Electronic System Formulae |
B
C E f G h I j L P Q |
susceptance capacitance voltage source frequency conductance h-operator current j-operator inductance active power reactive power |
[siemens, S] [farads, F] [volts, V] [hertz, Hz] [siemens, S] [1Ð120°] [amps, A] [1Ð90°] [henrys, H] [watts, W] [VAreactive, VArs] |
R
S t V W X Y Z f w |
resistance apparent power time voltage drop energy reactance admittance impedance phase angle angular frequency |
[ohms, W] [volt-amps, VA] [seconds, s] [volts, V] [joules, J] [ohms, W] [siemens, S] [ohms, W] [degrees, °] [rad/sec] |
Rectangular and polar forms of impedance Z:
Z = R + jX = (R2 + X2)½Ðtan-1(X / R)
= |Z|Ðf = |Z|cosf + j|Z|sinf
Addition of impedances Z1 and Z2:
Z1 + Z2 = (R1 + jX1) + (R2 + jX2)
= (R1 + R2) + j(X1 + X2)
Subtraction of impedances Z1 and Z2:
Z1 - Z2 = (R1 + jX1) - (R2 + jX2)
= (R1 - R2) + j(X1 - X2)
Multiplication of impedances Z1 and Z2:
Z1 * Z2 = |Z1|Ðf1
* |Z2|Ðf2
= ( |Z1| * |Z2| )Ð(f1
+ f2)
Division of impedances Z1 and Z2:
Z1 / Z2 = |Z1|Ðf1
/ |Z2|Ðf2
= ( |Z1| / |Z2| )Ð(f1
- f2)
In summary:
- use the rectangular form for addition and subtraction,
- use the polar form for multiplication and division.
Conversely, an admittance Y comprising a conductance G in parallel with a susceptance B
can be converted to an impedance Z comprising a resistance R in series with a reactance
X:
Z = Y -1 = 1 / (G - jB) = (G + jB) / (G2 + B2)
= G / (G2 + B2) + jB / (G2 + B2) = R + jX
R = G / (G2 + B2) = G / |Y|2
X = B / (G2 + B2) = B / |Y|2
Using the polar form of admittance Y:
Z = 1 / |Y|Ð-f = |Y| -1Ðf
= |Z|Ðf
= |Z|cosf + j|Z|sinf
The total impedance ZS of impedances Z1, Z2,
Z3,... connected in series is:
ZS = Z1 + Z1 + Z1 +...
The total admittance YP of admittances Y1, Y2,
Y3,... connected in parallel is:
YP = Y1 + Y1 + Y1 +...
In summary:
- use impedances when operating on series circuits,
- use admittances when operating on parallel circuits.
For a sinusoidal current i of amplitude I and angular frequency
w:
i = I sinwt
If sinusoidal current i is passed through an inductance L, the voltage e across the
inductance is:
e = L di/dt = wLI coswt
= XLI coswt
The current through an inductance lags the voltage across it by 90°.
Capacitive Reactance
The capacitive reactance XC of a capacitance C at angular frequency
w and frequency f is:
XC = 1 / wC = 1 / 2pfC
For a sinusoidal voltage v of amplitude V and angular frequency
w:
v = V sinwt
If sinusoidal voltage v is applied across a capacitance C, the current i through the
capacitance is:
i = C dv/dt = wCV coswt
= V coswt / XC
The current through a capacitance leads the voltage across it by 90°.
At resonance, the imaginary part of ZS is zero:
XC = XL
ZSr = R
wr = (1 / LC)½
= 2pfr
Parallel resonance
A parallel circuit comprising an inductance L with a series resistance R, connected in parallel
with a capacitance C, has an admittance YP of:
YP = 1 / (R + jXL) + 1 / (- jXC)
= (R / (R2 + XL2))
- j(XL / (R2 + XL2) - 1 / XC)
where XL = wL and
XC = 1 / wC
At resonance, the imaginary part of YP is zero:
XC = (R2 + XL2) / XL
= XL + R2 / XL
= XL(1 + R2 / XL2)
ZPr = YPr-1 = (R2 + XL2) / R
= XLXC / R = L / CR
wr = (1 / LC - R2 / L2)½
= 2pfr
Note that for the same values of L, R and C, the parallel resonance frequency is lower than the series resonance frequency, but if the ratio R / L is small then the parallel resonance frequency is close to the series resonance frequency.
