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

Sinusoidal alternating-current features u(t) u(t) = U m cos( ω t + ϕ ) Period T = 2 π / ω (s)  U m Frequency f = 1/T (Hz)  ω t Angular frequency  ϕ w = 2 π f (rad/s) T Phase wt+ ϕ  Initial phase ϕ (degrees or radians) (phase at t=0)  Amplitude=Maximum voltage U m (V)  U m U is the root mean square value. Is that measured by the  = rms 2 measurement devices on A.C. f Europe: 50 Hz f North America: 60 Hz

Phasor diagram 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.  u(t) = U m cos( ω t + ϕ ) Horizontal : U cos( t ) ω + ϕ m  Pr ojections  Vertical : U sin( t ) u(t) ω + ϕ m U U m U m ωt+φ ω t ϕ ω T

Phasor diagram 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: u(t) = U m cos( ω t + ϕ ) Phasor diagram u(t) U m U U m φ ω t ϕ T

Initial phase. Examples. u(t) = U m cos( ω t + ϕ u ) u(t) u(t) U U ω t ω t ϕ u =0 ϕ ϕ ϕ ϕ u =90º ( π π /2 rad) ϕ ϕ ϕ π π u(t) u(t) U U ω t ω t ϕ u =-45º (- π π /4 rad) ϕ u =-90º (- π π /2 rad) ϕ ϕ ϕ π π ϕ ϕ ϕ π π

Phase lag between two waves (voltage and intensity) i(t) = I m cos( ω t+ ϕ i ) u(t) = U m cos( ω t + ϕ u ) To be compared, both functions Phase lag is defined as ϕ = ϕ − ϕ must be sin or cos and with u i equal angular frequency ϕ < 0 ϕ u <0 ϕ ϕ ϕ ϕ i =0 ϕ ϕ ϕ Phasor diagram I ω t φ u ϕ U Voltage u(t) goes behind intensity i(t) Intensity i(t) goes ahead voltage u(t)

Phase lag between two waves (voltage and intensity) ϕ = ϕ − ϕ u i ϕ i =0 ϕ ϕ ϕ ϕ u >0 ϕ ϕ ϕ U φ =φ u ω t I ϕ ϕ > 0 ϕ i <0 ϕ ϕ ϕ ϕ u =0 ϕ ϕ ϕ U φ =φ i ω t ϕ I 0 ϕ >

Behaviour of basic dipoles. Resistor u R = iR Resistor u R u(t) i(t) i ω t i(t) = I m cos ω t u(t) = R i(t) = RI m cos ω t = U m cos ω t U I U m = R I m ϕ = 0 Tipler, chapter 29.1

Behaviour of basic dipoles. Inductor di(t) u L dt = L Inductor u i u(t) L i(t) ω t i(t) = I m cos ω t di(t) π π u(t) L L I sen t L I cos( t ) U cos( t ) = = − ω ω = ω ω + = ω + U m m m dt 2 2 I U m = L ω I m X L = L ω Inductive reactance ( Ω ) ϕ = π /2 Tipler, chapter 29.1

Behaviour of basic dipoles. Capacitor d(u ) C i(t) C dt = Capacitor u i u(t) C i(t) ω t u(t) = U m cos ω t q = Cu dq(t) Cdu(t) π π i(t) CU sen t CU cos( t ) I cos( t ) = = = − ω ω = ω ω + = ω + I m m m dt dt 2 2 U I m U = m C ω X C = 1/C ω Capacitive Reactance ( Ω ) φ = - π /2 Tipler, chapter 29.1

Behaviour of basic dipoles. Review R  R U = R I m m  u = I R cos( wt + ϕ ) R m u  ϕ = 0 Voltage and intensity go on phase  L L U = X I m L m  u = I Lw cos( wt + ϕ ) L m u π  ϕ = + 2 Voltage goes ahead intensity 90º  C C U = X I I m C m  m u = cos( wt + ϕ ) π C u  ϕ = − Cw 2 Voltage goes behind intensity 90º

