R, L and C components in AC circuits

tdtvdCti )()( =Capacitor

dttdiLtv )()( =Inductor

Resistor iRv =

General I-V dependencies

I-V dependencies

for sine-waveforms

iRv =

?

?

Objective:to find the mathematical functions or transformationsreplacing the time derivatives with algebraic actions,

like multiplication, division, etc.

tdy e ydt

= =

Example:if y(t) = et, then:

Issue: the actual electrical signals are sinusoidal, not et types; there is no simple transformation from cos(t) or sin(t) type waveforms into et function and back

Eulers formula

The fundamental relationship bridging complex numbers and

AC signals is provided by the Eulers formula:

)sin()cos( je j +=

Eulers formula

)sin()cos( je j +=

x

ej

cos()

jy

j*sin()

21

21 xyN +=The modulus of a complex number N(the length of the arrow)

( ) ( ) 122 =+= cossinNIf N = ej, then using Eulers formula

Geometrical interpretation

Eulers formula

)sin()cos( je j +=

( ) ( ) 122 =+= cossinNN = ej

Geometrical interpretation

ej

cos()

jy

j*sin()

1

Any complex number can be easily defined using Eulers formula:

)sin()cos( jRReR j +=

x

Rej

Rcos()

jy

j*Rsin()

R is the modulus; is the argument

The modulus

of Rej is R

R

The Eulers form describes a polar form of a complex number and hence a phasor:

Phasor describing AC current or voltage: N = R L Complex number in the Eulers form:

( |N| L ) = N e jx

Rej

Rcos()

jy

j*Rsin() Describes the complex

number with the modulus R and the angle

R

Phasor of an AC current or voltage

Complex number in Eulers form

An important case of the Eulers formula: = pi/2 = pi/2 = pi/2 = pi/2( ) ( ) ( ) jjje j =+=+= 10222 /sin/cos/ pipipi

( )2/pi=

jej

Example: find j

[ ] )/()/()/(/ 421221 pipi jj eejj ===

Multiplication and division of complex numbersusing Eulers formula:

;; 21 2211 jj

eRNeRN ==Following regular algebraic rules,

( ) )(212121 2121 +== jjj eRReeRRNNThe modulus of the product = the product of the moduli;

The argument = the sum of the arguments

( ) )(/ 2121 212121 == jjj eRReReRNNThe modulus of the quotient = the quotient of the moduli;

The argument = the difference of the arguments

Using Eulers formula to find time derivatives of complex numbers:

the time derivative becomes

Suppose the argument of a complex number N is a linear function of time: = t:

j tdN R j edt

=

j tN R e =

or:

dN j Ndt

=

( )( ) xkxk

xx

ekxd

ed

exd

ed

=

=

AC circuit analysis using complex numbers

The approach:

1. Bridge the actual waveform to the complex variable in the Eulers form; i.e. create a complex image of a real waveform.2. Apply the KVL, KCL and the I-V relationships to the complex images of voltages and currents in the AC circuit. (The math involved is much simpler than that required to solve the actual circuit).3. Find the actual current or voltage waveforms by taking the real part of the resulting complex variable.

The simplicity of time derivatives using complex numbers in the Eulers form opens up a simple way to analyze AC circuits.

Complex images of resistor voltage and current

Mv t V t( ) cos( )=The voltage across the resistor:

1. The phasor corresponding to the resistor voltage

v(t) = VMe j t Phasor design rules:

The phasor modulus = the real voltage amplitude. The phasor argument = t

tjM eR

VR == /v(t)iR

2. Find the complex images corresponding to the current through resistor. Use the same rules that apply to the actual voltage current relationship:

3. Find the actual resistor current by taking the real part of thecomplex current:

j tM M MV V Ve t j tR R R

cos( ) sin( ) = +

MR

Vi t t

R( ) cos( )=The resistor current

Phasors of resistor voltage and current (2)

Mv t V t( ) cos( )=The voltage across the capacitor:

Note that the actual capacitor current can be found as:

2

C M

M

d vi C C V tdt

C V t

sin( )

cos( / )

pi

= = =

= +

We will now find the current using the complex image technique.

Complex images of capacitor voltage and current

1. Given the actual voltage

The complex voltage corresponding to the actual voltage v(t)(shown as bold v): v(t) = VMe j t

Mv t V t( ) cos( )=

2. Find the complex capacitor currentusing the same rules that apply to the actual voltage and current

j tC M

v(ti C C j V et

= =

)

Using j = e jpipipipi/22

2

j t j j tC M M

j tM

i j C V e C V e eC V e

/

( / )

pi

pi

+

= = =

=

Complex images of capacitor voltage and current (2)

3. Take the real part of the complex current :

2 2j tC M Mi C V e C V t( / )Re cos( / ) pi pi+ = = +

2C Mi C V tcos( / ) pi= +

Compare to the current found by taking the time derivative of the capacitor voltage:

Complex images of capacitor voltage and current (3)

Mi t I t( ) cos( )=The current across the inductor:

Note that the actual inductor voltage can be found as:

2

L M

M

d iv L L I t

dtL I t

sin( )

cos( / )

pi

= = =

= +

We will now find the voltage using the complex image technique.

Complex images of inductor voltage and current

1. Given the actual current

The complex current corresponding to the actual current i(t)(shown as bold v): i(t) = IMe j t

Mi t I t( ) cos( )=

2. Find the complex inductor voltageusing the same rules that apply to the actual voltage and current

j tL M

i(tv L L j I e

t

= =

)

Using j = e jpipipipi/22

2

j t j j tL M M

j tM

v j L I e L I e eL I e

/

( / )

pi

pi

+

= = =

=

Complex images of inductor voltage and current (2)

3. Take the real part of the complex voltage:

2 2j tL M Mv L I e L I t( / )Re cos( / ) pi pi+ = = +

2L Mv L I tcos( / ) pi= +

Compare to the current found by taking the time derivative of the capacitor voltage:

Complex images of inductor voltage and current (3)

I-Vs in time-domain and on the complex plane

tdtvdCti )()( =Capacitor

dttdiLtv )()( =Inductor

Resistor )()( tiRtv =

Time domain(real variables)

Complex plane(rotating phasors)

( ) i tM MV R I e = j t

Mj C e = CI V

j tMj L e = LV I

Differential equations

Linear V-I dependencies similar to the Ohms law

Assuming v(t) and i(t) are the sinusoidal signals with the angular frequency ::::

Note that the term ejt can be omitted: they simply remind you what the angular frequency is; the remaning phasors are called the complex amplitudes.

Capacitor

Inductor

Resistor

Complex amplitudes

M MV R I =

MCj VIC =

MLj IVL =

R,C and L I-Vs on the complex plane

Complex planeequivalent circuit

Resistor with the resistance R

Quazi-Ohmic (i.e. linear) component with the complex

impedance ZC = 1/(jC)( )1 Mj C / = CV I

Quazi-Ohmic (i.e. linear) component with the complex

impedance ZL = jL