When a voltage V (taken as reference) is applied across the reactive load Z, the current
I is:
I = VY = V(R / |Z|2 - jX / |Z|2) = VR / |Z|2 - jVX / |Z|2
= IP - jIQ
The active current IP and the reactive current IQ are:
IP = VR / |Z|2 = |I|cosf
IQ = VX / |Z|2 = |I|sinf
The apparent power S, active power P and reactive power Q are:
S = V|I| = V2 / |Z| = |I|2|Z|
P = VIP = IP2|Z|2 / R = V2R / |Z|2
= |I|2R
Q = VIQ = IQ2|Z|2 / X = V2X / |Z|2
= |I|2X
The power factor cosf and reactive factor
sinf are:
cosf = IP / |I| = P / S = R / |Z|
sinf = IQ / |I| = Q / S = X / |Z|
Resistance and Shunt Reactance
The impedance Z of a reactive load comprising resistance R and shunt reactance X
is found from:
1 / Z = 1 / R + 1 / jX
Converting to the equivalent admittance Y comprising conductance G and shunt susceptance
B:
Y = 1 / Z = 1 / R - j / X = G - jB = |Y|Ð-f
When a voltage V (taken as reference) is applied across the reactive load Y, the current
I is:
I = VY = V(G - jB) = VG - jVB = IP - jIQ
The active current IP and the reactive current IQ are:
IP = VG = V / R = |I|cosf
IQ = VB = V / X = |I|sinf
The apparent power S, active power P and reactive power Q are:
S = V|I| = |I|2 / |Y| = V2|Y|
P = VIP = IP2 / G = |I|2G / |Y|2
= V2G
Q = VIQ = IQ2 / B = |I|2B / |Y|2
= V2B
The power factor cosf and reactive factor
sinf are:
cosf = IP / |I| = P / S = G / |Y|
sinf = IQ / |I| = Q / S = B / |Y|
Inductive Load
Z = R + jXL
I = IP - jIQ
cosf = R / |Z| (lagging)
I* = IP + jIQ
S = P + jQ
An inductive load is a sink of lagging VArs (a source of leading VArs).
Capacitive Load
Z = R - jXC
I = IP + jIQ
cosf = R / |Z| (leading)
I* = IP - jIQ
S = P - jQ
A capacitive load is a source of lagging VArs (a sink of leading VArs).
For a balanced delta connected load with line voltage Vline and line current
Iline:
Vdelta = Vline
Idelta = Iline / Ö3
Zdelta = Vdelta / Idelta
= Ö3Vline / Iline
Sdelta = 3VdeltaIdelta
= Ö3VlineIline
= 3Vline2 / Zdelta = Iline2Zdelta
The apparent power S, active power P and reactive power Q are related by:
S2 = P2 + Q2
P = Scosf
Q = Ssinf
where cosf is the power factor and
sinf is the reactive factor
Note that for equivalence between balanced star and delta connected loads:
Zdelta = 3Zstar
Select rated values as base values, usually rated power in MVA and rated phase voltage in kV:
Sbase = Srated
= Ö3ElineIline
Ebase = Ephase = Eline/ Ö3
The base values for line current in kA and per-phase star impedance in ohms/phase are:
Ibase = Sbase / 3Ebase
( = Sbase / Ö3Eline)
Zbase = Ebase / Ibase
= 3Ebase2 / Sbase
( = Eline2 / Sbase)
Note that selecting the base values for any two of Sbase, Ebase, Ibase or Zbase fixes the base values of all four. Note also that Ohm's Law is satisfied by each of the sets of actual, base and per-unit values for voltage, current and impedance.
