PERIODIC STEADY STATE ANALYSIS, EFFECTIVE VALUE,

Transcription

1 PERIODIC SEADY SAE ANALYSIS, EFFECIVE VALUE, DISORSION FACOR, POWER OF PERIODIC CURRENS t + Effective value of current (general definition) IRMS i () t dt Root Mean Square, in Czech boo denoted I he value of the DC current I, which generates an equivalent aount of heat lie alternating current p t P RI WRI t Dt i / Periodic current Fourier series it () I + I sin( ω t+ ψ ) Effective value I I I sin( ωt ψ ) dt + + Even phase shifted sin is orthogonal function, because ( ω ψ ) sin( ω φ ) B sin t+ B l t+ dt l t B B { cos ( l) ω t+ ψ φ cos ( + l) ω t+ ψ + φ } dt l l l B B B B l cos( ψ φ) cosϕ W li p t i i i Pavel Máša, X3EO, lecture 3 page

2 o the effective value contributes only haronics of the sae frequency, there is no interaction between -th and l-th haronics (i.e. if l). hen effective value in periodic steady state is I I + I I + I I + I + I + L Doing actual calculations we can consider just liited nuber of haronics Error depends on wavefor of the circuit variable: voltage source, first 3 haronics u [V] t [s] 5 x current passing RC series circuit 5 i [A] aplitude spectru of voltage.3.35.t [s] u [V].5 i [A] x 5 aplitude spectru of current Graphical construction of the effective value: I + I + I + I 3 I + I + I + I + I 3 I + I + I I I 3 I I + I I I Pavel Máša, X3EO, lecture 3 page

3 Distortion factor it describes the difference of wavefor fro sinus: I + I + I + L I + I + I + L d I I I L I Nuerator effective value of residual curve Another definition: I + I3 + I + L ds I Distortion factor for the -th haronic: I d I Units often in %. Exaple: U Distortion factor of the rectangular function u() t sin ωt π U U U, U π U U U 8 d.35 U U π U U π ds d U U π Instantaneous power pt utit () ()() Power of periodic voltages and currents Active power P p() t dt u()() t i t dt Pavel Máša, X3EO, lecture 3 page 3

4 If the periodic voltage and current will be expressed in the for of Fourier series () + sin( ω + ψ ) it () I + I sin( ωt+ η ) u t U U t, then (orthogonality!) (copare first ter with DC, the rest with AC) Active power, expressed by coplex exponential for of the Fourier series j t () U e ω j, it () I e ω ut U I P UI + UI + UI W + Coefficients of the Fourier series F j t + ω ω U () I () F u t e dt F j t F ite dt In the general definition we substitute voltage by the Fourier series + jωt P utitdt ()() it () Fe dt U If we change the order of integration and suation, and after substitution of Fourier coefficients of the current + + j t UF F F U I P i t e dt Because I ( ) I F, we obtain F cos ω () ( ) ( ψ η ) cosϕ [ ] t + P U I F F It is one of the fors of the Parseval s theore. he power could be coputed either by integration in tie or suing ultiplies of the coefficients of the Fourier series. It will be also e.g. () + ut dt + U, () it dt I Pavel Máša, X3EO, lecture 3 page

5 As well, P UI+ Re UI (another for of first definition); copare it with Parseval s theore why here ust be Re[] operator, and not in Parseval s theore? Exaple: π he wavefor of the voltage is ut () cos π + Because π ut e e e We obtain coefficients of the Fourier series: π π j j jπ jπ () cos π e j π 5 j π U, U e he result is (utilizing coefficients) π cos 5 5 π dt Reactive power Q UIsinϕ [ var] Apparent power S UI U + U I + I VA [ ] In the apparent power could not be applied orthogonality, U, I are constants, so each eleent is ultiplied with each other eleent, in contrast to P a Q, so then S P + Q Pavel Máša, X3EO, lecture 3 page 5

6 We introduce deforation power [ ] D S P Q VA Power factor P λ S cosϕ ev. ϕ ev. it is only fictive phase shift between equivalent sinusoidal voltage and current, exhibiting the sae effective values and producing the sae active power S D j ev. P Q Exaple: ut () sinπ π π π it ().5 + sin π + +.5sin 3π + +.sin 5π 6 6 π P cos.77 W π Q sin.77var S + ( + + ) D VA VA Pavel Máša, X3EO, lecture 3 page 6

7 Periodic steady state in linear circuits Linear circuit analysis is based on validity of the superposition theore: A u (t) u (t) u n (t) u(t) B A u(t) hen it is possible perfor analysis lie superposition of sinusoidal steady states, haronics by haronics. Approach:. Haronic analysis (it eans to find Fourier coefficients of the exciting periodic wavefor). Coputation of distinct haronics of resulting circuit variable using sinusoidal steady state analysis 3. Haronic synthesis we add resulting coponents together B Exaple: RC circuit is supplied fro the source of triangular voltage according to the figure:.5.5 R Ω, C -6 µf Fourier series a Mean value is zero, he wavefor is odd it contains only sinus coponents, a he wavefor is antiperiodic it contains just odd coponents Pavel Máša, X3EO, lecture 3 page 7

8 It could be deterined just in a quarter-period only: U 3U sin ω sin ω b t t dt t t dt u t v sin ωt 3U t cosω t cosωt dt u v cosωt ω ω ω + 8U + ( ) π For current it is valid: Iˆ 8U ˆ ( U ) π Zˆ R + j ω C + + π ω π. haronics 8 I ˆ π j j π j.5 3. haronics 8 I ˆ 9 3 π j j 3 π j haronics 8 I ˆ 5 5 π.9 + j j 5 π j.7 Pavel Máša, X3EO, lecture 3 page 8 e e e