### Lecture 4

```Power System Fundamentals
EE-317
Lecture 3
06 October 2010
Aims
 Finish Chapter 1 –
Real and Reactive Power
Power Triangle
 Chapter 2 – Three Phase Circuits
 Chapter 3 – Transformers
Sine Wave Basics (Review)

RMS – a method for computing the effective value of a time-varying e-m
wave, equivalent to the energy under the area of the voltage waveform
.
Real, Reactive and Apparent
Power in AC Circuits
in DC circuits: P=VI but…= in AC circuits: average
power supplied to the load will be affected by the phase
angle  between the voltage and the current.
 If load is inductive the phase angle (also called
impedance angle) is positive; (i.e, phase angle of
current will lag the phase angle of the voltage) and the
load will consume both real and positive reactive power
 If the load is capacitive the impedance angle will be
negative (the phase angle of the current will lead the
phase angle of the voltage) and the load will consume
real power and supply reactive power.

Impedance Angle, Current Angle
& Power
 Inductive loads  positive impedance angle
current angle lags voltage angle
 Capacitive loads  negative impedance angle
 Both types of loads consume real power
 One (inductive) consumes reactive as well while
the other (capacitive) supplies reactive power
Useful Equations
 First term is average or Real power (P)
 Second term is power transferred back and forth
between source and load (Reactive power- Q)
More equations

Real term averages to P = VI cos (+)
Reactive term averages to Q = VI sin (+/-)

Reactive power is the power that is first stored and then released

in the magnetic field of an inductor or in the electric field of a capacitor
 Apparent Power (S) is just = VI

V = IZ
Substituting…
P = I2Z cos 
 Q = I2Z sin 
 S= I2Z

Since… Z = R + jX = Z cos  + jZ sin 
 P = I2R and Q = I2X

Complex Power and Key
Relationship of Phase Angle to V&I
 S = P + jQ
 S = VI(complex conjugate operator)
 If V = V30o
and I = I15o
THEN….. COMPLEX POWER SUPPLIED TO
LOAD = S = (V30o)(I-15o) = VI (30o-15o )
= VI cos(15o ) + jVI sin(15o )
 NOTE: Since Phase Angle  =
V - I
S = VI cos() + jVI sin() = P + jQ
Review V, I, Z
 If load is inductive then the Phase Angle
(Impedance Angle Z) is positive, If phase
angle is positive, the phase angle of the current
flowing through the load will lag the voltage
phase angle across the load by the impedance
angle Z.
The Power Triangle
Example
 V = 2400o V
 Z = 40-30o 
 Calculate current I, Power Factor (is it leading or
lagging), real, reactive, apparent and complex
 HW Assignment 2:
 Problems 1-9, 1-15, 1-18, 1-19, 2-4
Example Problem
 HW 1-19 (a)
Chapter 2

Three-Phase (3-) Circuits
What are they?
Benefits of 3- Systems
Generating 3- Voltages and Currents
Wye (Y) and delta () connections
Balanced systems
One-Line Diagrams
What does Three-Phase mean?
A 3- circuit is a 3- AC-generation system
 3 - 1- AC generators with equal voltage but
phase angle differing from the others by 120o

Multiple poles….
Benefits of 3- circuits
 GENERATION SIDE:
 More power per kilogram
 Constant power out (vs. pulsating sinusoidal)
 Induction Motors (no starters required)
Common Neutral
A 3- circuit can have the negative ends of the
3- generators connected to the negative ends
of the 3- AC loads and one common neutral
wire can complete the system
 If the three loads are equal (or balanced) what
will the return current be in the common neutral?

the return current can be calculated to be…
 ZERO!
 (see trig on p. 59 for more detail)
 Neutral is actually unnecessary in a balanced
three-phase system (but is provided since
circumstances may change)

Wye (Y) and delta () connection
Delta ()
Y and 
 Y-connection
IL = I
VLL = 3 V
 -connection
VLL = V
IL = 3 I
Balanced systems
One-Line Diagrams
since all phases are same (except for phase
angle) and loads are typically balanced only one
of the phases is usually shown on an electrical
diagram… it is called a one-line diagram
 Typically include all major components of the
system (generators, transformers, transmission

Chapter 3

Transformers
Benefits of Transformers
Types and Construction, The Ideal Transformer
Transformer Efficiency and Voltage Regulation
Transformer Taps
Autotransformers
3- Transformer connections
– Y-Y,
Y-,
-Y,
-
Benefits







Range of Power Systems
Power Levels
Seamless Converter of Power (Voltage)
Reduced Transmission Losses
Efficient Converter
Low Maintenance (min. moving parts)
Enables Utilization of Power at nearly all levels
```