قوانين ومعادلات كهربية مهمة

هذه مجموعة من المعادلات الكهربية الاساسية وفى الغالب تضم جميع الاساسيات 

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 = (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.

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) / (R² + X²) = R / (R² + X²) - jX / (R² + X²) = G - jB
G = R / (R² + X²) = R / |Z|²
B = X / (R² + X²) = X / |Z|²
Using the polar form of impedance Z:
Y = 1 / |Z|Ðf = |Z| ^-1Ð-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) / (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 Z_S of impedances Z₁, Z₂, Z₃,... connected in series is:
Z_S = Z₁ + Z₁ + Z₁ +...
The total admittance Y_P of admittances Y₁, Y₂, Y₃,... connected in parallel is:
Y_P = Y₁ + Y₁ + Y₁ +...

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_LI 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

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 / (R² + X_L²)) - j(X_L / (R² + X_L²) - 1 / X_C)
where X_L = wL and X_C = 1 / wC

At resonance, the imaginary part of Y_P is zero:
X_C = (R² + X_L²) / X_L = X_L + R² / X_L = X_L(1 + R² / X_L²)
Z_Pr = Y_Pr^-1 = (R² + X_L²) / R = X_LX_C / R = L / CR
w_r = (1 / LC - R² / L²)^½ = 2pf_r

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) / (R² + X²) = 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| = V² / |Z| = |I|²|Z|
P = VI_P = I_P²|Z|² / R = V²R / |Z|² = |I|²R
Q = VI_Q = I_Q²|Z|² / X = V²X / |Z|² = |I|²X

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| = V²|Y|
P = VI_P = I_P² / G = |I|²G / |Y|² = V²G
Q = VI_Q = I_Q² / B = |I|²B / |Y|² = V²B

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_starI_star = Ö3V_lineI_line = V_line² / Z_star = 3I_line²Z_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_deltaI_delta = Ö3V_lineI_line = 3V_line² / Z_delta = I_line²Z_delta

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:
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_lineI_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_base² / S_base ( = E_line² / 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_1lineI_1line = S = Ö3E_2lineI_2line

Converting to base (per-phase star) values:
3E_1baseI_1base = S_base = 3E_2baseI_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 + h²I_c) / 3
I_a2 = (I_a + h²I_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 = h²I_a1 + hI_a2 + I_a0
I_c = I_c1 + I_c2 + I_c0 = hI_a1 + h²I_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°
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°

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
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

Z_net = Z₁ + Z₂Z₀ / (Z₂ + Z₀) and S_zz = Z₁Z₂ + Z₂Z₀ + Z₀Z₁ = (Z₂ + Z₀)Z_net

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

Note that the single phase fault current is greater than the three phase fault current if Z₀ is less than (2Z₁ - Z₂).

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 if the system is earthed through an impedance Z_n (carrying current 3I₀) then an impedance 3Z_n (carrying current I₀) 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 / Ö3|Z_S|

The three phase fault level S_sc of the power system is:
S_sc = 3I_sc²|Z_S| = 3E_phaseI_sc = 3E_phase² / |Z_S| = E_line² / |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, |Z_S| » X_S.

Power Factor Correction

If an inductive load with an active power demand P has an uncorrected power factor of cosf₁ lagging, and is required to have a corrected power factor of cosf₂ lagging, the uncorrected and corrected reactive power demands, Q₁ and Q₂, are:
Q₁ = P tanf₁
Q₂ = P tanf₂
where tanf_n = (1 / cos²f_n - 1)^½

The leading (capacitive) reactive power demand Q_C which must be connected across the load is:
Q_C = 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:
Q_C = Q₁ = P tanf₁
I₂ / I₁ = S₂ / S₁ = cosf₁ = P / S₁

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_line² / X_Cstar = 2pfC_starV_line²
I_line = Q_Cstar / Ö3V_line = V_line / Ö3X_Cstar
C_star = Q_Cstar / 2pfV_line²

