Electrical Formulas
- Notation
- Impedance
- Admittance
- Reactance
- Resonance
- Reactive Loads and Power Factor
- Complex Power
- Three Phase Power
- Per-unit System
- Symmetrical Components
- Fault Calculations
- Three Phase Fault Level
- Thermal Short-time Rating
- Instrument Transformers
- Power Factor Correction
- Reactors
- Harmonic Resonance
Notation | |||||
The symbol font is used for some notation and formulae. If the Greek symbols for alpha beta delta do not appear here [ a b d ] the symbol font needs to be installed for correct display of notation and formulae. | |||||
BCE f G h I j L P Q | .susceptancecapacitance.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] | QRS t V W X Y Z f w | .quality factorresistance.apparent power .time .voltage drop .energy .reactance .admittance .impedance .phase angle .angular frequency | .[number][ohms, W].[volt-amps, VA] .[seconds, s] .[volts, V] .[joules, J] .[ohms, W] .[siemens, S] .[ohms, W] .[degrees, °] .[rad/sec] |
Impedance
The impedance Z of a resistance R in series with a reactance X is:
Z = R + jX
Rectangular and polar forms of impedance Z:
Z = R + jX = (R^{2} + X^{2})^{½}Ðtan^{-1}(X / R) = |Z|Ðf = |Z|cosf + j|Z|sinf
Addition of impedances Z_{1} and Z_{2}:
Z_{1} + Z_{2} = (R_{1} + jX_{1}) + (R_{2} + jX_{2}) = (R_{1} + R_{2}) + j(X_{1} + X_{2})
Subtraction of impedances Z_{1} and Z_{2}:
Z_{1} – Z_{2} = (R_{1} + jX_{1}) – (R_{2} + jX_{2}) = (R_{1} – R_{2}) + j(X_{1} – X_{2})
Multiplication of impedances Z_{1} and Z_{2}:
Z_{1} * Z_{2} = |Z_{1}|Ðf_{1} * |Z_{2}|Ðf_{2} = ( |Z_{1}| * |Z_{2}| )Ð(f_{1} + f_{2})
Division of impedances Z_{1} and Z_{2}:
Z_{1} / Z_{2} = |Z_{1}|Ðf_{1} / |Z_{2}|Ðf_{2} = ( |Z_{1}| / |Z_{2}| )Ð(f_{1} – f_{2})
In summary:
- use the rectangular form for addition and subtraction,
- use the polar form for multiplication and division.
Admittance
An impedance Z comprising a resistance R in series with a reactance X can be converted to an admittance Y comprising a conductance G in parallel with a susceptance B:
Y = Z^{ -1} = 1 / (R + jX) = (R – jX) / (R^{2} + X^{2}) = R / (R^{2} + X^{2}) – jX / (R^{2} + X^{2}) = G – jB
G = R / (R^{2} + X^{2}) = R / |Z|^{2}
B = X / (R^{2} + X^{2}) = X / |Z|^{2}
Using the polar form of impedance Z:
Y = 1 / |Z|Ðf = |Z|^{ -1}Ð-f = |Y|Ð-f = |Y|cosf – j|Y|sinf
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) / (G^{2} + B^{2}) = G / (G^{2} + B^{2}) + jB / (G^{2} + B^{2}) = R + jX
R = G / (G^{2} + B^{2}) = G / |Y|^{2}
X = B / (G^{2} + B^{2}) = B / |Y|^{2}
Using the polar form of admittance Y:
Z = 1 / |Y|Ð-f = |Y|^{ -1}Ðf = |Z|Ðf = |Z|cosf + j|Z|sinf
The total impedance Z_{S} of impedances Z_{1}, Z_{2}, Z_{3},… connected in series is:
Z_{S} = Z_{1} + Z_{1} + Z_{1} +…
The total admittance Y_{P} of admittances Y_{1}, Y_{2}, Y_{3},… connected in parallel is:
Y_{P} = Y_{1} + Y_{1} + Y_{1} +…
In summary:
- use impedances when operating on series circuits,
- use admittances when operating on parallel circuits.
Reactance
Inductive Reactance
The inductive reactance X_{L} of an inductance L at angular frequency w and frequency f is:
X_{L} = wL = 2pfL
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 = X_{L}I coswt
The current through an inductance lags the voltage across it by 90°.
