Ac steady state analysis

AC Power Engineering Review
Presented by
Brenden Motte, Alissa Olson & Long Pham

Why not just use DC?

New York City
utility lines,
1980

AC vs DC
• DC was first use for electricity transmission.
• Then “The war of current.” NY, 1880s:

▫ AC Vs DC ~ Testla Vs Edison ~ Employee Vs Boss.
▫ Edison’s least favorite of Tesla’s “impractical” ideas was the concept of using alternating current (AC) to bring
electricity to the people. Edison insisted that his own direct current (DC) system was superior, in that it maintained
a lower voltage from power station to consumer, and was, therefore, safer. “Direct current is like a river flowing
peacefully to sea, while alternating current is like a torrent rushing violently over a precipice.” - Tom
▫ Tesla insisted that he could increase the efficiency of Edison’s prototypical dynamos (by changing from DC to AC
generator). Edison promised him $50,000 if he succeeded. Tesla worked around the clock for several months and
made a great deal of progress. When he demanded his reward, Edison claimed the offer was a joke, saying, “When
you become a full-fledged American, you will appreciate an American joke.” Ever prideful, Tesla quit, and spent the
next few months picking up odd jobs across New York City. Nikola Tesla: ditch digger.
▫ Tesla eventually raised enough money to found the Tesla Electric Light Company, where he developed several
successful patents including AC generators, wires, transformers, lights, and a 100 horsepower AC motor. The most
significant contribution to the early success of ac was his patenting of the poly-phase ac motor in 1888.
▫ Always more of a visionary than a businessman, Tesla ended up selling most of his patents (for the healthy but
finite sum of $1 million) to George Westinghouse, an inventor, entrepreneur, and engineer who had himself been
feuding with Edison for years. Their partnership, made the eventual popularizing of AC that much more bitter for
Edison.

• AC electrical energy has been the most convenient form of energy to be generated, transmitted, and
distributed.

DC vs AC
• While AC was perfectly adequate for the conditions of much of the 19th & 20th
century, the needs of the 21st century are showing its limits.
• We are facing a revolution in the way our electricity is produced and used. More
and more electricity is being generated from renewable sources of energy in remote
areas: hydropower plants (mountains far from urban centers), wind farms (people
tend not to live in windy areas), offshore wind farms (have higher capacity factor,
better alignment with peak demand), etc. DC is the only technology that allows
power to be transmitted economically over very long distances, and DC is the type
of power produced by photovoltaic panels.
• On the consumer side, more and more equipment runs on DC: computers, cell
phones, LED lights, CFLs, high efficiency motor drives which found in new HVAC
systems, industrial, etc. ABB estimates the savings from using DC instead of AC in
buildings could be in the order of 10 to 20 percent.
Reference: http://www.abb.com/cawp/seitp202/c646c16ae1512f8ec1257934004fa545.aspx

AC Power: Today’s topics
• Part 1: AC Circuits Analysis
▫ Steady-state sinusoidal response, Impedance Model.

• Part 2: AC Power Analysis
▫ Power in AC circuits, Power Factor, Power factor corrections, Poly-Phase
Circuits.

• Part 3: Elements of the AC Power Systems
▫ Basics elements comprising the AC grid: Power Generator, Transformers,
Capacitor Banks, Transmission circuits, etc.

Part 1: AC Circuits
• Complex Number: A Quick Review
• Resistor, capacitor, Inductor: A practical explanation
• Wave form of a signal & characteristic of Sinusoids signal
▫ Period, frequency, radian frequency (or angular frequency), phase leading
& lagging, etc.

•
•
•
•
•

Sinusoidal response of RC Network: usual approach
Phasor domain analysis or Frequency domain analysis
Sinusoidal response of RC Network: Impedance model
Using Impedance Model to solve AC circuits
Examples

Complex Number: A Quick Review
Rectangular form

Polar form

Exponential form

Im


A

b
|A|


0

Re
a

Note: Most calculator can
do complex arithmetic &
convert between
Rectangular form to/from
polar form

Complex Number
Example:

• Evaluate the following complex numbers
into rectangular form:
a.