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_line² / X_Cdelta = 6pfC_deltaV_line²
I_line = Q_Cdelta / Ö3V_line = Ö3V_line / X_Cdelta
C_delta = Q_Cdelta / 6pfV_line²

Note that for the same leading reactive power Q_C:
X_Cdelta = 3X_Cstar
C_delta = C_star / 3

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_line² / X_Lstar = V_line² / 2pfL_star
I_line = Q_Lstar / Ö3V_line = V_line / Ö3X_Lstar
L_star = V_line² / 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_line² / X_Ldelta = 3V_line² / 2pfL_delta
I_line = Q_Ldelta / Ö3V_line = Ö3V_line / X_Ldelta
L_delta = 3V_line² / 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_lineX_Lseries = 2pfL_seriesI_line
Q_Lseries = 3V_drop² / X_Lseries = 3V_dropI_line = 3I_line²X_Lseries = 6pfL_seriesI_line²
L_series = Q_Lseries / 6pfI_line²

Note that the apparent power rating S_rating of the set of three line reactors is based on the line voltage V_line and not the voltage drop V_drop:
S_rating = Ö3V_lineI_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 = 3E² / |Z| = 3E² / |R + jX_L|
If the ratio X_L / R of the source impedance Z is sufficiently large, |Z| » X_L so that:
S_sc » 3E² / 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 = 3E² / 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

Dielectric Dissipation Factor

If an alternating voltage V of frequency f is applied across an insulation system comprising capacitance C and equivalent series loss resistance R_S, then the voltage V_R across R_S and the voltage V_C across C due to the resulting current I are:
V_R = IR_S
V_C = IX_C
V = (V_R² + V_C²)^½

The dielectric dissipation factor of the insulation system is the tangent of the dielectric loss angle d between V_C and V:
tand = V_R / V_C = R_S / X_C = 2pfCR_S
R_S = X_Ctand = tand / 2pfC
Note that an increase in the dielectric losses of a insulation system (from an increase in the series loss resistance R_S) results in an increase in tand. Note also that tand increases with frequency.

The dielectric power loss P is related to the capacitive reactive power Q_C by:
P = I²R_S = I²X_Ctand = Q_Ctand

The power factor of the insulation system is the cosine of the phase angle f between V_R and V:
cosf = V_R / V
so that d and f are related by:
d + f = 90°

tand and cosf are related by:
tand = 1 / tanf = cosf / sinf = cosf / (1 - cos²f)^½
so that when cosf is close to zero, tand » cosf

Note that the series loss resistance R_S is not related to the shunt leakage resistance of the insulation system (which is measured using direct current).

Notation

The library uses the symbol font for some of the notation and formulae. If the symbols for the letters 'alpha beta delta' do not appear here [a b d] then the symbol font needs to be installed before all notation and formulae will be displayed correctly.

C E e G I i k L M N P

capacitance voltage source instantaneous E conductance current instantaneous I coefficient inductance mutual inductance number of turns power

[farads, F] [volts, V] [volts, V] [siemens, S] [amps, A] [amps, A] [number] [henrys, H] [henrys, H] [number] [watts, W]

Q q R T t V v W F Y y

charge instantaneous Q resistance time constant instantaneous time voltage drop instantaneous V energy magnetic flux magnetic linkage instantaneous Y

[coulombs, C] [coulombs, C] [ohms, W] [seconds, s] [seconds, s] [volts, V] [volts, V] [joules, J] [webers, Wb] [webers, Wb] [webers, Wb]

Resistance

The resistance R of a circuit is equal to the applied direct voltage E divided by the resulting steady current I:
R = E / I

Resistances in Series

When resistances R₁, R₂, R₃, ... are connected in series, the total resistance R_S is:
R_S = R₁ + R₂ + R₃ + ...

