Z|. Plot I(max) vs. w. How big is the "width" of the resonance?

Analog between L,C,R and mechanical system

• A series LCR circuit is analogous to a mass-spring system with friction

  mass-spring system	LCR circuit
  m	L
  k (spring constant)	1/C
  x(t) (displacement from equilibrium)	Q(t)
  P.E. = 1/2 k x**2	1/2 Q**2/C
  K.E. = 1/2 m v**2	1/2 L I**2
  natural freq. = sqrt(k/m)	natural freq. = sqrt(1/LC)
  friction or viscous force	RdI/dt
  If there is no friction (and there is no external force), then there is conservation of energy: 1/2 k x(t)**2 + 1/2 m v(t)**2 = constant =1/2 k x(max)**2 = 1/2 m v(max)**2	If R=0 and EMF =0, then 1/2 Q(t)**2/C + 1/2 L I(t)**2 = constant =1/2Q(max)**2/C = 1/2 L I(max)**2

Review

• Write a brief review here

Go to top of page

Ph272 homepage