90200 Phasors p

Revision Lecture 2: Phasors

Revision Lecture 2:Phasors

Basic Concepts

Reactive Components

Phasors

Phasor Diagram

Complex Power

Complex Power inComponents

E1.1 Analysis of Circuits (2014-4472) Revision Lecture 2 1 / 7

Basic Concepts


Basic ConceptsReactive Components

Phasors

Phasor Diagram

Complex Power



Phasors and Complex impedances are only relevant to sinusoidalsources.

A DC source is a special case of a cosine wave with = 0.

For two sine waves, the leading one reaches its peak first, the laggingone reaches its peak second. So sint lags cost.

If A cos (t+ ) = F costG sint, then A =

F 2 +G2, = tan1 G

F.

F = A cos , G = A sin . In CMPLX mode, Casio fx-991ES can do complex arithmetic and

can switch between the two forms with SHIFT,CMPLX,3 orSHIFT,CMPLX,4

Reactive Components


Basic Concepts

ReactiveComponents

Phasors

Phasor Diagram

Complex Power



Impedances: R, jL, 1jC

= jC

.

Admittances: 1R, 1

jL= j

L, jC

In a capacitor or inductor, the Current and Voltage are 90 apart :

CIVIL: Capacitor - current leads voltage; Inductor - current lagsvoltage

Average current (or DC current) through a capacitor is always zero

Average voltage across an inductor is always zero

Average power absorbed by a capacitor or inductor is always zero

Phasors


Basic Concepts

Reactive Components

PhasorsPhasor Diagram

Complex Power



A phasor represents a time-varying sinusoidal waveform by a fixed complexnumber.

Waveform Phasor

x(t) = F costG sint X = F + jG [Note minus sign]x(t) = A cos (t+ ) X = Aej = A

max (x(t)) = A |X| = A

x(t) is the projection of a rotating rod ontothe real (horizontal) axis.

X = F + jG is its starting position at t = 0.

Phasor Diagram


Basic Concepts

Reactive Components

Phasors

Phasor DiagramComplex Power



Draw phasors as vectors. Join vectors end-to-end to show how voltages inseries add up (or currents in parallel).

Find y(t) if x(t) = cos 300t .

YX

=1

jC

R+ 1jC

= 1jRC+1

= 11+3j

= 0.1 0.3j = 0.32 72

x(t) = cos 300t X = 1Y = X Y

X

= 0.1 0.3j = 0.32 72

y(t) = 0.1 cos 300t+ 0.3 sin 300t

= 0.32 cos (300t 1.25)= 0.32 cos (300 (t 4.2ms))

0 0.5 1

-0.4

-0.2

0

Real

0 20 40 60 80 100 120-1

-0.5

0

0.5

1x

y

time (ms)x=

blue

, y=

red

w = 300 rad/s, Gain = 0.32, Phase = -72

Complex Power


Basic Concepts

Reactive Components

Phasors

Phasor Diagram

Complex PowerComplex Power inComponents


If V = |V |V is a phasor, we define V = 12 V to be the corresponding

r.m.s. phasor. The r.m.s. voltage isV

= 12 |V |.

Power Factor cos = cos (V I)Complex Power S = V I = P + jQ [ = complex conjugate]Apparent Power |S| =

V

I unit = VAs

Average Power P = (S) = |S| cos unit = Watts heatReactive Power Q = (S) = |S| sin unit = VARs

Conservation of power (Tellegens theorem): in any circuit the totalcomplex power absorbed by all components sums to zero P = average power and Q = reactive power sum separately to zero.VARs are generated by capacitors and absorbed by inductors.

> 0 for inductive impedance. < 0 for capacitive impedance.

Complex Power in Components


Basic Concepts

Reactive Components

Phasors

Phasor Diagram

Complex Power



S = V I =|V |2Z

=I2

Z S = ZResistor: positive real (absorbs watts)Inductor: positive imaginary (absorbs VARs)Capacitor: negative imaginary (generates VARs)

Power Factor Correction

V = 230. Motor is 5||7j .I = 46 33j = 56.5 36S = V I = 1336 kVAcos = P|S|= cos 36 = 0.81

Add parallel capacitor: 300FI = 46 11j = 47 14S = V I = 10.914 kVAcos = P|S| = cos 14

= 0.97

Current decreases by factor of 0.810.97

. Lower power transmission losses.

RevisionLecture2: PhasorsBasic ConceptsReactive ComponentsPhasorsPhasor DiagramComplex PowerComplex Power in Components

Documents

90200 Phasors p