Voltage Division by Series Resistances

When a total voltage E_S is applied across series connected resistances R₁ and R₂, the current I_S which flows through the series circuit is:
I_S = E_S / R_S = E_S / (R₁ + R₂)

The voltages V₁ and V₂ which appear across the respective resistances R₁ and R₂ are:
V₁ = I_SR₁ = E_SR₁ / R_S = E_SR₁ / (R₁ + R₂)
V₂ = I_SR₂ = E_SR₂ / R_S = E_SR₂ / (R₁ + R₂)

In general terms, for resistances R₁, R₂, R₃, ... connected in series:
I_S = E_S / R_S = E_S / (R₁ + R₂ + R₃ + ...)
V_n = I_SR_n = E_SR_n / R_S = E_SR_n / (R₁ + R₂ + R₃ + ...)
Note that the highest voltage drop appears across the highest resistance.

Resistances in Parallel

When resistances R₁, R₂, R₃, ... are connected in parallel, the total resistance R_P is:
1 / R_P = 1 / R₁ + 1 / R₂ + 1 / R₃ + ...

Alternatively, when conductances G₁, G₂, G₃, ... are connected in parallel, the total conductance G_P is:
G_P = G₁ + G₂ + G₃ + ...
where G_n = 1 / R_n

For two resistances R₁ and R₂ connected in parallel, the total resistance R_P is:
R_P = R₁R₂ / (R₁ + R₂)
R_P = product / sum

The resistance R₂ to be connected in parallel with resistance R₁ to give a total resistance R_P is:
R₂ = R₁R_P / (R₁ - R_P)
R₂ = product / difference

Current Division by Parallel Resistances

When a total current I_P is passed through parallel connected resistances R₁ and R₂, the voltage V_P which appears across the parallel circuit is:
V_P = I_PR_P = I_PR₁R₂ / (R₁ + R₂)

The currents I₁ and I₂ which pass through the respective resistances R₁ and R₂ are:
I₁ = V_P / R₁ = I_PR_P / R₁ = I_PR₂ / (R₁ + R₂)
I₂ = V_P / R₂ = I_PR_P / R₂ = I_PR₁ / (R₁ + R₂)

In general terms, for resistances R₁, R₂, R₃, ... (with conductances G₁, G₂, G₃, ...) connected in parallel:
V_P = I_PR_P = I_P / G_P = I_P / (G₁ + G₂ + G₃ + ...)
I_n = V_P / R_n = V_PG_n = I_PG_n / G_P = I_PG_n / (G₁ + G₂ + G₃ + ...)
where G_n = 1 / R_n
Note that the highest current passes through the highest conductance (with the lowest resistance).

Capacitance

When a voltage is applied to a circuit containing capacitance, current flows to accumulate charge in the capacitance:
Q = òidt = CV

Alternatively, by differentiation with respect to time:
dq/dt = i = C dv/dt
Note that the rate of change of voltage has a polarity which opposes the flow of current.

The capacitance C of a circuit is equal to the charge divided by the voltage:
C = Q / V = òidt / V

Alternatively, the capacitance C of a circuit is equal to the charging current divided by the rate of change of voltage:
C = i / dv/dt = dq/dt / dv/dt = dq/dv

Capacitances in Series

When capacitances C₁, C₂, C₃, ... are connected in series, the total capacitance C_S is:
1 / C_S = 1 / C₁ + 1 / C₂ + 1 / C₃ + ...

For two capacitances C₁ and C₂ connected in series, the total capacitance C_S is:
C_S = C₁C₂ / (C₁ + C₂)
C_S = product / sum

Voltage Division by Series Capacitances

When a total voltage E_S is applied to series connected capacitances C₁ and C₂, the charge Q_S which accumulates in the series circuit is:
Q_S = òi_Sdt = E_SC_S = E_SC₁C₂ / (C₁ + C₂)

The voltages V₁ and V₂ which appear across the respective capacitances C₁ and C₂ are:
V₁ = òi_Sdt / C₁ = E_SC_S / C₁ = E_SC₂ / (C₁ + C₂)
V₂ = òi_Sdt / C₂ = E_SC_S / C₂ = E_SC₁ / (C₁ + C₂)