RLC series circuit. Impedance of dipole Let’s take a circuit with resistor, inductor and capacitor in series. If a  sinusoidal intensity i(t)=I m cos(wt) is flowing through such devices, voltage on terminals of circuit will be the addition of voltages on each device: i(t)= I m cos ( wt ) R L C u R u L u C Addition of sinusoidal functions is another sinusoidal function u(t) u(t) = u L (t)+ u R (t)+ u C (t)= U m cos ( wt+ ϕ )

RLC series circuit. Impedance of dipole U m cos ( wt+ ϕ ) = LwIm cos (wt + π /2)+RIm cos (wt)+(1/Cw)Im cos (wt - π /2) U = U L + U R + U C U L -U C (Lω-1/Cω) I m U L U ϕ U m U m ϕ I I U R RI m U R U C 1 1 U   2 2 2 2 2 2 2 2 2 m U ( RI ) (( Lw ) I ) U I ( R ( Lw ) R ( X X ) R X Z = + − = + − = + − = + = m m m m m L C Cw Cw I m 1 Z Is called Impedance of dipole ( Ω ) Lw − X X X − Cw ϕ is phase lag of dipole L C tg tg ϕ = = = = ϕ R R R π π ϕ is ranging between - and 2 2 Z and ϕ are depending not only on parameters of R, L and C, but also on frequency of applied current.

Impedance triangle. All the equations of a RLC dipole can be summarized on Impedance Triangle  of a dipole for a given frequency: R ϕ ϕ ϕ ϕ Z X X ϕ ϕ ϕ ϕ Z R X<0 ( ϕ ϕ <0) ϕ ϕ X>0 ( ϕ ϕ ϕ ϕ >0) 1 ( ( ) 2 2 2 2 Z = R + Lw − = R + X X=X L -X C =Lw-1/Cw Cw Dipole 1 − Lw Reactance − X X X Cw L C tg ϕ = = = R R R

RLC series circuit. Resonance 1 ) U Drawing Z v.s frequency ( ( 2 2 m Z = = R + Lw − I Cw m There is a frequency where X L =X C and then the impedance gets its minimum value (Z=R). This frequency is called Frequency of resonance (f 0 ) and can be easily computed: 1 1 1 1   L f ω = ω = = 0 0 0 C LC LC ω π 2 0 Z v.s. freq 600 Example taking: R = 80 Ω L = 100 mH C = 20 μF 500 400 On resonance, impedance of circuit is minimum, and Z (Ohm) amplitude of intensity reaches a maximum (for a given 300 voltage) . Intensity and voltage on terminals of RLC 200 circuit go then on phase. 100 0 0 500 1000 1500 2000 2500 3000 3500 4000 frequency (Hz) Resonance: f 0 =707 Hz Z=80 Ω

RLC series circuit as a Bandpass filter C U RU m m U U RI R = = = = Input Output ouput R m Z 1 2 2 R ( L ) + ω − u ( t ) u R ( t ) R C ω L Bandwith [f 1 , f 2 ] U U R output U 1 R = = R = U U 1 input m 2 2 R ( L ) U + ω − 2 m C f 1 f , ω 2 The tunning circuit of a 1 L Q = radio is a Bandpass filter R C

RLC series circuit as a Highpass filter U L U ω m m Input Output U U L I L = = ω = ω = ouput L m Z 1 2 2 R ( L ) + ω − C ω U U L ω ouput L = = U U 1 input m 2 2 R ( L ) + ω − C ω 1 L Q = R C U 1 Bandpass [f 1 , ∞ ] L = U 2 m f 1

RLC series circuit as a Lowpass filter Input Output 1 1 U U m m U U I = = = = ouput C m C C Z ω ω 1 2 2 C R ( L ) ω + ω − C ω U U 1 ouput C = = U U 1 input m 2 2 C R ( L ) ω + ω − C ω 1 L Q = R C U 1 C Bandpass [ ∞ , f 1 ] = U 2 m f 1