Announcement

1
Lecture 11

• Announcement
• Midterm 1 will cover Chap 1 to Chap 5.3
• HW schedule is changed due on Thursdays instead
• HW 5 will be due on Thursday 3/10 instead (posted
  on line on 3/3)
• Prof. Chang will be out of town 2/24-25.
• Wed 11-12 OH will be kept
• There will be no OH on Friday
• Prof. Roger Howe will substitute.
• OUTLINE
• Phasors
• Complex impedances
• Circuit analysis using complex impdenaces
• Reading Chap 5.1-5.3 (skip 5.4-5.7)

2
Complex Numbers (1)

• x is the real part
• y is the imaginary part
• z is the magnitude
• q is the phase

• Rectangular Coordinates
• Z x jy
• Polar Coordinates
• Z z ? q
• Exponential Form

3
Complex Numbers (2)
Eulers Identities
Exponential Form of a complex number
4
Arithmetic Operations of Complex Numbers

• Add and Subtract it is easiest to do this in
  rectangular format
• Add/subtract the real and imaginary parts
  separately
• Multiply and Divide it is easiest to do this in
  exponential/polar format
• Multiply (divide) the magnitudes
• Add (subtract) the phases

5
Phasors

• Assuming a source voltage is a sinusoid
  time-varying function
• v(t) V cos (wt q)
• We can write
• Similarly, if the function is v(t) V sin (wt
  q)

6
Complex Exponentials

• We represent a real-valued sinusoid as the real
  part of a complex exponential.
• Complex exponentials
• provide the link between time functions and
  phasors.
• make solving for AC steady state an algebraic
  problem.
• Phasors allow us to express current-voltage
  relationships for inductors and capacitors much
  like we express the current-voltage relationship
  for a resistor.
• A complex exponential is the mathematical tool
  needed to obtain this relationship.

7
Capacitor Impedance (1)
8
Capacitor Impedance (2)
Phasor definition
9
Inductor Impedance

i(t)
v(t)
L
-

• V jwL I

10
Phase
Voltage
inductor current
capacitor current
11
Phasor Diagrams

• A phasor diagram is just a graph of several
  phasors on the complex plane (using real and
  imaginary axes).
• A phasor diagram helps to visualize the
  relationships between currents and voltages.
• Capacitor I leads V by 90o
• Inductor V leads I by 90o

12
Impedance

• AC steady-state analysis using phasors allows us
  to express the relationship between current and
  voltage using a formula that looks likes Ohms
  law
• V I Z
• Z is called impedance.

13
Some Thoughts on Impedance

• Impedance depends on the frequency w.
• Impedance is (often) a complex number.
• Impedance allows us to use the same solution
  techniques for AC steady state as we use for DC
  steady state.

14
Example Single Loop Circuit
20kW


VC
1mF
10V ? 0?
-
-

• f60 Hz, VC?

How do we find VC? First compute impedances for
resistor and capacitor ZR R 20kW 20kW ? 0?
ZC 1/j (2pf x 1mF) 2.65kW ? -90?
15
Impedance Example
20kW ? 0?


VC
2.65kW ? -90?
10V ? 0?
-
-

• Now use the voltage divider to find VC

16
What happens when w changes?
20kW


VC
1mF
10V ? 0?
-
-

• w 10
• Find VC

17
Circuit Analysis Using Complex Impedances

• Suitable for AC steady state.
• KVL
• Phasor Form KCL
• Use complex impedances for inductors and
  capacitors and follow same analysis as in chap 2.

Phasor Form KVL
18
Steady-State AC Analysis

• Find v(t) for w2p 3000

19
Find the Equivalent Impedance
20
Change the Frequency

• Find v(t) for w2p 455000

21
Find an Equivalent Impedance
22
Series Impedance

• Zeq Z1 Z2 Z3

For example
Zeq jw(L1L2)
23
Parallel Impedance

• 1/Zeq 1/Z1 1/Z2 1/Z3

For example
24
Steady-State AC Node-Voltage Analysis
C
I1cos(wt)
I0sin(wt)
R
L

• Nodal analysis or mesh?
• What are the nodes (or meshes)?
• What happens if the sources are at different
  frequencies?