In general terms, for capacitances C₁, C₂, C₃, ... connected in series:
Q_S = òi_Sdt = E_SC_S = E_S / (1 / C_S) = E_S / (1 / C₁ + 1 / C₂ + 1 / C₃ + ...)
V_n = òi_Sdt / C_n = E_SC_S / C_n = E_S / C_n(1 / C_S) = E_S / C_n(1 / C₁ + 1 / C₂ + 1 / C₃ + ...)
Note that the highest voltage appears across the lowest capacitance.

Capacitances in Parallel

When capacitances C₁, C₂, C₃, ... are connected in parallel, the total capacitance C_P is:
C_P = C₁ + C₂ + C₃ + ...

Charge Division by Parallel Capacitances

When a voltage E_P is applied to parallel connected capacitances C₁ and C₂, the charge Q_P which accumulates in the parallel circuit is:
Q_P = òi_Pdt = E_PC_P = E_P(C₁ + C₂)

The charges Q₁ and Q₂ which accumulate in the respective capacitances C₁ and C₂ are:
Q₁ = òi₁dt = E_PC₁ = Q_PC₁ / C_P = Q_PC₁ / (C₁ + C₂)
Q₂ = òi₂dt = E_PC₂ = Q_PC₂ / C_P = Q_PC₂ / (C₁ + C₂)

In general terms, for capacitances C₁, C₂, C₃, ... connected in parallel:
Q_P = òi_Pdt = E_PC_P = E_P(C₁ + C₂ + C₃ + ...)
Q_n = òi_ndt = E_PC_n = Q_PC_n / C_P = Q_PC_n / (C₁ + C₂ + C₃ + ...)
Note that the highest charge accumulates in the highest capacitance.

Inductance

When the current changes in a circuit containing inductance, the magnetic linkage changes and induces a voltage in the inductance:
dy/dt = e = L di/dt
Note that the induced voltage has a polarity which opposes the rate of change of current.

Alternatively, by integration with respect to time:
Y = òedt = LI

The inductance L of a circuit is equal to the induced voltage divided by the rate of change of current:
L = e / di/dt = dy/dt / di/dt = dy/di

Alternatively, the inductance L of a circuit is equal to the magnetic linkage divided by the current:
L = Y / I

Note that the magnetic linkage Y is equal to the product of the number of turns N and the magnetic flux F:
Y = NF = LI

Mutual Inductance

The mutual inductance M of two coupled inductances L₁ and L₂ is equal to the mutually induced voltage in one inductance divided by the rate of change of current in the other inductance:
M = E_2m / (di₁/dt)
M = E_1m / (di₂/dt)

If the self induced voltages of the inductances L₁ and L₂ are respectively E_1s and E_2s for the same rates of change of the current that produced the mutually induced voltages E_1m and E_2m, then:
M = (E_2m / E_1s)L₁
M = (E_1m / E_2s)L₂
Combining these two equations:
M = (E_1mE_2m / E_1sE_2s)^½ (L₁L₂)^½ = k_M(L₁L₂)^½
where k_M is the mutual coupling coefficient of the two inductances L₁ and L₂.

If the coupling between the two inductances L₁ and L₂ is perfect, then the mutual inductance M is:
M = (L₁L₂)^½

Inductances in Series

When uncoupled inductances L₁, L₂, L₃, ... are connected in series, the total inductance L_S is:
L_S = L₁ + L₂ + L₃ + ...

When two coupled inductances L₁ and L₂ with mutual inductance M are connected in series, the total inductance L_S is:
L_S = L₁ + L₂ ± 2M
The plus or minus sign indicates that the coupling is either additive or subtractive, depending on the connection polarity.

Inductances in Parallel

When uncoupled inductances L₁, L₂, L₃, ... are connected in parallel, the total inductance L_P is:
1 / L_P = 1 / L₁ + 1 / L₂ + 1 / L₃ + ...

