Complex Numbers, Phasors and Circuits - Amanogawaamanogawa.com/archive/docs/A-tutorial.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 complex plane, and can be represented in Cartesian coordinates

1z a jb j= + = −

or in polar (exponential) form

( ) ( )( )( )

exp( ) cos sincossin

z A j A jAa Ab A

= φ = φ + φ

= φ

= φ

r

imagina

ea

ry

l part

part

where

2 2 1tan bA a ba

− = + φ =

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exp( ) exp( 2 )z A j A j j n= φ = φ ± πNote :

φ

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 2cos sin

z A j A jA j jA jA

= φ = − φ

= π − φ

= φ − φ

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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

( ) ( ) ( )nz a jb a jb a jb= + ⋅ + +…

is cumbersome, and impractical for non−integer exponents. In polar form, instead, the result is immediate

[ ] ( )exp( ) expnn nz A j A jn= φ = φ

In the case of roots, one should remember to consider φ + 2kπ as argument of the exponential, with k = integer, otherwise possible roots are skipped:

( ) 2exp 2 expnn n kz A j j k A j jn nφ π = φ + π = +

The results corresponding to angles up to 2π are solutions of the root 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 jπ π = − = −

It is also useful to remember the following expressions for trigonometric functions

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

jz jz jz jzz zj

+ − − −= =

resulting from Euler’s 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

A j j t

A j t

ω + φ = ω + φ

= φ ω

= ω

The complex quantity

( )expA A j= φ

contains all the information about amplitude and phase of the signal and is called the phasor of

( )cosA tω + φ If it is known that the signal is time-harmonic with frequency ω, the phasor completely characterizes its behavior.

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

( )sinA tω + φ 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

A j j j t

A j j t

A j t

ω + φ = − ω + φ + ω + φ = − ω + φ

= − π φ ω

= φ − π ω = ω

with phasor ( )( )exp / 2A A j= φ − π

This result is not surprising, since

cos( / 2) sin( )t tω + φ − π = ω + φ

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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= ω + φ = φwith 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 Vt

∂= −ω ω + φ

∂

= ω φ ω

∂⇒ ω φ = ω∂

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 simple circuit 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

R L

v (t) i (t)

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Upon time-differentiation we can eliminate the integral as

( )2

2( ) 1 ( )

d i td v t d iL R i tdt dt Cdt

= + +

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

0 0

0 0

( ) cos( ) exp( )( ) cos( ) epha

phxp( )o

a ors rsV V

I I

v t V t V V ji t I t I I j

= ω + α = α

= ω + α = α⇒

⇒

If V0 and αV are given,

⇒ I0 and αI are the unknowns of the problem.

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The differential equation can be rewritten using phasors

( ){ } ( ){ }

( ){ } ( ){ }

ω ω ω ω

ω ω ω

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 IC

= + ω − = ω

where

1Z R j LC

= + ω − ω Impedance ResistanceReactance

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The result for the phasor current is simply obtained as

( )0 exp1 I

V VI I jZ R j L j

C

= = = α + ω − ω

which readily yields the unknowns I0 and αI . The time dependent current is then obtained from

( ) ( ){ }( )

0

0

( ) Re exp exp

cosI

I

i t I j j t

I t

= α ω

= ω + α

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The phasor formalism provides a convenient way to solve time-harmonic problems in steady state, without having to solve directly a differential equation. The key to the success of phasors is that with the exponential representation one can immediately separate frequency and phase information. Direct solution of the time-dependent differential equation is only necessary for transients.

Anti- Transform

Direct Solution( Transients )

Solution

TransformIntegro-differential

equations

i ( t ) = ?

i ( t )

Algebraic equations based 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 is not explicitly a function of frequency. The reactance components are instead linear functions of frequency: Inductive component ⇒ proportional to ω Capacitive component ⇒ inversely proportional to ω Because of this frequency dependence, for specified values of L and C , one can always find a frequency at which the magnitudes of the inductive and capacitive terms are equal

1 1r r

rL

C LCω = ⇒ ω =

ω

This is a resonance condition. The reactance cancels out and the impedance becomes purely resistive.

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The peak value of the current phasor is maximum at resonance

00 2

2 1

VI

R LC

= + ω − ω

IM

ωr

| I0|

ω

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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

21

1in

j LZ R j C Rj L LC

− ω = + + ω = + ω − ω

CV

I L R

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When 0

1in

in

in

Z R

ZLC

Z R

ω = =

ω = →∞

ω→∞ =

At the resonance condition

1r LC

ω =

the part of the circuit containing the reactance components behaves like an open circuit, and no current can flow. The voltage at the terminals of the parallel circuit is the same as the input voltage V.

Complex Numbers, Phasors and Circuits - Amanogawaamanogawa.com/archive/docs/A-tutorial.pdf ·...

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