The Vip uses a variety of algorithms to compute RMS voltage and current, real, reactive, and apparent power, true and displacement power factor, phase angle, total harmonic distortion, and harmonic magnitudes and phases.The formulas for these algorithms are detailed here. 1 Introduction TheVip samples four pairs of voltages and currents. From these samples it computes RMS voltage and current; real, reactive, and apparent power; power factor and displacement power factor, phase angle, volt- all measurements are in volts or amperes, with infinite precision, and perfectly synchronized such that 256 samples is exactly one power line cycle (hereafter called a 60 Hz cycle, though the actual frequency may be from 46 to 70 Hz). The formulas given here are not necessarily those performed by the Vip, but are numerically equivalent expressions. The Vip samples four channels of voltage and four channels of current. Let v1[n],v2[n] ,v3[n] ,v4[n] andi1[n],i2[n] ,i3[n] ,i4[n] represent the sampled voltages and currents for the four channels. In a single 60Hz cycle, the samples are indexed in the range0≤n≤ 255. Where the channel number is not relevant, the subscript may be dropped. Where multiple cycles of data are needed,a superscript is added:vjm[n] is the nth voltage sample for the jth channel for the mth cycle, where0≤n≤ 255,1≤j≤ 4, andm> 0.22 Independent Channels/ Single Phase. In this monitoring mode, each pair of voltage and current channels are used independently. Three phase wye and delta calculations are extensions to the formulas for the single phase case. 2.1RMS Voltage and Current Therms value is computed once per cycle for each channel of voltage and current. The voltage rms value is computed by 25521age and current THD, and harmonic magnitudes and VRMS=v[n]. (1)256phases. The raw waveforms are sampled at a rate of n=02256 samples per power line cycle (usually 60Hz). Here Similarly, the current rms value is given by the complications of A/D quantization, scaling, finite 255 precision math, gain and offset correction, hardware temperature drift compensation, harmonic magnitude21 IRMS=i[n](2)and phase corrections, and synchronization with the.256 powerline frequency are not discussed. Thus, assume n=0 2.2Real Power Real power is computed once per cycle for each pair of voltage and current channels. The real power value is computed by2255 256n=01 W=v[n]i[n].(3) 1.1Notation and Sampled Data Notethat real power is signed to indicate direction of power flow. 2.3Apparent Power Apparent power is computed once per cycle for each pair of voltage and current channels. The apparent power value is computed by VA=VRMS × IRMS.(4) 2.4HarmonicsAn FFT of each voltage and current channel is computed every cycle. Since harmonics only to the 51st are required, the anti-aliased, sampled data is smoothed and down sampled byafactor of two before a128-point FFT is performed. The smoothing is done by averaging each pair of data points. The complex FFT result, including the smoothing and downsampling, is given by 12721 −j2 πkn/ 128V[k]=(x [2n]+x[2n+ 1])e(5)2n=0 √ fork=0, .. ., 63.Here j represents − 1. Since the FFT is done onasingle 60Hz cycle of data, the index k also represents the harmonic number. The 128 point FFT gives a decomposition into 64 harmonics of 60Hz. For specific channels and cycle numbers, the notation Vjm[k] andIjm[k] denote the FFT value forjth channel, for themth cycle number, for thekth harmonic. The real and imaginary parts ofV[k] are denoted byVx[k]andVy[k ], respectively. The real and imaginary parts for channel j areVjx[k]andVjy[k ]. The harmonic magnitudes and phases are computed once per second, to provide some averaging and to reduce transient effects. The one-cycle FFT values are averaged over theMcycles which comprise each second, to formM12V[k ]=Vm[k]. (6)Mm=1 Thek th harmonic magnitude is then given by_2 __2V[k]_ VMAG[k]==Vx[k ]+Vy[k ], (7) and the rawkth harmonic phase angle isVy[k]V θ[k ]=V[k]=arctan.(8)Vx[k]The arctan function is the four quadrant inverse tangent, witharange of − 180 to +180 degrees. The current magnitudes and phase angles are computed in the same manner. The voltage harmonic phase angles are referred to the first voltage channel’s first harmonic phase angle. The current harmonic phase angles are then referred to their corresponding voltage 60Hz phase angles. This two-step algorithm proceeds as follows for thejth channel: 1)Vθj[k] =V θj[k] −kV θ1[1],k=1, .. ., 51 2)Iθj[k] =I θj[k] −kV θj[1],k=1, .. ., 51.2.5Phase Angle The phase angle, θ , is the angular phase shift between the 60Hz voltage and current sinusoids. It is computed every cycle, and is simply θ =I θ [1] −Vθ [1], (9) whereIθ [1] andVθ [1] are the phase angles for the 1st harmonic (60Hz). These phase angles are computed using (8) on the raw FFT outputs instead of the one second average, withk= 1. 2.6Reactive Power Reactive power is computed every cycle for each pair of voltage and current channels. The result is given by 512VAR=(Vx[k]Iy[k]−Ix[k]Vy[k]).(10)k=1EachVx[k]Iy[k]−Ix[k]Vy[k] term is the reactive power contributed by harmonic k. 2.7Power Factor Powerfactor is computed once per cycle for each pair of voltage and current channels. The result is given by PF=W VA, no suffix, lead, lag, for θ=0 or θ=± 180 for0θ 180 for − 180 θ0(11) This expression is also known as true power factor, since it includes the effects of harmonics. 