Capacitive Reactance
The capacitive reactance X_{C} of a capacitance C at angular frequency w and frequency f is:
X_{C} = 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 / X_{C}
The current through a capacitance leads the voltage across it by 90°.
Resonance
Series Resonance
A series circuit comprising an inductance L, a resistance R and a capacitance C has an impedance Z_{S} of:
Z_{S} = R + j(X_{L} – X_{C})
where X_{L} = wL and X_{C} = 1 / wC
At resonance, the imaginary part of Z_{S} is zero:
X_{C} = X_{L}
Z_{Sr} = R
w_{r} = (1 / LC)^{½} = 2pf_{r}
The quality factor at resonance Q_{r} is:
Q_{r} = w_{r}L / R = (L / CR^{2})^{½} = (1 / R )(L / C)^{½} = 1 / w_{r}CR
Parallel resonance
A parallel circuit comprising an inductance L with a series resistance R, connected in parallel with a capacitance C, has an admittance Y_{P} of:
Y_{P} = 1 / (R + jX_{L}) + 1 / (- jX_{C}) = (R / (R^{2} + X_{L}^{2})) – j(X_{L} / (R^{2} + X_{L}^{2}) – 1 / X_{C})
where X_{L} = wL and X_{C} = 1 / wC
At resonance, the imaginary part of Y_{P} is zero:
X_{C} = (R^{2} + X_{L}^{2}) / X_{L} = X_{L} + R^{2} / X_{L} = X_{L}(1 + R^{2} / X_{L}^{2})
Z_{Pr} = Y_{Pr}^{-1} = (R^{2} + X_{L}^{2}) / R = X_{L}X_{C} / R = L / CR
w_{r} = (1 / LC – R^{2} / L^{2})^{½} = 2pf_{r}
The quality factor at resonance Q_{r} is:
Q_{r} = w_{r}L / R = (L / CR^{2} – 1)^{½} = (1 / R )(L / C – R^{2})^{½}
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.
Reactive Loads and Power Factor
Resistance and Series Reactance
The impedance Z of a reactive load comprising resistance R and series reactance X is:
Z = R + jX = |Z|Ðf
Converting to the equivalent admittance Y:
Y = 1 / Z = 1 / (R + jX) = (R – jX) / (R^{2} + X^{2}) = R / |Z|^{2} – jX / |Z|^{2}
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} = I_{P} – jI_{Q}
The active current I_{P} and the reactive current I_{Q} are:
I_{P} = VR / |Z|^{2} = |I|cosf
I_{Q} = VX / |Z|^{2} = |I|sinf
The apparent power S, active power P and reactive power Q are:
S = V|I| = V^{2} / |Z| = |I|^{2}|Z|
P = VI_{P} = I_{P}^{2}|Z|^{2} / R = V^{2}R / |Z|^{2} = |I|^{2}R
Q = VI_{Q} = I_{Q}^{2}|Z|^{2} / X = V^{2}X / |Z|^{2} = |I|^{2}X
The power factor cosf and reactive factor sinf are:
cosf = I_{P} / |I| = P / S = R / |Z|
sinf = I_{Q} / |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 = I_{P} – jI_{Q}
The active current I_{P} and the reactive current I_{Q} are:
I_{P} = VG = V / R = |I|cosf
I_{Q} = VB = V / X = |I|sinf
The apparent power S, active power P and reactive power Q are:
S = V|I| = |I|^{2} / |Y| = V^{2}|Y|
P = VI_{P} = I_{P}^{2} / G = |I|^{2}G / |Y|^{2} = V^{2}G
Q = VI_{Q} = I_{Q}^{2} / B = |I|^{2}B / |Y|^{2} = V^{2}B
The power factor cosf and reactive factor sinf are:
cosf = I_{P} / |I| = P / S = G / |Y|
sinf = I_{Q} / |I| = Q / S = B / |Y|
Complex Power
When a voltage V causes a current I to flow through a reactive load Z, the complex power S is:
S = VI* where I* is the conjugate of the complex current I.
Inductive Load
Z = R + jX_{L}
I = I_{P} – jI_{Q}
cosf = R / |Z| (lagging)
I* = I_{P} + jI_{Q}
S = P + jQ
An inductive load is a sink of lagging VArs (a source of leading VArs).
