Unit 10: Alternating-current circuits Introduction. Alternating - - PowerPoint PPT Presentation

Unit 10: Alternating-current circuits

• Introduction. Alternating current features.
• Phasor diagram.
• Behaviour of basic dipoles (resistor, inductor,

capacitor) to an alternating current.

• RLC series circuit. Impedance and phase lag.
• Resonance. Filters

Niagara Falls Nikola Tesla 1856-1943

• Period T = 2π/ω (s)
• Frequency f = 1/T (Hz)
• Angular frequency

w = 2πf (rad/s)

• Phase wt+ϕ
• Initial phase ϕ (degrees or radians) (phase at t=0)
• Amplitude=Maximum voltage Um (V)
• is the root mean square value. Is that measured by the

measurement devices on A.C.

ωt T

ϕ

Um u(t) = Um cos(ωt + ϕ)

u(t) f Europe: 50 Hz f North America: 60 Hz

Sinusoidal alternating-current features

2

m rms

U U =

• To simplify the analysis of A.C. circuits, a graphical representation of sinusoidal

functions called phasor diagram can be used.

• A phasor is a vector whose modulus (length) is proportional to the amplitude of

sinusoidal function it represents.

• The vector rotates counterclockwise at an angular speed equal to ω. The angle

made up with the horizontal axis is the phase (ωt+φ).

• Therefore, depending if we are working with the function sinus or the function

cosinus, this function will be represented by the vertical projection or the horizontal projection of the rotating vector.

ωt T

ϕ

Um

u(t) = Um cos(ωt + ϕ)

u(t)

Phasor diagram

U

ωt+φ Um ω

   + + ) sin( : ) cos( : Pr ϕ ω ϕ ω t U Vertical t U Horizontal

• jections

m m

• As the position of phasor is different for any time considered, the graphical

representations are done on time t=0 and then, the initial phase φ is the angle between vector and horizontal axis. In this way, the phasor is a unique vector (not changing on time) for a given function:

ωt T

ϕ

Um

u(t) = Um cos(ωt + ϕ)

u(t)

Phasor diagram

U

φ Um

Phasor diagram

ωt

ϕ ϕ ϕ ϕu =0

u(t) = Um cos(ωt + ϕu)

u(t)

Initial phase. Examples.

ωt

u(t) ϕ ϕ ϕ ϕu=90º (π π π π/2 rad)

ωt

u(t) ϕ ϕ ϕ ϕu=-90º (-π π π π/2 rad)

ωt

u(t) ϕ ϕ ϕ ϕu=-45º (-π π π π/4 rad) U U U U

u(t) = Um cos(ωt + ϕu ) i(t) = Im cos(ωt+ϕi)

ϕ

ωt

i u

ϕ ϕ ϕ − =

Phase lag between two waves (voltage and intensity)

Phase lag is defined as ϕ ϕ ϕ ϕi=0 ϕ ϕ ϕ ϕu<0

< ϕ

Voltage u(t) goes behind intensity i(t) Intensity i(t) goes ahead voltage u(t)

To be compared, both functions must be sin or cos and with equal angular frequency

U

φu

Phasor diagram

I

ϕ

ωt

i u

ϕ ϕ ϕ − =

Phase lag between two waves (voltage and intensity)

ϕ ϕ ϕ ϕi=0 ϕ ϕ ϕ ϕu>0

> ϕ

ϕ

ωt ϕ ϕ ϕ ϕi<0 ϕ ϕ ϕ ϕu=0

> ϕ

U

φ =φu

I U

φ =φi

I

Behaviour of basic dipoles. Resistor

Resistor

ωt

i u u(t) = R i(t) = RIm cosωt = Um cosωt i(t) = Im cosωt

R i(t) u(t)

Um = R Im ϕ = 0

Tipler, chapter 29.1

uR = iR

U I

Behaviour of basic dipoles. Inductor

i(t) = Im cosωt

L i(t) u(t)

Um = LωIm ϕ = π/2

Tipler, chapter 29.1

Inductor

ωt

i u XL = Lω Inductive reactance (Ω)

U I

=

L

di(t) u L dt

m m m

di(t) u(t) L L I sen t L I cos( t ) U cos( t ) dt π π ω ω ω ω ω = = − = + = + 2 2

Behaviour of basic dipoles. Capacitor

C i(t) u(t)

φ = - π/2

Tipler, chapter 29.1

Capacitor

ωt

i u u(t) = Um cosωt XC = 1/Cω Capacitive Reactance (Ω)