[(5  j2)( 1  j4)  5 60 o ]

b.

10  j5  340 o
 10 30 o
 3  j4

Solution:
a. –15.5 + j13.67
b. 8.293 + j2.2

Using complex number Solving Trigonometric nightmare
• Problem:
v1  t   20cos t  45  V

v2  t   10sin t  60  V

Find vs  v1  v2  ?

• Solution: V1  20  45 V
V2  10  30 V


 
Vs  V1  V2

Why can we do this?

 20  45  10  30
 14.14  j14.14  8.660  j5
 23.06  j19.14
 29.97  39.7 V
Complex number doesn’t make life
more complicated but more simple!

Resistors
• Purely resistive loads are almost
non-existent in AC networks. AC
network loads consist primarily out
of inductive and to lesser extent
capacitive
loads,
both
in
combination with resistive loads. It
is therefore VERY important to
understand the characteristics of
capacitors and inductors in an AC
environment.
• Inductors and capacitors are energy
storage elements. The difference
lies in how and the type of energy
that is stored by each.

Inductors
• When current flows, a magnetic field is
created.
• Energy provided by the current is stored in
the magnetic field. The stronger the current
or higher the number of coils (higher
inductance), the greater the stored amount
of energy will be.
• Inductor does however have a limit as to the
amount of energy it can store and the rate
at which it can store the energy.
(~Saturated)
• Energy stored in inductor’s magnetic field
can be retrieved.
• Inductance is represented by: L and
measured in Henry: (H)

It is clear that an inductor stores energy in the
form of a magnetic field created by current
flowing through the inductor coil.

Inductors

Large 50 MVAR three-phase iron-core loading
inductor at German utility substation

Inductors characteristics
• When you pushing a car, it will slowly increase it’s velocity (~kinetic
energy). Similarly, when you apply a voltage across an inductor, it will
slowly increase it’s current (~magnetic energy). Inductor’s current
cannot change instantly just as car’s velocity. (voltage can change
instantly)
• When you stop pushing, the car will continue to run. Similarly, when you
stop applying voltage, the inductor’s current will continue to flow.
• What happens if something block the car from running? What happens
if something block the inductor’s current from flowing?
• The car will try its best to run until it can’t run anymore; all the kinetic
energy stored in the car will dissipate to the blocker. A big force will
created by the car’s effort to try to maintain it’s velocity.
• Similarly, the inductor will try its best to flow the current until it can’t
run anymore; all the magnetic energy store in the field will dissipate to
the blocker. A big voltage is created by the inductor in its effort to try to
maintain it’s current.
• Mathematically, the relation between inductor’s current & voltage is the
same as the relation between car’s velocity & force:

Capacitors
• When apply a voltage to a capacitor, a electrical
field is created between the capacitor’s plates.
• Energy provided by a voltage difference is stored
in the electric field. The higher the voltage or
bigger capacitor (higher capacitance), the
stronger the electric field and more energy is
stored inside the capacitor.
• Capacitor does however have a limit as to the
amount of energy it can store.
• Energy stored in capacitor’s magnetic field can be
retrieved.
• Capacitance is represented by: C and measured
in Farad: (F)

It is clear that a capacitor stores energy in the
form of a electric field created by potential
difference across the capacitor plates.

Waveform of a signal
• A waveform is the shape or form of a signal
against time, physical medium or an abstract
representation.
• Common periodic waveforms are: Sine, Square,
Triangle, etc.
Example:
▫
▫
▫
▫
▫
▫

Electrocardiogram: 1Hz
Main power: 50Hz
Audio signal: 20Hz
Wifi signal: 2.4Ghz
GPS signal L1 band: 1575.42Mhz
GLONASS L1 signal: 1602Mhz

• Sinusoids’ important because signals can be
represented as a sum of sinusoids. Response to
sinusoids of various frequencies -- aka frequency
response -- tells s a lot about the system.

Sinusoidal response of RC Network

Usual approach
Current flow out of node i equal 0:

That was easy!