Capacitive Load
Z = R – jX_{C}
I = I_{P} + jI_{Q}
cosf = R / |Z| (leading)
I* = I_{P} – jI_{Q}
S = P – jQ
A capacitive load is a source of lagging VArs (a sink of leading VArs).
Three Phase Power
For a balanced star connected load with line voltage V_{line} and line current I_{line}:
V_{star} = V_{line} / Ö3
I_{star} = I_{line}
Z_{star} = V_{star} / I_{star} = V_{line} / Ö3I_{line}
S_{star} = 3V_{star}I_{star} = Ö3V_{line}I_{line} = V_{line}^{2} / Z_{star} = 3I_{line}^{2}Z_{star}
For a balanced delta connected load with line voltage V_{line} and line current I_{line}:
V_{delta} = V_{line}
I_{delta} = I_{line} / Ö3
Z_{delta} = V_{delta} / I_{delta} = Ö3V_{line} / I_{line}
S_{delta} = 3V_{delta}I_{delta} = Ö3V_{line}I_{line} = 3V_{line}^{2} / Z_{delta} = I_{line}^{2}Z_{delta}
The apparent power S, active power P and reactive power Q are related by:
S^{2} = P^{2} + Q^{2}
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:
Z_{delta} = 3Z_{star}
Per-unit System
For each system parameter, per-unit value is equal to the actual value divided by a base value:
E_{pu} = E / E_{base}
I_{pu} = I / I_{base}
Z_{pu} = Z / Z_{base}
Select rated values as base values, usually rated power in MVA and rated phase voltage in kV:
S_{base} = S_{rated} = Ö3E_{line}I_{line}
E_{base} = E_{phase} = E_{line}/ Ö3
The base values for line current in kA and per-phase star impedance in ohms/phase are:
I_{base} = S_{base} / 3E_{base} ( = S_{base} / Ö3E_{line})
Z_{base} = E_{base} / I_{base} = 3E_{base}^{2} / S_{base} ( = E_{line}^{2} / S_{base})
Note that selecting the base values for any two of S_{base}, E_{base}, I_{base} or Z_{base} 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:
Ö3E_{1line}I_{1line} = S = Ö3E_{2line}I_{2line}
Converting to base (per-phase star) values:
3E_{1base}I_{1base} = S_{base} = 3E_{2base}I_{2base}
E_{1base} / E_{2base} = I_{2base} / I_{1base}
Z_{1base} / Z_{2base} = (E_{1base} / E_{2base})^{2}
The impedance Z_{21pu} referred to the primary side, equivalent to an impedance Z_{2pu} on the secondary side, is:
Z_{21pu} = Z_{2pu}(E_{1base} / E_{2base})^{2}
The impedance Z_{12pu} referred to the secondary side, equivalent to an impedance Z_{1pu} on the primary side, is:
Z_{12pu} = Z_{1pu}(E_{2base} / E_{1base})^{2}
Note that per-unit and percentage values are related by:
Z_{pu} = Z_{%} / 100
Symmetrical Components
In any three phase system, the line currents I_{a}, I_{b} and I_{c} may be expressed as the phasor sum of:
- a set of balanced positive phase sequence currents I_{a1}, I_{b1} and I_{c1} (phase sequence a-b-c),
- a set of balanced negative phase sequence currents I_{a2}, I_{b2} and I_{c2} (phase sequence a-c-b),
- a set of identical zero phase sequence currents I_{a0}, I_{b0} and I_{c0} (cophasal, no phase sequence).
The positive, negative and zero sequence currents are calculated from the line currents using:
I_{a1} = (I_{a} + hI_{b} + h^{2}I_{c}) / 3
I_{a2} = (I_{a} + h^{2}I_{b} + hI_{c}) / 3
I_{a0} = (I_{a} + I_{b} + I_{c}) / 3
The positive, negative and zero sequence currents are combined to give the line currents using:
I_{a} = I_{a1} + I_{a2} + I_{a0}
I_{b} = I_{b1} + I_{b2} + I_{b0} = h^{2}I_{a1} + hI_{a2} + I_{a0}
I_{c} = I_{c1} + I_{c2} + I_{c0} = hI_{a1} + h^{2}I_{a2} + I_{a0}
The residual current I_{r} is equal to the total zero sequence current:
I_{r} = I_{a0} + I_{b0} + I_{c0} = 3I_{a0} = I_{a} + I_{b} + I_{c} = I_{e}
which is measured using three current transformers with parallel connected secondaries.
