what is this all about? Introduction three-phase diode bridge rectifier input voltages input voltages, waveforms normalization of voltages voltages?

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1 what is this all about? v A Introduction three-phase diode bridge rectifier D1 D D D4 D5 D6 i OUT + v OUT v B i 1 i i v 1 v v input voltages input voltages, waveforms v 1 = V m cos ω 0 t v = V m cos ω 0 t v = V m cos ω 0 t 4 v k = V m cos ω 0 t k 1, k {1,, } normalization of voltages v k = V m cos ω 0 t k 1, k {1,, } m X v X V m m 1 = cos ω 0 t m = cos ω 0 t m = cos ω 0 t 4 v k = V m cos ω 0 t k 1, k {1,, } v k = V m cos ω 0 t k 1, k {1,, }

2 v k = V m cos ω 0 t k 1, k {1,, } v k = V m cos ω 0 t k 1, k {1,, } v k = V m cos ω 0 t k 1, k {1,, } v 1, spectrum v, spectrum v, spectrum voltages, quantitative characterization T HD And what is THD? all graphs and data PyLab processed k V k RMS T HDv k V.4 % V.77 % V.06 % Parseval s identity: results in I RMS = T HD k= I k RMS I 1 RMS Ik RMS assumed I 0 = 0 T HD I RMS I 1 RMS I 1 RMS simple, but important computational issues, finite sums...

3 normalization of currents and time how does it work? part 1: theory j X i X I OUT va unless otherwise noted D1 D D5 + iout vout ϕ ω 0 t D D4 D6 vb i1 i i good: physical dimensions lost, reduced number of variables, results are generalized, core of the problem focused bad: physical dimensions lost, perfect double-check tool is lost v1 v v one of the three: D1, D, D5 v A v A, analytical v A, spectrum m A = max m 1, m, m m A = k+1 9k 1 cos kω 0t what about v B? one of the three, again: D, D4, D6 v A D1 D D5 + i OUT v OUT D D4 D6 v B i 1 i i v 1 v v

4 v B v B, analytical m B = min m 1, m, m v B, spectrum the output voltage, v OUT m B = k 1 cos kω 0t m OUT = m A m B = max m 1, m, m min m 1, m, m v OUT, spectrum currents? i 1 t = d 1 t d t I OUT i t = d t d 4 t I OUT i t = d 5 t d 6 t I OUT m OUT = 1 1 6k 1 cos 6kω 0t states of the diodes the input currents

5 consider i 1 spectra of the input currents spectra of the input currents, analytical numerical verification, Gibbs phenomenon J 1C, k = j 1 t = + J 1C, k cos kω 0 t 1 k, k = 6n 1 1 k, k = 6n + 1 0, otherwise for n N 0, k > 0 double-check: P IN = 1 = = P OUT obtained using wxmaxima THD of the input currents voltages and currents I k RMS = I OUT 6 I k RMS, 1 = I OUT Ik RMS I k RMS, 1 T HD I k RMS, 1 T HD = % Parseval s identity based formula turned out to be useful some more parameters 1 X RMS already used for the THD 0 xω 0 t dω 0 t, x {i, v} and if the voltages are sinusoidal... S = V RMS I RMS P = V RMS I 1, RMS cos φ 1 P 1 S I RMS V RMS 0 vω 0 t iω 0 t dω 0 t P F P S DP F cos φ 1 and φ1 is... P F = P S = I 1, RMS I RMS T HD = cos φ 1 = I 1, RMS I RMS DP F = cos ϕ 1 DP F IRMS I 1, RMS IRMS = 1 I 1, RMS I 1, RMS i.e. everything depends on the current waveform and its position

6 some more parameters, plain rectifier back to the rectifier: how does it work? part : experiment I k RMS = I OUT V k RMS = 1 V m S = I OUT 1 V m = V m I OUT P = V OUT I OUT = V m I OUT D1 D D D4 v A D5 D6 i OUT + v OUT P F = 95.5% DP F = 1 actually, not so bad; T HD is the problem i 1 i i v 1 v v v B input, at I OUT = A input, at I OUT = A input, at I OUT = A input, at I OUT = A input, at I OUT = 6 A input, at I OUT = 6 A

7 input, at I OUT = 6 A input, at I OUT = 6 A input, at I OUT = 9 A input, at I OUT = 9 A input, at I OUT = 9 A input, at I OUT = 9 A output, at I OUT = A output, at I OUT = A

8 output, at I OUT = 6 A output, at I OUT = 6 A output, at I OUT = 9 A output, at I OUT = 9 A in quantitative terms, input, 1st in quantitative terms, input, nd I OUT k I k RMS [A] V k RMS [V] S k [VA] P k [W] 0 A A A A I OUT k P F k T HDi k [%] T HDv k [%] 0 A A A A in quantitative terms, output overall impressions I OUT [A] V OUT [V] P OUT [W] P IN [W] η [%] pretty good rectifier simple, robust, cheap good symmetry excellent DP F acceptable P F poor T HD but not that poor up to this point: diode bridge rectifier analyzed measurement tools developed is there a way to do something with the T HD?