Cu q =

U I

C

d(u ) i(t) C dt =

m m m

dq(t) Cdu(t) i(t) CU sen t CU cos( t ) I cos( t ) dt dt π π ω ω ω ω ω = = = − = + = + 2 2

m m

I U Cω =

R L C

) cos(

u m L

wt Lw I u ϕ + = ) cos(

u m R

wt R I u ϕ + = ) cos(

u m C

wt Cw I u ϕ + =    + = =

2 π

ϕ

m L L m

I X U    = = ϕ

m R m

I R U    − = =

2 π

ϕ

m C C m

I X U

Behaviour of basic dipoles. Review

Voltage and intensity go on phase Voltage goes ahead intensity 90º Voltage goes behind intensity 90º

L R C

uL uR uC

i(t)= Im cos (wt)

u(t) = uL (t)+ uR (t)+ uC (t)= Um cos (wt+ϕ)

• Let’s take a circuit with resistor, inductor and capacitor in series. If a

sinusoidal intensity i(t)=Imcos(wt) is flowing through such devices, voltage on terminals of circuit will be the addition of voltages on each device:

RLC series circuit. Impedance of dipole

u(t)

Addition of sinusoidal functions is another sinusoidal function

RLC series circuit. Impedance of dipole

Um cos (wt+ϕ) = LwIm cos (wt +π/2)+RIm cos (wt)+(1/Cw)Im cos (wt -π/2)

UL I UR UC UL-UC I UR U

ϕ Um

(Lω-1/Cω) Im RIm

ϕ

Um

Z X R X X R I U Cw Lw R I U I Cw Lw RI U

C L m m m m m m m

= + = − + =  − + =  − + =

2 2 2 2 2 2 2 2 2

) ( ) 1 ( ( ) ) 1 (( ) (

ϕ ϕ tg R X R X X R Cw Lw tg

C L

= = − = − = 1

Z Is called Impedance of dipole (Ω) ϕ is phase lag of dipole

Z and ϕ are depending not only on parameters of R, L and C, but also on frequency of applied current.

ϕ is ranging between - and

2 π 2 π

U = UL + UR + UC

R Z X ϕ ϕ ϕ ϕ X<0 (ϕ ϕ ϕ ϕ<0) R Z X ϕ ϕ ϕ ϕ

Impedance triangle.

• All the equations of a RLC dipole can be summarized on Impedance Triangle
• f a dipole for a given frequency:

2 2 2 2

1 X R Cw Lw R Z + = − + = ) ( ( R X R X X R Cw Lw tg

C L

= − = − = 1 ϕ

X=XL-XC=Lw-1/Cw

Dipole Reactance

X>0 (ϕ ϕ ϕ ϕ>0)

2 2

1 ) ( ( Cw Lw R I U Z

m m

− + = =

RLC series circuit. Resonance

Drawing Z v.s frequency

Z v.s. freq 100 200 300 400 500 600 500 1000 1500 2000 2500 3000 3500 4000 frequency (Hz) Z (Ohm)

Example taking: R = 80 Ω L = 100 mH C = 20 μF

Resonance: f0=707 Hz Z=80 Ω

On resonance, impedance of circuit is minimum, and amplitude of intensity reaches a maximum (for a given

voltage). Intensity and voltage on terminals of RLC

circuit go then on phase. There is a frequency where XL=XC and then the impedance gets its minimum value (Z=R). This frequency is called Frequency of resonance (f0) and can be easily computed:

LC f LC C L 1 2 1 1 1 π ω ω ω =  =  =

RLC series circuit as a Bandpass filter

C R u(t) L uR(t)

Input Output

2 2 m m m R

• uput

) C 1 L ( R RU Z U R RI U U ω ω − + = = = =

2 2 m R input

• utput

) C 1 L ( R R U U U U ω ω − + = =

2 1 U U

2 1 f

, f m R

=

Bandwith [f1 , f2]

1 L Q R C =

The tunning circuit of a radio is a Bandpass filter

RLC series circuit as a Highpass filter

Input Output

1

1 2

L m f

U U =

Bandpass [f1 , ∞]

1 L Q R C =

2 2

1 ( )

m m

• uput

L m

U L U U U L I L Z R L C ω ω ω ω ω = = = = + −

2 2

1 ( )

• uput

L input m

U U L U U R L C ω ω ω = = + −

RLC series circuit as a Lowpass filter

Input Output

1

1 2

C m f

U U =

Bandpass [∞, f1]

1 L Q R C =

2 2

1 1 1 ( )

m m

• uput

C m

U U U U I C C Z C R L C ω ω ω ω ω = = = = + −

2 2

1 1 ( )

• uput

C input m

U U U U C R L C ω ω ω = = + −