a-complex numbers phasors and circuts.pdf

7/29/2019 A-Complex Numbers Phasors and Circuts.pdf

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Amanogawa, 2006 Digital Maestro Series 1

Complex Numbers, Phasors and Circuits

Complex numbers are defined by points or vectors in the complexplane, and can be represented in Cartesian coordinates

1z a jb j+ = or in polar (exponential) form

( ) ( )( )( )

exp( ) cos sin

cos

sin

z A j A jA

a A

b A

= = + = =

r

imagina

ea

ry

l part

part

where

2 2 1tanb

A a ba

= + =


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exp( ) exp( 2 )z A j A j j n = Note :

z

Re

Im

A

b

a


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Every complex number has a complex conjugate

( )* *z a jb a jb+ = so that

22 2 2

* ( ) ( )z z a jb a jb

a b z A

= + = + = =

In polar form we have

( )( )( ) ( )

* exp( ) * exp( )

exp 2

cos sin

z A j A j

A j j

A jA

= = = =


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The polar form is more useful in some cases. For instance, when

raising a complex number to a power, the Cartesian form

( ) ( ) ( )n

z a jb a jb a jb+ + + is cumbersome, and impractical for noninteger exponents. Inpolar form, instead, the result is immediate

[ ] ( )exp( ) expnn nz A j A jn = In the case ofroots, one should remember to consider + 2k asargument of the exponential, with k= integer, otherwise possibleroots are skipped:

( ) 2exp 2 expnn n kz A j j k A j jn n = + = + The results corresponding to angles up to 2 are solutions of theroot operation.


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In electromagnetic problems it is often convenient to keep in mind

the following simple identities

exp exp2 2

j j j j = =

It is also useful to remember the following expressions fortrigonometric functions

( ) ( )exp( ) exp( ) exp( ) exp( )cos ; sin2 2

jz jz jz jzz z

j

+ = = resulting from Eulers identity

exp( ) cos( ) sin( )jz z j z=


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Complex representation is very useful for time-harmonic functions

of the form ( ) ( )]( ) ( )]( )]

cos Re exp

Re exp exp

Re exp

A t A j t j

j j t

A j t

+ = + = =

The complex quantity

( )expA j contains all the information about amplitude and phase of thesignal and is called the phasorof

( )cosA t+ If it is known that the signal is time-harmonic with frequency , thephasor completely characterizes its behavior.


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Often, a time-harmonic signal may be of the form:

( )sinA t+ and we have the following complex representation

( ) ( ) ( ))( )]

( ) ( ) ( )]( )) ( )

( )]

sin Re cos sin

Re exp

Re exp / 2 exp exp

Re exp / 2 exp

Re exp

A t jA t j t

jA j t j

j j j t

A j j t

A j t

+ = + + + = + = = =

with phasor ( ))exp / 2A A j This result is not surprising, since

cos( / 2) sin( )t t+ = +


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Time differentiation can be greatly simplified by the use of phasors.Consider for instance the signal

( ) ( )0 0( ) cos expV t V t V V j + = with phasor The time derivative can be expressed as

( )( ) ( )}

( )

0

0

0

( )

sin

Re exp exp

( )exp

V t

V tt

j V j j t

V tj V j j V

t

= + = = is the phasor of


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With phasors, time-differential equations for time harmonic signals

can be transformed into algebraic equations. Consider the simplecircuit below, realized with lumped elements

This circuit is described by the integro-differential equation

( ) 1( ) ( )

td i tv t L Ri i t dt

dt C + +

C

RL

v t i(t)


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Upon time-differentiation we can eliminate the integral as

( )22

( ) 1( )

d i td v t d i L R i t

dt dt C dt= + +

If we assume a time-harmonic excitation, we know that voltage andcurrent should have the form

0 0

0 0

( ) cos( ) exp( )

( ) cos( ) epha

ph

xp( )o

a or

s r

sV V

I I

v t V t V V j

i t I t I I j = + = = + =

If V0 and V are given,

I0 and Iare the unknowns of the problem.


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The differential equation can be rewritten using phasors

( ){ } ( ){ }

( ){ } ( ){ }

2Re exp Re exp

1

Re exp Re exp

+

+ =

L I j t R j I j t

I j t j V j tC

Finally, the transform phasor equation is obtained as

1V R j L j I Z I C

= + = where

1Z R j L

C

= + Impedance Resistance

Reactance


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The result for the phasor current is simply obtained as

( )0 exp1

IV V

I I jZ

R j L j

C

= = = +

which readily yields the unknownsI0 and I.

The time dependent current is then obtained from

( ) ( )}( )

0

0

( ) Re exp exp

cos

I

I

i t I j j t

I t

= = +


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The phasor formalism provides a convenient way to solve time-

harmonic problems in steady state, without having to solve directlya differential equation. The key to the success of phasors is thatwith the exponential representation one can immediately separatefrequency and phase information. Direct solution of the time-dependent differential equation is only necessary fortransients.

Anti-

Transform

Direct SolutionTransients

Solution

TransformIntegro-differential

equations

i ( t ) = ?

i ( t )

Algebraic equationsbased on phasors

I = ?

I


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The phasor representation of the circuit example above has

introduced the concept of impedance. Note that the resistance isnot explicitly a function of frequency. The reactance componentsare instead linear functions of frequency:

Inductive component proportional to Capacitive component inversely proportional to Because of this frequency dependence, for specified values ofL

and C, one can always find a frequency at which the magnitudes ofthe inductive and capacitive terms are equal

1 1r r

r

L

C LC

= = This is a resonance condition. The reactance cancels out and theimpedance becomes purely resistive.


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The peak value of the current phasoris maximum at resonance

00

22 1

VI

R L C

= +

IM

r

|I0|


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Consider now the circuit below where an inductor and a capacitor

are in parallel

The input impedance of the circuit is

1

2

1

1in

j LZ R j C R

j L LC

= + + = +

C

V

IL


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When

01

in

in

in

Z R

ZLC

Z R

= = = =

At the resonance condition

1r

LC =

the part of the circuit containing the reactance componentsbehaves like an open circuit, and no current can flow. The voltageat the terminals of the parallel circuit is the same as the input

voltage V.

a-complex numbers phasors and circuts.pdf

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