Digital Filter Examples

Transcription

1 Power Systems Protecton and Relayng Sesson ; Page /22 Fall 28 Defne samplng rate per cycle 6 Dgtal Flter Examples Defne length of sample data set, n cycles CY 8 Total number of samples: M CY n M Δt 6Hz Δt.42 ms t n Δt nδt Create nput data sgnal, sampled at per cycle I measn cos 2πn I meas2n cos 2πn 8 I meas3n cos 2πn 5 I meas4n cos 2πn n 2 e 2 I measn I meas2n I meas4n n

2 Power Systems Protecton and Relayng Sesson ; Page 2/22 Fall 28 Lets look at the Cosne Flter Coeffcents: k 4 ( 4 ) k 8 ( 8 ) k 6 ( 6 ) k 32 ( 32 ) cos coef k 4 4 cos coef ( k) cos 2πk cos coef k cos coef k k 4 cos coef k cos coef k k 32

3 Power Systems Protecton and Relayng Sesson ; Page 3/22 Fall 28 Now lets look at the Sne Flter Coeffcents: sn coef k 4 4 sn coef ( k) sn 2πk sn coef k sn coef k k 4 sn coef k sn coef k k 32

4 Power Systems Protecton and Relayng Sesson ; Page 4/22 Fall 28 Now defne Cosne and Sn flters COSF( A q) 2 k cos coef ( k) A [ q( ) ] k SINF( A q) 2 k sn coef ( k) A [ q( ) ] k Create a flter ndex, "" (whch ncludes samples of past hstory (so t starts at ( - )) ( ) M Create a flter ndex, "v" (whch ncludes /4 samples of past hstory for delayng cosne flter output put a quarter cycle (so t starts at (/4 - )) v 4 M COSF I meas SINF I meas

5 Power Systems Protecton and Relayng Sesson ; Page 5/22 Fall 28 I meas COSF I meas SINF I meas Phasor 2 COSF I meas jsinf I meas Mag Angle Phasor argphasor I meas Mag I meas Angle

6 Power Systems Protecton and Relayng Sesson ; Page 6/22 Fall 28 So we need to compare ths angle to a reference. In the case wth only one measurement, we compare t to tself. θ Angle Angle Fltered_ 2Mag cos 2π θ I meas Fltered_ Now repeat wth the second sgnal, whch s phase shfted I meas2 COSF I meas2 SINF I meas2 Phasor2 2 COSF I meas2 jsinf I meas2 Mag2 Phasor2 Angle2 argphasor2

7 Power Systems Protecton and Relayng Sesson ; Page 7/22 Fall 28 I meas2 Mag2 I meas2 Angle2 Tracks new angle So we need to compare ths angle to a reference. In the case we'll use the frst sgnal as a reference θ 2 Angle2 Angle Fltered_2 2Mag2 cos 2π θ 2

8 Power Systems Protecton and Relayng Sesson ; Page 8/22 Fall 28 I meas2 I meas Fltered_2 Tracks the phase shft Now plot the angle θ θ 2 2 We have a problem n the angle due to the dfference n reset tmes Fx for the reset tme ssue: Phase_Case2 Angle2 Angle f Angle2 Angle π Angle2 Angle 2π f Angle2 Angle Angle2 Angle 2π f Angle2 Angle π ( π)

9 Power Systems Protecton and Relayng Sesson ; Page 9/22 Fall 28 θ Phase_Case2 5 5 Now repeat wth the thrd sgnal, whch has a constant dc offset I meas3 COSF I meas3 SINF I meas3 Rejects DC offset Phasor3 2 COSF I meas3 jsinf I meas3 Mag3 Phasor3 Angle3 argphasor3

10 Power Systems Protecton and Relayng Sesson ; Page /22 Fall 28 I meas3 Mag3 RMS magntude of fundamental component I meas3 Angle3 Agan need to compare ths angle to a reference. In the case we'll use the frst sgnal as a reference θ 3 Angle3 Angle Fltered_3 2Mag3 cos 2π θ 3

11 Power Systems Protecton and Relayng Sesson ; Page /22 Fall 28 I meas3 I meas Fltered_ Fltered_3 Note the DC offset not n fltered results Now repeat wth the fourth sgnal, whch has a decayng DC offset. I meas4 COSF I meas4 SINF I meas4 Sne fler passng some DC offset, but not cosne Phasor4 2 COSF I meas4 jsinf I meas4 Mag4 Angle4 Phasor4 argphasor4

12 Power Systems Protecton and Relayng Sesson ; Page 2/22 Fall 28 I meas4 Mag4 Magntude has error wth decayng offset I meas4 Angle4 θ 4 Angle4 Angle I meas4 θ 4 2 As does angle, but we have a second problem due to reset tmes Fx for the reset tme ssue:

13 Power Systems Protecton and Relayng Sesson ; Page 3/22 Fall 28 Phase_Case4 Angle4 Angle f Angle4 Angle π Angle4 Angle 2π f Angle4 Angle Angle4 Angle 2π f Angle4 Angle π ( π) 5 Phase_Case4 5 So stll see decayng dc offset problem n angle calculaton. Alternatve to usng Sne Flter: Note that delayng a cosne by 9 rees (/4 cycle) gve a sne functon COSF I meas4 v SINF I meas4 v COSFI meas4 v v Note I'm changng ndex to "v" nstead of "" due to dfferent startng pont

14 Power Systems Protecton and Relayng Sesson ; Page 4/22 Fall 28 Phasor4_alt v 2 COSF I meas4 v jcosf I meas4 v 4 Mag4_alt v Angle4_alt v Phasor4_alt v argphasor4_alt v 8 75 Mag4 v 7 Mag4_alt v 65 6 Magntude has less error due to DC offset, but cosne sn't perfect rejecton ether v Phase_Case4_alt v Angle4_alt v Angle v f Angle4_alt v Angle v π Angle4_alt v Angle v 2π f Angle4_alt v Angle v Angle4_alt v Angle v 2π f Angle4_alt v Angle v π ( π) 6 4 Phase_Case4 v 2 Phase_Case4_alt v 2 4 Agan, much better, but not perfect. 6 v

15 Power Systems Protecton and Relayng Sesson ; Page 5/22 Fall 28 Fltered_4 v 2Mag4_alt v cos 2πv Phase_Case4_alt v I meas4 I meas Fltered_4 DC offset largely removed, but not entrely A few more cases: I meas5n cos 2πn f n 8cos 2π( n 2) 8 3 otherwse 3 5 I meas5n n Phasor5 v 2 COSF I meas5 v jcosf I meas5 v 4 Mag5 v Phasor5 v Angle5 v arg Phasor5 v

16 Power Systems Protecton and Relayng Sesson ; Page 6/22 Fall Mag5 v 3 2 v Notce the flter response tme Mag5 v Takes a lttle over a cycle v Phase_Case5 v Angle5 v Angle v f Angle5 v Angle v π Angle5 v Angle v 2π f Angle5 v Angle v Angle5 v Angle v 2π f Angle5 v Angle v π ( π) Phase_Case5 v v

17 Power Systems Protecton and Relayng Sesson ; Page 7/22 Fall 28 Fltered_5 v 2Mag5 v cos 2πv Phase_Case5 v 3 5 I meas5 Fltered_5 5 3 Notce delay n flter response Now add some harmoncs. Frst, nteger harmoncs I meas6n cos 2πn 8sn 2 2πn 4cos 3 2πn 2cos 7 2πn 2 I meas6n n

18 Power Systems Protecton and Relayng Sesson ; Page 8/22 Fall 28 Phasor6 v 2 COSF I meas6 v jcosf I meas6 v 4 Mag6 v Phasor6 v Angle6 v arg Phasor6 v Mag6 v v Phase_Case6 v Angle6 v Angle v f Angle6 v Angle v π Angle6 v Angle v 2π f Angle6 v Angle v Angle6 v Angle v 2π f Angle6 v Angle v π ( π) Phase_Case6 v v Fltered_6 v 2Mag6 v cos 2πv Phase_Case6 v

19 Power Systems Protecton and Relayng Sesson ; Page 9/22 Fall 28 2 I meas6 Fltered_6 Fltered_ 2 Only fundamental component s passed. Now, how about a non-nterger harmonc I meas7n cos 2πn 4cos 5.5 2πn 2 I meas7n n Phasor7 v 2 COSF I meas7 v jcosf I meas7 v 4

20 Power Systems Protecton and Relayng Sesson ; Page 2/22 Fall 28 Mag7 v Phasor7 v Angle7 v arg Phasor7 v 8 75 Mag7 v v Not so good. Flter doesn't have a gan of at non-nteger harmoncs Phase_Case7 v Angle7 v Angle v f Angle7 v Angle v π Angle7 v Angle v 2π f Angle7 v Angle v Angle7 v Angle v 2π f Angle7 v Angle v π ( π) 6 Phase_Case7 v Fltered_7 v 2Mag7 v cos v 2πv Phase_Case7 v 2 I meas7 Fltered_7 2

21 Power Systems Protecton and Relayng Sesson ; Page 2/22 Fall 28 2 Fltered_7 2 Dstorton defntely attenuated, but not elmnated. A low pass flter s needed USe a smple flter for the moment 3 LPW floor LPW 3 6HzΔt LPW cel CY 6HzΔt LP ( ) 2 LPW k I meas LPWk LPW I measn LP ( ) 2 5 nlpw

22 Power Systems Protecton and Relayng Sesson ; Page 22/22 Fall 28 LP 7 ( ) LPW k I meas7 LPWk LPW 2 I meas7n LP 7 ( ) 2 5 nlpw 2 I measn LP 7 ( ) 2 5 nlpw