I_{e} is the earth fault current of the system.
Similarly, for phase-to-earth voltages V_{ae}, V_{be} and V_{ce}, the residual voltage V_{r} is equal to the total zero sequence voltage:
V_{r} = V_{a0} + V_{b0} + V_{c0} = 3V_{a0} = V_{ae} + V_{be} + V_{ce} = 3V_{ne}
which is measured using an earthed-star / open-delta connected voltage transformer.
V_{ne} 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°
h^{2} = – 1 / 2 – jÖ3 / 2 = 1Ð240° = 1Ð-120°
Some useful properties of h are:
1 + h + h^{2} = 0
h + h^{2} = – 1 = 1Ð180°
h – h^{2} = jÖ3 = Ö3Ð90°
h^{2} – h = – jÖ3 = Ö3Ð-90°
Fault Calculations
The different types of short-circuit fault which occur on a power system are:
- single phase to earth,
- double phase,
- double phase to earth,
- three phase,
- three phase to earth.
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
I_{e} = earth fault current
E_{a} = normal phase voltage at the fault location
Z_{1} = positive phase sequence network impedance to the fault
Z_{2} = negative phase sequence network impedance to the fault
Z_{0} = zero phase sequence network impedance to the fault
Single phase to earth – fault from phase ‘a’ to earth:
V_{a} = 0I_{b} = I_{c} = 0 I_{ f} = I_{a} = I_{e} | I_{a1} = I_{a2} = I_{a0} = I_{a} / 3V_{a1} + V_{a2} + V_{a0} = 0 _{ } | I_{a1} = E_{a} / (Z_{1} + Z_{2} + Z_{0})I_{a2} = I_{a1} I_{a0} = I_{a1} | I_{ f} = 3I_{a0} = 3E_{a} / (Z_{1} + Z_{2} + Z_{0}) = I_{e}I_{a} = I_{ f} = 3E_{a} / (Z_{1} + Z_{2} + Z_{0}) _{ } |
Double phase – fault from phase ‘b’ to phase ‘c’:
V_{b} = V_{c}I_{a} = 0 I_{ f} = I_{b} = – I_{c} | I_{a1} + I_{a2} = 0I_{a0} = 0 V_{a1} = V_{a2} | I_{a1} = E_{a} / (Z_{1} + Z_{2})I_{a2} = – I_{a1} I_{a0} = 0 | I_{ f} = – jÖ3I_{a1} = – jÖ3E_{a} / (Z_{1} + Z_{2})I_{b} = I_{ f} = – jÖ3E_{a} / (Z_{1} + Z_{2}) I_{c} = – I_{ f} = jÖ3E_{a} / (Z_{1} + Z_{2}) |
Double phase to earth – fault from phase ‘b’ to phase ‘c’ to earth:
V_{b} = V_{c} = 0I_{a} = 0 I_{ f} = I_{b} + I_{c} = I_{e} | I_{a1} + I_{a2} + I_{a0} = 0V_{a1} = V_{a2} = V_{a0} _{ } | I_{a1} = E_{a} / Z_{net}I_{a2} = – I_{a1}Z_{0} / (Z_{2} + Z_{0}) I_{a0} = – I_{a1}Z_{2} / (Z_{2} + Z_{0}) | I_{ f} = 3I_{a0} = – 3E_{a}Z_{2} / S_{zz} = I_{e}I_{b} = I_{ f} / 2 – jÖ3E_{a}(Z_{2} / 2 + Z_{0}) / S_{zz} I_{c} = I_{ f} / 2 + jÖ3E_{a}(Z_{2} / 2 + Z_{0}) / S_{zz} |
Z_{net} = Z_{1} + Z_{2}Z_{0} / (Z_{2} + Z_{0}) and S_{zz} = Z_{1}Z_{2} + Z_{2}Z_{0} + Z_{0}Z_{1} = (Z_{2} + Z_{0})Z_{net}
Three phase (and three phase to earth) – fault from phase ‘a’ to phase ‘b’ to phase ‘c’ (to earth):
V_{a} = V_{b} = V_{c} (= 0)I_{a} + I_{b} + I_{c} = 0 (= I_{e}) I_{ f} = I_{a} = hI_{b} = h^{2}I_{c} | V_{a0} = V_{a} (= 0)V_{a1} = V_{a2} = 0 _{ } | I_{a1} = E_{a} / Z_{1}I_{a2} = 0 I_{a0} = 0 | I_{ f} = I_{a1} = E_{a} / Z_{1} = I_{a}I_{b} = E_{b} / Z_{1} I_{c} = E_{c} / Z_{1} |
The values of Z_{1}, Z_{2} and Z_{0} 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 Z_{0} is less than (2Z_{1} – Z_{2}).
