Solving Alternating
Current Circuits Using
Complex Numbers
Electromagnetism 1, PHYS1090
Dr Joachim Rose
Department of Physics and Astronomy
University of Leeds
EM1-L20-1
Overview
The plan for todays lecture
• RLC circuit
Calculation of current amplitude and phase
using complex numbers
• Complex impedances
RLC circuits and analogous DC circuits
• Summary
EM1-L20-2
RLC circuit and
complex numbers
EM1-L20-3
Differential equation for RLC circuit
Kirchhoff’s loop rule
L·
dI
Q
+ R · I + = Emax · cos (ωt)
dt
C
R
Use Q = (dQ/dt) dt =
dI
1
L·
+R·I + ·
dt
C
Z
R
I dt
I dt = Emax · cos (ωt)
Note that the right side of the above equation
is the real part of the complex function
Emax · eiωt = Emax · (cosωt + i sin ωt)
RLC circuit and complex numbers
EM1-L20-4
Complex differential equation
A very similar differential equation but for a
complex function z:
dz
1
L·
+R·z+ ·
dt
C
Z
z dt = Emax · eiωt
The real part of the solution z is the current I
I = Re(z)
As a solution for the differential equation try
z = I0 · eiωt
which then implies
dz
=iωz
dt
Z
1
z dt =
z
iω
Substituting into the differential equation gives
iωL z + R z +
1
Emax
z = Emax eiωt =
z
iωC
I0
RLC circuit and complex numbers
EM1-L20-5
Complex differential equation (continued)
Divide each term by z, write ωL = XL and (1/ωC) = XC ,
and use XC /i = iXC /i2 = −iXC then
iωL z + R z +
1
Emax
z = Emax eiωt =
z
iωC
I0
becomes
i(XL − XC ) + R =
Emax
I0
The complex constant I0 is then
I0 =
Emax
i(XL − XC ) + R
The denominator expressed in polar form is
q
i(XL − XC ) + R = (XL − XC )2 + R2 · eiδ = Z · eiδ
where Z is the impedance and δ the phase.
tan δ =
XL − XC
R
RLC circuit and complex numbers
EM1-L20-6
Current in a RCL circuit
Using the polar form the constant I0 is
Emax
Emax −iδ
I0 =
=
e
Z eiδ
Z
and the complex solution z is
z = I0eiωt =
Emax −iδ iωt
Emax i(ωt−δ)
e
e
=
e
Z
Z
The current I is thus given by
I = Re(z) =
Emax
cos (ωt − δ)
Z
RLC circuit and complex numbers
EM1-L20-7
Complex Impedances
EM1-L20-8
Generalized Ohms law
Differential equation for complex function z
dz
1
L·
+R·z+ ·
z dt = Emax · eiωt
dt
C
As a solution for the differential equation try
Z
z = I0 · eiωt
This gives
iωL z + R z +
Emax
1
z=
z
iωC
I0
Multiply by I0/z
1
I0 · iωL + R +
iωC
= Emax
Complex impedances: iωL, R, 1/iωC
Complex impedances
EM1-L20-9
Complex impedances
Resistor:
Zeq = R
Inductor:
Zeq = i ω L = i XL
Capacitor:
i
1
Zeq =
=−
= −i XC
iωC
ωC
Complex impedances
EM1-L20-10
Complex impedance for RCL circuit
Complex impedance:
Zeq = iωL + R +
1
iωC
= i XL + R − i XC
Complex impedances
EM1-L20-11
Current, impedance and phase
Current
I=
Emax
cos (ωt − δ)
Z
Phase
XL − XC
tan δ =
R
Impedance
Z=
Complex impedances
q
(XL − XC )2 + R2
EM1-L20-12
Summary
• Resistor
Zeq = R
• Inductor
Zeq = i ω L = i XL
• Capacitor
1
i
Zeq =
=−
= −i XC
iωC
ωC
• Complex equivalent impedance:
Add up the complex impedances in a circuit
with external generator. Similar to finding
equivalent resistance for a DC circuit.
• Current
I=
Emax
cos (ωt − δ)
Z
Reading: Tipler, chapter 31, page 980-981
Summary
EM1-L20-13