Transformers
The primary and secondary MVA ratings of a transformer are equal, but the voltages and currents in the
primary (subscript 1) and the secondary (subscript 2) are usually
different:
Ö3E1lineI1line = S
= Ö3E2lineI2line
Converting to base (per-phase star) values:
3E1baseI1base = Sbase = 3E2baseI2base
E1base / E2base = I2base / I1base
Z1base / Z2base = (E1base / E2base)2
The impedance Z21pu referred to the primary side, equivalent to an impedance
Z2pu on the secondary side, is:
Z21pu = Z2pu(E1base / E2base)2
The impedance Z12pu referred to the secondary side, equivalent to an impedance
Z1pu on the primary side, is:
Z12pu = Z1pu(E2base / E1base)2
Note that per-unit and percentage values are related by:
Zpu = Z% / 100
The positive, negative and zero sequence currents are calculated from the line currents using:
Ia1 = (Ia + hIb + h2Ic) / 3
Ia2 = (Ia + h2Ib + hIc) / 3
Ia0 = (Ia + Ib + Ic) / 3
The positive, negative and zero sequence currents are combined to give the line currents using:
Ia = Ia1 + Ia2 + Ia0
Ib = Ib1 + Ib2 + Ib0
= h2Ia1 + hIa2 + Ia0
Ic = Ic1 + Ic2 + Ic0
= hIa1 + h2Ia2 + Ia0
The residual current Ir is equal to the total zero sequence current:
Ir = Ia0 + Ib0 + Ic0 = 3Ia0
= Ia + Ib + Ic = Ie
which is measured using three current transformers with parallel connected secondaries.
Ie is the earth fault current of the system.
Similarly, for phase-to-earth voltages Vae, Vbe and
Vce, the residual voltage Vr is equal to the total zero sequence
voltage:
Vr = Va0 + Vb0 + Vc0 = 3Va0
= Vae + Vbe + Vce = 3Vne
which is measured using an earthed-star / open-delta connected voltage transformer.
Vne is the neutral displacement voltage of the system.
The h-operator
The h-operator (1Ð120°) is the complex cube root of unity:
h = - 1 / 2 + jÖ3 / 2
= 1Ð120° = 1Ð-240°
h2 = - 1 / 2 - jÖ3 / 2
= 1Ð240° = 1Ð-120°
Some useful properties of h are:
1 + h + h2 = 0
h + h2 = - 1 = 1Ð180°
h - h2 = jÖ3
= Ö3Ð90°
h2 - h = - jÖ3
= Ö3Ð-90°
For each type of short-circuit fault occurring on an unloaded system:
- the first column states the phase voltage and line current conditions at the fault,
- the second column states the phase 'a' sequence current and voltage conditions at the fault,
- the third column provides formulae for the phase 'a' sequence currents at the fault,
- the fourth column provides formulae for the fault current and the resulting line currents.
By convention, the faulted phases are selected for fault symmetry with respect to reference phase 'a'.
I f = fault current
Ie = earth fault current
Ea = normal phase voltage at the fault location
Z1 = positive phase sequence network impedance to the fault
Z2 = negative phase sequence network impedance to the fault
Z0 = zero phase sequence network impedance to the fault
Single phase to earth - fault from phase 'a' to earth:
Va = 0
Ib = Ic = 0 I f = Ia = Ie |
Ia1 = Ia2 = Ia0 = Ia / 3
Va1 + Va2 + Va0 = 0 |
Ia1 = Ea / (Z1 + Z2 + Z0)
Ia2 = Ia1 Ia0 = Ia1 |
I f = 3Ia0
= 3Ea / (Z1 + Z2 + Z0) = Ie
Ia = I f = 3Ea / (Z1 + Z2 + Z0) |
Double phase - fault from phase 'b' to phase 'c':
Vb = Vc
Ia = 0 I f = Ib = - Ic |
Ia1 + Ia2 = 0
Ia0 = 0 Va1 = Va2 |
Ia1 = Ea / (Z1 + Z2)
Ia2 = - Ia1 Ia0 = 0 |
I f = - jÖ3Ia1
= - jÖ3Ea / (Z1 + Z2)
Ib = I f = - jÖ3Ea / (Z1 + Z2) Ic = - I f = jÖ3Ea / (Z1 + Z2) |
Double phase to earth - fault from phase 'b' to phase 'c' to earth:
Vb = Vc = 0
Ia = 0 I f = Ib + Ic = Ie |
Ia1 + Ia2 + Ia0 = 0
Va1 = Va2 = Va0 |
Ia1 = Ea / Znet
Ia2 = - Ia1Z0 / (Z2 + Z0) Ia0 = - Ia1Z2 / (Z2 + Z0) |
I f = 3Ia0 = - 3EaZ2
/ Szz = Ie
Ib = I f / 2 - jÖ3Ea(Z2 / 2 + Z0) / Szz Ic = I f / 2 + jÖ3Ea(Z2 / 2 + Z0) / Szz |
Three phase (and three phase to earth) - fault from phase 'a' to phase 'b' to phase 'c'
(to earth):
Va = Vb = Vc (= 0)
Ia + Ib + Ic = 0 (= Ie) I f = Ia = hIb = h2Ic |
Va0 = Va (= 0)
Va1 = Va2 = 0 |
Ia1 = Ea / Z1
Ia2 = 0 Ia0 = 0 |
I f = Ia1 = Ea / Z1 = Ia
Ib = Eb / Z1 Ic = Ec / Z1 |
The values of Z1, Z2 and Z0 are each determined from the respective positive, negative and zero sequence impedance networks by network reduction to a single impedance.