Note also that if the system is earthed through an impedance Z_{n} (carrying current 3I_{0}) then an impedance 3Z_{n} (carrying current I_{0}) must be included in the zero sequence impedance network.
Three Phase Fault Level
The symmetrical three phase short-circuit current I_{sc} of a power system with no-load line and phase voltages E_{line} and E_{phase} and source impedance Z_{S} per-phase star is:
I_{sc} = E_{phase} / Z_{S} = E_{line} / Ö3Z_{S}
The three phase fault level S_{sc} of the power system is:
S_{sc} = 3I_{sc}^{2}Z_{S} = 3E_{phase}I_{sc} = 3E_{phase}^{2} / Z_{S} = E_{line}^{2} / Z_{S}
Note that if the X / R ratio of the source impedance Z_{S} (comprising resistance R_{S} and reactance X_{S}) is sufficiently large, then Z_{S} » X_{S}.
Transformers
If a transformer of rating S_{T} (taken as base) and per-unit impedance Z_{Tpu} is fed from a source with unlimited fault level (infinite busbars), then the per-unit secondary short-circuit current I_{2pu} and fault level S_{2pu} are:
I_{2pu} = E_{2pu} / Z_{Tpu} = 1.0 / Z_{Tpu}
S_{2pu} = I_{2pu} = 1.0 / Z_{Tpu}
If the source fault level is limited to S_{S} by per-unit source impedance Z_{Spu} (to the same base as Z_{Tpu}), then the secondary short-circuit current I_{2pu} and fault level S_{2pu} are reduced to:
I_{2pu} = E_{2pu} / (Z_{Tpu} + Z_{Spu}) = 1.0 / (Z_{Tpu} + Z_{Spu})
S_{2pu} = I_{2pu} = 1.0 / (Z_{Tpu} + Z_{Spu})
where Z_{Spu} = S_{T} / S_{S}
Thermal Short-time Rating
If a conductor which is rated to carry full load current I_{load} continuously is rated to carry a maximum fault current I_{limit} for a time t_{limit}, then a lower fault current I_{fault} can be carried for a longer time t_{fault} according to:
( I_{limit} – I_{load} )^{2} t_{limit} = ( I_{fault} – I_{load} )^{2} t_{fault}
Rearranging for I_{fault} and t_{fault}:
I_{fault} = ( I_{limit} – I_{load} ) ( t_{limit} / t_{fault} )^{½} + I_{load}
t_{fault} = t_{limit} ( I_{limit} – I_{load} )^{2} / ( I_{fault} – I_{load} )^{2}
If I_{load} is small compared with I_{limit} and I_{fault}, then:
I_{limit}^{2} t_{limit} » I_{fault}^{2} t_{fault}
I_{fault} » I_{limit} ( t_{limit} / t_{fault} )^{½}
t_{fault} » t_{limit} ( I_{limit} / I_{fault} )^{2}
Note that if the current I_{fault} is reduced by a factor of two, then the time t_{fault} is increased by a factor of four.
