11. complex ac

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Lecture 12 2Lecture 11

Complex Notation for AC Quantities:Complex Impedance

Aims:To appreciate:Use of complex quantities.Role of impedance and j.Influence of power factor.Appreciate operation of low, high and band pass filters

To be able:To analyse some basic circuits.


Revision of Complex Number Arithmetic

2

3

1

1

1

1

j

j

j j

j j

= = = =

=

2 2

is part; is part

(modulus) : tan (argument)

(cos sin )

j

Z R jX R X

X Z R X

R Z Z e

Z Z j

= +

= + =

== +

real imaginary

( ) ( )( )

( )

1 2

1 2

1 2 1 2 1 2

1 2 1 2

11

2 2

2 2

j

j

j j

Z Z R R j X X

Z Z Z Z e

Z Z e

Z Z

Z jZ Z e Z e

j

+

+

+ = + + +

=

=

= =

*

* 2 2

(complex conjugate) j Z R jX Z e

ZZ R X

= == +

what is j? three ways to express a complex number

complex arithmetic

complex conjugate

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Phasors as Complex NumbersIf we plot phasors on an Arganddiagram we can use complexnumber representation:

(cos sin )V t V t j t +

Which means that we can use thepowerful tools of complex algebrato manipulate AC quantities.

Note that we use j for -1 and not iThis is to avoid confusion with i as asymbol for currents.

Real

Imaginary

V

t

Vsin t

Vcos t


Complex AC quantitiesIn general, all AC quantities are complex numbers containing amplitude and phase:

j RE IM V jV V e

= + =VVoltage j

RE IM I jI I e = + =ICurrent

j R jX Z e = + =ZImpedanceThe complex quantities obey all the laws and techniques that we

have derived for DC networks:

Kirchhoffs Current Law Kirchhoffs Voltage Law Ohms Law V=IZ Impedances in series: Z = Z 1+Z1 Impedances in parallel: Y = Y 1+Y 2

The physical significance of the real and imaginary parts of current and voltage: Real currents and voltages are associated with energy dissipation (power

averaged over one cycle is positive). Measurable Imaginary parts are associated with energy storage (power averaged over

one cycle is zero charging and discharging). Not measurable

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Look Back at Inductive Reactance:

t jo L e I I

=

The reciprocal of impedance is ADMITTANCE, symbol Y, units Siemens

dt dI

LV L L =Let and we know that

What is V L ?


Look Back at Capacitative Reactance:

t joC eV V

=

The reciprocal of impedance is ADMITTANCE, symbol Y, units Siemens

dt dV

C I C C =Let and we know that

What is I C ?

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Power in AC CircuitsPower in an AC circuit is given by W=VI 0

( )0

(2 )0 0

j t

j t

j t

V V e I I e

W V I e

+

+

===

This reduces to:

( )2 20 0 cos sin j t j t W V I e e + = +

If we average over one cycle (from t =0 to t =2/):e+ j2t averages to and e- j2t averages to 0, so

0 0

1cos or cos

2 RMS RMSW V I W V I = =

The cos term is called the power factor

Resistive power Reactive power


Power Factor:cos RMS RMSW V I =

This tells us that when the current and voltage are /2 out of phase(e.g.in a pure L or pure C), the power dissipated is zero.

Power factor is a big issue for electrical engineers.

Many industrial loads have a high inductance inseries with the resistance (e.g. heating coils forlarge tanks)

This can affect the power factor and reduce thepower dissipated in the resistor.

In many cases a capacitor is used to correct thepower factor

R

Power factor correction

L

50 Hz

Power ratings of industrial equipment are often quoted inkVA kilo-volt-amp rather than kW to indicate thatthe power factor may not be 1

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RC Network with Complex Numbers

C

I

V R

VC

VR


RCL Network in Series

R

C

L

VR

V

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RCL Network in SeriesR C L

1. Complex impedance is given by

1where

Z R jX

X LC

= +

=

2. Convert to exponential form for multiplication:

222

2

1

1tan

j

LC Z RC

X LC R CR

Z Z e

= +

= =

=

3. Get the current

( ) j t

j t A A j

V e V I e

Z e Z

= =

4. Get the voltage across the resistor

( )

221

11

j t A R

j R

j R

RV V IR e

Z

V Re

V Z

V eV LC

CR

= =

=

= +


RCL Network in Series

R

C

L

VR

VA

22

1

11

j R j t

A

V e

V e LC CR

= +

when =0, denominator , V R 0when , denominator , V R 0when = 0, denominator = 1, V R=V A

01

LC

=

0

VR

This is a series resonant circuit.At resonance X C=-X L so thereactance in the circuit is zero

11. complex ac

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