Note that the single phase fault current is greater than the three phase fault current if Z0 is less than (2Z1 - Z2).
Note also that if the system is earthed through an impedance Zn (carrying current 3I0) then an impedance 3Zn (carrying current I0) must be included in the zero sequence impedance network.
The three phase fault level Ssc of the power system is:
Ssc = 3Isc2ZS
= 3EphaseIsc = 3Ephase2 / ZS
= Eline2 / ZS
Note that if the X / R ratio of the source impedance ZS (comprising resistance RS and reactance XS) is sufficiently large, then ZS » XS.
Transformers
If a transformer of rating ST (taken as base) and per-unit impedance
ZTpu is fed from a source with unlimited fault level (infinite busbars),
then the per-unit secondary short-circuit current I2pu and fault level
S2pu are:
I2pu = E2pu / ZTpu = 1.0 / ZTpu
S2pu = I2pu = 1.0 / ZTpu
If the source fault level is limited to SS by per-unit source impedance
ZSpu (to the same base as ZTpu), then the secondary
short-circuit current I2pu and fault level S2pu are
reduced to:
I2pu = E2pu / (ZTpu + ZSpu)
= 1.0 / (ZTpu + ZSpu)
S2pu = I2pu = 1.0 / (ZTpu + ZSpu)
where ZSpu = ST / SS
Rearranging for Ifault and tfault:
Ifault = ( Ilimit - Iload ) ( tlimit
/ tfault )½ + Iload
tfault = tlimit ( Ilimit - Iload )2
/ ( Ifault - Iload )2
If Iload is small compared with Ilimit and Ifault, then:
Ilimit2 tlimit »
Ifault2 tfault
Ifault » Ilimit ( tlimit
/ tfault )½
tfault » tlimit ( Ilimit
/ Ifault )2
Note that if the current Ifault is reduced by a factor of two, then the time tfault is increased by a factor of four.
Current Transformer
For a measurement current transformer of voltampere rating S, rated primary current
IP and rated secondary current IS, the maximum secondary voltage
VSmax, maximum secondary burden resistance RBmax and maximum primary
voltage VPmax are:
VSmax = S / IS
RBmax = VSmax / IS = S / IS2
VPmax = S / IP = VSmaxIS / IP
For a protection current transformer of voltampere rating S, rated primary current IP,
rated secondary current IS and rated accuracy limit factor F, the rated secondary
reference voltage VSF, maximum secondary burden resistance RBmax and
equivalent primary reference voltage VPF are:
VSF = SF / IS
RBmax = VSF / ISF = S / IS2
VPF = SF / IP = VSFIS / IP
Impedance Measurement
If the primary voltage Vpri and the primary current Ipri are measured at
a point in a system, then the primary impedance Zpri at that point is:
Zpri = Vpri / Ipri
If the measured voltage is the secondary voltage Vsec of a voltage transformer of
primary/secondary ratio NV and the measured current is the secondary current
Isec of a current transformer of primary/secondary ratio NI, then the
primary impedance Zpri is related to the secondary impedance Zsec by:
Zpri = Vpri / Ipri = VsecNV
/ IsecNI = ZsecNV / NI
= ZsecNZ
where NZ = NV / NI
If the no-load (source) voltage Epri is also measured at the point, then the source impedance
ZTpri to the point is:
ZTpri = (Epri - Vpri) / Ipri
= (Esec - Vsec)NV / IsecNI
= ZTsecNV / NI = ZTsecNZ
The leading (capacitive) reactive power demand QC which must be connected across the load is:
QC = Q1 - Q2
= P (tanf1 - tanf2)
The uncorrected and corrected apparent power demands, S1 and S2, are related
by:
S1cosf1 = P
= S2cosf2
Comparing corrected and uncorrected load currents and apparent power demands:
I2 / I1 = S2 / S1
= cosf1 / cosf2
If the load is required to have a corrected power factor of unity, Q2 is zero and:
QC = Q1 = P tanf1
I2 / I1 = S2 / S1 = cosf1
= P / S1
Shunt Capacitors
For star-connected shunt capacitors each of capacitance Cstar on a three phase system
of line voltage Vline and frequency f, the leading reactive power demand
QCstar and the leading reactive line current Iline are:
QCstar = Vline2 / XCstar
= 2pfCstarVline2
Iline = QCstar / Ö3Vline
= Vline / Ö3XCstar
Cstar = QCstar / 2pfVline2
For delta-connected shunt capacitors each of capacitance Cdelta on a three phase system
of line voltage Vline and frequency f, the leading reactive power demand
QCdelta and the leading reactive line current Iline are:
QCdelta = 3Vline2 / XCdelta
= 6pfCdeltaVline2
Iline = QCdelta / Ö3Vline
= Ö3Vline / XCdelta
Cdelta = QCdelta / 6pfVline2
Note that for the same leading reactive power QC:
XCdelta = 3XCstar
Cdelta = Cstar / 3
Series Capacitors
For series line capacitors each of capacitance Cseries carrying line current
Iline on a three phase system of frequency f, the voltage drop
Vdrop across each line capacitor and the total leading reactive power demand
QCseries of the set of three line capacitors are:
Vdrop = IlineXCseries
= Iline / 2pfCseries
QCseries = 3Vdrop2 / XCseries
= 3VdropIline = 3Iline2XCseries
= 3Iline2 / 2pfCseries
Cseries = 3Iline2 / 2pfQCseries
Note that the apparent power rating Srating of the set of three series line capacitors is
based on the line voltage Vline and not the voltage drop Vdrop:
Srating = Ö3VlineIline
For delta-connected shunt reactors each of inductance Ldelta on a three phase system
of line voltage Vline and frequency f, the lagging reactive power demand
QLdelta and the lagging reactive line current Iline are:
QLdelta = 3Vline2 / XLdelta
= 3Vline2 / 2pfLdelta
Iline = QLdelta / Ö3Vline
= Ö3Vline / XLdelta
Ldelta = 3Vline2 / 2pfQLdelta
Note that for the same lagging reactive power QL:
XLdelta = 3XLstar
Ldelta = 3Lstar
Series Reactors
For series line reactors each of inductance Lseries carrying line current
Iline on a three phase system of frequency f, the voltage drop
Vdrop across each line reactor and the total lagging reactive power demand
QLseries of the set of three line reactors are:
Vdrop = IlineXLseries
= 2pfLseriesIline
QLseries = 3Vdrop2 / XLseries
= 3VdropIline = 3Iline2XLseries
= 6pfLseriesIline2
Lseries = QLseries / 6pfIline2
Note that the apparent power rating Srating of the set of three series line reactors is
based on the line voltage Vline and not the voltage drop Vdrop:
Srating = Ö3VlineIline
The series resonance angular frequency wr of an inductance L
with a capacitance C is:
wr = (1 / LC)½
= w(XC / XL)½
The three phase fault level Ssc at the node for no-load phase voltage E and
source impedance Z per-phase star is:
Ssc = 3E2 / |Z| = 3E2 / |R + jXL|
If the ratio XL / R of the source impedance Z is sufficiently large,
|Z| » XL so that:
Ssc » 3E2 / XL
The reactive power rating QC of the power factor correction capacitors for a capacitive
reactance XC per phase at phase voltage E is:
QC = 3E2 / XC
The harmonic number fr / f of the series resonance of XL with
XC is:
fr / f = wr / w
= (XC / XL)½
» (Ssc / QC)½
Note that the ratio XL / XC which results in a harmonic number
fr / f is:
XL / XC = 1 / ( fr / f )2
so for fr / f to be equal to the geometric mean of the third and fifth harmonics:
fr / f = Ö15 = 3.873
XL / XC = 1 / 15 = 0.067
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