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Transcript
```AC Circuits (quoting from E. M. Purcell, Electricity and Magnetism, 2nd. Ed, Sect. 8.3)
1. An alternating current or voltage can be represented by a complex number.
2. Any one branch or element of the circuit can be characterized, at a given frequency, by the
relation between the voltage and the current in that branch.
Adopt the following rule for the representation:
1. An alternating current I o cos(ωt + φ ) is to be represented by the complex number I o e iφ , that
is, the number whose real part is I o cos φ and whose imaginary part is I o sin φ .
2. Going the other way, if the complex number x + iy represents a current I, then the current as a
function of time is given by the real part of the product ( x + iy ) e iωt .
Circuit Element
Resistor
Inductor
Capacitor
I=VY
V=IZ
Impedance Z
1
R
1
−i
=
iωL ωL
iω C
R
iωL
1
−i
=
iωC ωC
Note that impedances add for elements in series, since voltages add for elements in series
REVIEW OF COMPLEX NUMBERS
1
i = − 1 , = −i , e iθ = cos θ + i sin θ
i
z = x + iy, z* = x − iy, z = zz * = x 2 + y 2
y
x
A + iB
A − iB
z=
, z* =
C + iD
C − iD
z = z e iθ , tan θ =
```
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