2.8Displacement Power Factor Displacement power factor is computed once per cycle for each pair of voltage and current channels. This quantity represents only the 60Hz contribution to the true power factor. The result is computed by no suffix, for θ=0 or θ=± 180 dPF=| cos θ |, lead, for0<θ 180 lag, for − 180 <θ0(12) 2.9THD Total harmonic distortion, computed every second for each channel of voltage and current, is given in percent by 512(VMAG[k])2k =2 VTHD× 100.(13) VMAG[1] Since this THD definition is referred to the fundamental (as opposed to the RMS value), it may be over 100%.3Three Phase Wye In a three phase wye hookup, each pair of voltage and current channels are handled in the same manner as the single phase hookup. The first three pairs are also grouped together to form total power quantities. 3.1Total Powers Total real, reactive, and apparent power are computed and displayed but not recorded in wye mode. The three phase totals are the sum of the individual phases:WTOT =W1 +W2 +W3 (14)VARTOT= VAR1+ VAR2+ VAR3(15)VATOT= VA1+ VA2+ VA3. (16) All these totals are computed every second from one second averages. The values are displayed on the front panel and then discarded. 3.2Total Power Factors, Phase Angle These total quantities are computed as weighted averages of the three phases, weighted by apparent power: PF1VA1+ PF2VA2+ PF3VA3PFTOT= (17) VATOTdPF1VA1+dPF2VA2+ dPF3VA3dPFTOT= (18) VATOTθ1VA1+θ2VA2+ θ3VA3θTOT= (19) VATOT All these totals are computed every second from one second averages. The values are displayed on the front panel and then discarded.4Three Wire Delta Witha three wire delta circuit, individual phase powers and power factors cannot be computed without imposing assumptions such as a balanced load, balanced source, etc. The Vip only computes total quantities in this mode. These values are computed and recorded as channel one data. As in the wye case, these values are computed once per cycle. The fourth channel is treated as an extra single phase channel with power calculations as detailed in Section 2. Real and reactive power are calculated using the two-wattmeter method, using voltage channels1and 2, and current channels1and 3. The Vip is connected as a delta, with each voltage channel connected from phase to phase. 4.1Real Power Real power is computed using the two-wattmeter method. This requires two voltage and current channels to compute the three phase total. Voltage channels one and two are used with current channels one and three:WTOT=1 256 2552n=0v1[n]i1[n] − 2552n=0v2[n]i3[n].(19) 4.2Reactive Power Reactive power is computed using the two-wattmeter method. This requires two voltage and current channels to compute the three phase total. Voltage channels one and two are used with current channels one and three: 512VARTOT=(V1x[k]I1y[k] −I1x[k]V1y[k]) (20)k=1512−(V2x[k]I3y[k]−I3x[k]V2y[k]).k=1 4.3Apparent Power Apparent power is computed by: VATOT= (WTOT)2+ (VARTOT)2. (21) 4.4Phase Angle The phase angle, θ, is the angular phase shift between the 60Hz voltage and current sinusoids. Since the actual phase current cannot be measured in a three wire delta hookup, the 60Hz component of the real and reactive powers must be used to compute a total three-phase phase angle. The 60Hz component of the reactive power, VAR TOT[1]is computed using (20) withk=1(since 60Hz is the 1st harmonic), giving VARTOT[1]=V1x[1]I1y[1] −I1x[1]V1y[1] (22) −V2x[1]I3y[1]+I3x [1]V2y[1].The 60Hz component of the real power,WTOT [1], can be obtained in an analogous fashion usingWTOT [1]=V1x[1]I1x[1]+I1y [1]V1y[1] (23) −V2x[1]I3x[1] −I3y[1]V2y[1].This results in the following expression for θTOT: VARTOT[1] θTOT= arctan.(24)WTOT[1] 4.5Power Factors Power factor and displacement power factor are computed with (11) and (12), with the use ofWTOT, VATO T, and θTOT instead of the single phase W, VA, and θ.5 Four Wire Delta With a four wire delta circuit, individual phase powers and power factors cannot be computed without imposing assumptions such as a balanced load, balanced source, etc. The Vip only computes total quantities in this mode. These values are computed and recorded as channel one. These computations happen once per cycle, as in the wye case. The fourth channel is treated as an extra single phase channel with power calculations as detailed in Section 2. Real and reactive power are calculated using the three-wattmeter method, which uses all three voltage and current channels. The Vip itself is connected asawye, with each voltage channel measuring from phase to neutral. 5.1Total Powers Real and reactive total power is computed as the sum of the individual channel ’ real and reactive powers, computed as if they were part ofawye circuit. Thus, (14) and (15) can be used, with (3) and (10) used to compute channel powers as in the wye case. Total apparent power is computed with (21). 5.2Phase Angle The phase angle is computed with (24). To compute the 60Hz real and reactive power used in (24), all three voltage and current channels are utilized, as per the three-wattmeter methodology. The expressions forWTOT [1]and VARTOT[1]become32WTOT [1]=Vjx [1]Ijx [1]+I jy [1]Vjy [1] (25)j=1and32VARTOT[1]=(Vjx [1]Ijy [1] −Ijx [1]Vjy [1]).(26)j=1