How to solve DE
•

Solving differential
equations is actually
quite easy –commonly
guesswork and applying
patterns

•
(Homogeneous solution for any linear constantcoefficient ODE is always of this form.)

•

It WORK!
But trig. Nightmare!
(is there an easier way to find Vp?)

Sinusoidal Steady State (SSS)
In AC Power, we are usually
interested only in the particular
solution for sinusoids, i.e. after
transients have died.

Phasor Analysis
“I have found the equation that will enable us to
transmit electricity through alternating current over
thousands of miles. I have reduced it to a simple
problem in algebra.”
Charles Proteus Steinmetz, 1893
The use of complex numbers to solve ac circuit problems was first done by German-Austrian mathematician and
electrical engineer Charles Proteus Steinmetz in a paper presented in 1893. He is noted also for the laws of hysteresis
and for his work in manufactured lighting.
Steinmetz was born in Breslau, Germany, the son of a government railway worker. He was deformed from birth
and lost his mother when he was 1 year old, but this did not keep him from becoming a scientific genius. Just as his
work on hysteresis later attracted the attention of the scientific community, his political activities while he was at the
University at Breslau attracted the police. He was forced to flee the country just as he had finished the work for his
doctorate, which he never received. He did electrical research in the United States, primarily with the General Electric
Company. His paper on complex numbers revolutionized the analysis of ac circuits, although it was said at the time
that no one but Steinmetz understood the method. In 1897 he also published the first book to reduce ac calculations
to a science.

Phasor: A useful way to think about sine wave
• A phase vector, or phasor is a complex
number that represents the amplitude
and phase of a sinusoidal signal whose
amplitude, frequency, and phase are timeinvariant.
• Frequency is common to all signal in our
analysis. Phasor allow this common to be
factored out, leaving just the amplitude
and phase features. The result is that
trigonometry reduces to algebra, and
linear differential equations become
algebraic ones.
• Phasor is a complex number. So it can be
represent in one of the three forms of
complex number: Rectangular, Polar,
Exponential.

A phasor can be considered
a vector rotating about the
origin in a complex plane

The sum of phasors
as
addition
of
rotating vectors

Time Domain Vs Phasor Domain representation of
sinusoidal signals
Time domain

Phasor domain

Example: Convert SS between Time Domain & Phasor Domain
Time domain

Phasor domain

Sinusoidal Signals: Arithmetic Comparisons
Time domain
( )=
−

Adding & Scaling trigonometry

Derivative and Integral of sin &cos

Phasor domain
= +

Adding & scaling complex numbers

Example: Solving DE using Phasor Domain
Find i(t):

di
4i  8 idt  3  50 cos(2t  75)
dt
- Convert to Phasor domain
- Solve algebraic equation
- Convert back to Time domain

Answer: i(t) = 4.642cos(2t + 143.2o) A

Voltage / Current relationships
Time domain

R
L
C
Complicated!

Phasor domain

The Impedance Model in Phasor domain

Back to the RC example…
Time domain

DONE!

Phasor domain

Calculator

Example
Time domain

Phasor domain

Another example of Sinusoidal Steady state response (SSS)
Remember, we want only the Steady-state response to sinusoid: SSS
SSS: Sinusoidal Steady-state response

Calculator

Solving AC Circuits The Easy Way

Impedance and Admittance
• The impedance Z of a circuit is the ratio of the
phasor voltage V to the phasor current I,
measured in ohms Ω.
V VM  v VM
Z 

( v   i ) | Z |  z  R  jX
I
I M  i I M

where R = Re(Z) is the resistance and X = Im(Z) is the
reactance. Positive X is for L and negative X is for C.

• The admittance Y is the reciprocal of impedance,
measured in siemens (S).

1 I
Y 
Z V

AC Equivalent Circuits
Thévenin and Norton equivalent circuits apply in AC analysis
• Equivalent voltage/current is complex and frequency dependent
I
Thévenin Equivalent

Source

+
V
–

Load

Norton Equivalent

I

I

VT(jω)
+
–

ZT

IN(jω)

+
V
–

Load

+
ZN

V
–

Load

Thevenin and Norton Transformation

Thevenin transform

Norton transform

Computation of Thévenin and Norton Impedances:
1.
2.