Instrument Transformers
Voltage Transformer
For a voltage transformer of voltampere rating S, rated primary voltage V_{P} and rated secondary voltage V_{S}, the maximum secondary current I_{Smax}, maximum secondary burden conductance G_{Bmax} and maximum primary current I_{Pmax} are:
I_{Smax} = S / V_{S}
G_{Bmax} = I_{Smax} / V_{S} = S / V_{S}^{2}
I_{Pmax} = S / V_{P} = I_{Smax}V_{S} / V_{P}
Current Transformer
For a measurement current transformer of voltampere rating S, rated primary current I_{P} and rated secondary current I_{S}, the maximum secondary voltage V_{Smax}, maximum secondary burden resistance R_{Bmax} and maximum primary voltage V_{Pmax} are:
V_{Smax} = S / I_{S}
R_{Bmax} = V_{Smax} / I_{S} = S / I_{S}^{2}
V_{Pmax} = S / I_{P} = V_{Smax}I_{S} / I_{P}
For a protection current transformer of voltampere rating S, rated primary current I_{P}, rated secondary current I_{S} and rated accuracy limit factor F, the rated secondary reference voltage V_{SF}, maximum secondary burden resistance R_{Bmax} and equivalent primary reference voltage V_{PF} are:
V_{SF} = SF / I_{S}
R_{Bmax} = V_{SF} / I_{S}F = S / I_{S}^{2}
V_{PF} = SF / I_{P} = V_{SF}I_{S} / I_{P}
Impedance Measurement
If the primary voltage V_{pri} and the primary current I_{pri} are measured at a point in a system, then the primary impedance Z_{pri} at that point is:
Z_{pri} = V_{pri} / I_{pri}
If the measured voltage is the secondary voltage V_{sec} of a voltage transformer of primary/secondary ratio N_{V} and the measured current is the secondary current I_{sec} of a current transformer of primary/secondary ratio N_{I}, then the primary impedance Z_{pri} is related to the secondary impedance Z_{sec} by:
Z_{pri} = V_{pri} / I_{pri} = V_{sec}N_{V} / I_{sec}N_{I} = Z_{sec}N_{V} / N_{I} = Z_{sec}N_{Z}
where N_{Z} = N_{V} / N_{I}
If the no-load (source) voltage E_{pri} is also measured at the point, then the source impedance Z_{Tpri} to the point is:
Z_{Tpri} = (E_{pri} – V_{pri}) / I_{pri} = (E_{sec} – V_{sec})N_{V} / I_{sec}N_{I} = Z_{Tsec}N_{V} / N_{I} = Z_{Tsec}N_{Z}
Power Factor Correction
If an inductive load with an active power demand P has an uncorrected power factor of cosf_{1} lagging, and is required to have a corrected power factor of cosf_{2} lagging, the uncorrected and corrected reactive power demands, Q_{1} and Q_{2}, are:
Q_{1} = P tanf_{1}
Q_{2} = P tanf_{2}
where tanf_{n} = (1 / cos^{2}f_{n} – 1)^{½}
The leading (capacitive) reactive power demand Q_{C} which must be connected across the load is:
Q_{C} = Q_{1} – Q_{2} = P (tanf_{1} – tanf_{2})
The uncorrected and corrected apparent power demands, S_{1} and S_{2}, are related by:
S_{1}cosf_{1} = P = S_{2}cosf_{2}
Comparing corrected and uncorrected load currents and apparent power demands:
I_{2} / I_{1} = S_{2} / S_{1} = cosf_{1} / cosf_{2}
If the load is required to have a corrected power factor of unity, Q_{2} is zero and:
Q_{C} = Q_{1} = P tanf_{1}
I_{2} / I_{1} = S_{2} / S_{1} = cosf_{1} = P / S_{1}
Shunt Capacitors
For star-connected shunt capacitors each of capacitance C_{star} on a three phase system of line voltage V_{line} and frequency f, the leading reactive power demand Q_{Cstar} and the leading reactive line current I_{line} are:
Q_{Cstar} = V_{line}^{2} / X_{Cstar} = 2pfC_{star}V_{line}^{2}
I_{line} = Q_{Cstar} / Ö3V_{line} = V_{line} / Ö3X_{Cstar}
C_{star} = Q_{Cstar} / 2pfV_{line}^{2}
For delta-connected shunt capacitors each of capacitance C_{delta} on a three phase system of line voltage V_{line} and frequency f, the leading reactive power demand Q_{Cdelta} and the leading reactive line current I_{line} are:
Q_{Cdelta} = 3V_{line}^{2} / X_{Cdelta} = 6pfC_{delta}V_{line}^{2}
I_{line} = Q_{Cdelta} / Ö3V_{line} = Ö3V_{line} / X_{Cdelta}
C_{delta} = Q_{Cdelta} / 6pfV_{line}^{2}
Note that for the same leading reactive power Q_{C}:
X_{Cdelta} = 3X_{Cstar}
C_{delta} = C_{star} / 3
Series Capacitors
For series line capacitors each of capacitance C_{series} carrying line current I_{line} on a three phase system of frequency f, the voltage drop V_{drop} across each line capacitor and the total leading reactive power demand Q_{Cseries} of the set of three line capacitors are:
V_{drop} = I_{line}X_{Cseries} = I_{line} / 2pfC_{series}
Q_{Cseries} = 3V_{drop}^{2} / X_{Cseries} = 3V_{drop}I_{line} = 3I_{line}^{2}X_{Cseries} = 3I_{line}^{2} / 2pfC_{series}
C_{series} = 3I_{line}^{2} / 2pfQ_{Cseries}
Note that the apparent power rating S_{rating} of the set of three series line capacitors is based on the line voltage V_{line} and not the voltage drop V_{drop}:
S_{rating} = Ö3V_{line}I_{line}
Reactors
Shunt Reactors
For star-connected shunt reactors each of inductance L_{star} on a three phase system of line voltage V_{line} and frequency f, the lagging reactive power demand Q_{Lstar} and the lagging reactive line current I_{line} are:
Q_{Lstar} = V_{line}^{2} / X_{Lstar} = V_{line}^{2} / 2pfL_{star}
I_{line} = Q_{Lstar} / Ö3V_{line} = V_{line} / Ö3X_{Lstar}
L_{star} = V_{line}^{2} / 2pfQ_{Lstar}
For delta-connected shunt reactors each of inductance L_{delta} on a three phase system of line voltage V_{line} and frequency f, the lagging reactive power demand Q_{Ldelta} and the lagging reactive line current I_{line} are:
Q_{Ldelta} = 3V_{line}^{2} / X_{Ldelta} = 3V_{line}^{2} / 2pfL_{delta}
I_{line} = Q_{Ldelta} / Ö3V_{line} = Ö3V_{line} / X_{Ldelta}
L_{delta} = 3V_{line}^{2} / 2pfQ_{Ldelta}
Note that for the same lagging reactive power Q_{L}:
X_{Ldelta} = 3X_{Lstar}
L_{delta} = 3L_{star}
Series Reactors
For series line reactors each of inductance L_{series} carrying line current I_{line} on a three phase system of frequency f, the voltage drop V_{drop} across each line reactor and the total lagging reactive power demand Q_{Lseries} of the set of three line reactors are:
V_{drop} = I_{line}X_{Lseries} = 2pfL_{series}I_{line}
Q_{Lseries} = 3V_{drop}^{2} / X_{Lseries} = 3V_{drop}I_{line} = 3I_{line}^{2}X_{Lseries} = 6pfL_{series}I_{line}^{2}
L_{series} = Q_{Lseries} / 6pfI_{line}^{2}
Note that the apparent power rating S_{rating} of the set of three series line reactors is based on the line voltage V_{line} and not the voltage drop V_{drop}:
S_{rating} = Ö3V_{line}I_{line}
Harmonic Resonance
If a node in a power system operating at frequency f has a inductive source reactance X_{L} per phase and has power factor correction with a capacitive reactance X_{C} per phase, the source inductance L and the correction capacitance C are:
L = X_{L} / w
C = 1 / wX_{C}
where w = 2pf
The series resonance angular frequency w_{r} of an inductance L with a capacitance C is:
w_{r} = (1 / LC)^{½} = w(X_{C} / X_{L})^{½}
The three phase fault level S_{sc} at the node for no-load phase voltage E and source impedance Z per-phase star is:
S_{sc} = 3E^{2} / |Z| = 3E^{2} / |R + jX_{L}|
If the ratio X_{L} / R of the source impedance Z is sufficiently large, |Z| » X_{L} so that:
S_{sc} » 3E^{2} / X_{L}
The reactive power rating Q_{C} of the power factor correction capacitors for a capacitive reactance X_{C} per phase at phase voltage E is:
Q_{C} = 3E^{2} / X_{C}
The harmonic number f_{r} / f of the series resonance of X_{L} with X_{C} is:
f_{r} / f = w_{r} / w = (X_{C} / X_{L})^{½} » (S_{sc} / Q_{C})^{½}
Note that the ratio X_{L} / X_{C} which results in a harmonic number f_{r} / f is:
X_{L} / X_{C} = 1 / ( f_{r} / f )^{2}
so for f_{r} / f to be equal to the geometric mean of the third and fifth harmonics:
f_{r} / f = Ö15 = 3.873
X_{L} / X_{C} = 1 / 15 = 0.067