Remove the load (open circuit at load terminal)
Zero all independent sources



3.

Voltage sources
Current sources

short circuit (v = 0)
open circuit (i = 0)

Compute equivalent impedance across load terminals (with load removed)

Z3

Z1
+
–
Vs(jω)

ZL

Z2

Z3

Z1

a

Z4

a
ZT

Z2

Z4
b

NB: same procedure as equivalent resistance

b

Computing Thévenin voltage:
1.
2.
3.


4.

Remove the load (open circuit at load terminals)
Define the open-circuit voltage (Voc) across the load terminals
Chose a network analysis method to find Voc
node, mesh, superposition, etc.

Thévenin voltage VT = Voc

Z3

Z1
+
–
Vs(jω)

+
–
Vs(jω)

Z2
Z4
b

Z3

Z1

a

Z2
Z4

a
+
VT
–
b

Computing Norton current:
1.
2.
3.


4.

Replace the load with a short circuit
Define the short-circuit current (Isc) across the load terminals
Chose a network analysis method to find Isc
node, mesh, superposition, etc.

Norton current IN = Isc

Z3

Z1

+
–
Vs(jω)

+
–
Vs(jω)

Z2
Z4
b

Z3

Z1

a

a
IN

Z2
Z4
b

• Example: find the Thévenin equivalent
• ω = 103 rads/s, Rs = 50Ω, RL = 50Ω, L = 10mH, C = 0.1uF

Rs

vs(t)

+
~
–

L

C

RL

+
vL
–


Rs

1.

L

Note frequencies of AC sources

Only one AC source: ω = 103 rad/s

vs(t)

+
~
–

C

RL

+
vL
–


Rs

vs(t)

+
~
–

1.
2.

L

Convert to phasor domain

ZL
Zs

C

RL

+
vL
–

Vs(jω)

+
~
–

ZC
ZLD


ZL

ZL

Zs

Vs(jω)
+
~
–

Zs

ZC
ZLD

1.
2.
3.

Find ZT
•
Remove load & zero
sources

Z T  Z S  Z C || Z L
ZC

 RS 

( jL)(1 / jC )
( jL)  (1 / jC )

 RS  j

L
1   2 LC

 50  j 65.414
 82.330.9182


ZL

ZL

Zs

Vs(jω)
+
~
–

4.

Zs

Vs(jω)

ZC
ZLD

+
~
–

ZC

1.
2.
3.

+
VT(jω)
–

Find ZT
•
Remove load & zero
sources
Find VT(jω)
•
Remove load

NB: Since no current flows in the
circuit once the load is removed:

VT  VS
ZT  82 .330.9182

ZT

ZL
Zs
VT(jω)
Vs(jω)
+
~
–

+
~
–

ZC

ZLD

ZLD

VT  VS

ZT  82 .330.9182

Thevenin and Norton Equivalent Circuits

Thevenin and Norton Equivalent Circuits
-j1

+

+

j2

AC

60 V

-j1

-j1

+
VOC
-

AC

20 A 60 V

-

-

20 A

I0

VOC  6  2(1  j )  8  j 2

-j1

8  j2
8  j 2 8.246  14.04
I0 


1  j1  j 2  2 3  j1
3.16218.43
I 0  2.608  32.47

+

8  j2

j2

AC

-

I0

ZTH  1  j1

Obtain current Io using Norton’s theorem at terminals a-b.

Obtain the Thevenin equivalent circuit at
terminals a-b:

Super position for AC Circuits
• Usual procedures for DC circuits apply.
• However, phasor transformation must be carefully carried out if the
circuit has sources operating at different frequencies
• A different phasor circuit for each source frequency because impedance
is a frequency-dependent quantity.
• For sources of different frequencies, the total response must be added
in the time domain
• DO NOT ADD INDIVIDUAL RESPONSES IN THE PHASOR DOMAIN IF THE
SOURCES HAVE DIFFERENT FREQUENCIES.

Circuit example
Time domain

Phasor domain

Ac steady state analysis

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