Time Constants

Capacitance and resistance
The time constant of a capacitance C and a resistance R is equal to CR, and represents the time to change the voltage on the capacitance from zero to E at a constant charging current E / R (which produces a rate of change of voltage E / CR across the capacitance).

Similarly, the time constant CR represents the time to change the charge on the capacitance from zero to CE at a constant charging current E / R (which produces a rate of change of voltage E / CR across the capacitance).

If a voltage E is applied to a series circuit comprising a discharged capacitance C and a resistance R, then after time t the current i, the voltage v_R across the resistance, the voltage v_C across the capacitance and the charge q_C on the capacitance are:
i = (E / R)e ^{- t / CR}
v_R = iR = Ee ^{- t / CR}
v_C = E - v_R = E(1 - e ^{- t / CR})
q_C = Cv_C = CE(1 - e ^{- t / CR})

If a capacitance C charged to voltage V is discharged through a resistance R, then after time t the current i, the voltage v_R across the resistance, the voltage v_C across the capacitance and the charge q_C on the capacitance are:
i = (V / R)e ^{- t / CR}
v_R = iR = Ve ^{- t / CR}
v_C = v_R = Ve ^{- t / CR}
q_C = Cv_C = CVe ^{- t / CR}

Inductance and resistance
The time constant of an inductance L and a resistance R is equal to L / R, and represents the time to change the current in the inductance from zero to E / R at a constant rate of change of current E / L (which produces an induced voltage E across the inductance).

If a voltage E is applied to a series circuit comprising an inductance L and a resistance R, then after time t the current i, the voltage v_R across the resistance, the voltage v_L across the inductance and the magnetic linkage y_L in the inductance are:
i = (E / R)(1 - e ^{- tR / L})
v_R = iR = E(1 - e ^{- tR / L})
v_L = E - v_R = Ee ^{- tR / L}
y_L = Li = (LE / R)(1 - e ^{- tR / L})

If an inductance L carrying a current I is discharged through a resistance R, then after time t the current i, the voltage v_R across the resistance, the voltage v_L across the inductance and the magnetic linkage y_L in the inductance are:
i = Ie ^{- tR / L}
v_R = iR = IRe ^{- tR / L}
v_L = v_R = IRe ^{- tR / L}
y_L = Li = LIe ^{- tR / L}

Rise Time and Fall Time
The rise time (or fall time) of a change is defined as the transition time between the 10% and 90% levels of the total change, so for an exponential rise (or fall) of time constant T, the rise time (or fall time) t_10-90 is:
t_10-90 = (ln0.9 - ln0.1)T » 2.2T

The half time of a change is defined as the transition time between the initial and 50% levels of the total change, so for an exponential change of time constant T, the half time t₅₀ is :
t₅₀ = (ln1.0 - ln0.5)T » 0.69T

Note that for an exponential change of time constant T:
- over time interval T, a rise changes by a factor 1 - e ^-1 (» 0.63) of the remaining change,
- over time interval T, a fall changes by a factor e ^-1 (» 0.37) of the remaining change,
- after time interval 3T, less than 5% of the total change remains,
- after time interval 5T, less than 1% of the total change remains.

Power

The power P dissipated by a resistance R carrying a current I with a voltage drop V is:
P = V² / R = VI = I²R

Similarly, the power P dissipated by a conductance G carrying a current I with a voltage drop V is:
P = V²G = VI = I² / G

The power P transferred by a capacitance C holding a changing voltage V with charge Q is:
P = VI = CV(dv/dt) = Q(dv/dt) = Q(dq/dt) / C

The power P transferred by an inductance L carrying a changing current I with magnetic linkage Y is:
P = VI = LI(di/dt) = Y(di/dt) = Y(dy/dt) / L

Energy

The energy W consumed over time t due to power P dissipated in a resistance R carrying a current I with a voltage drop V is:
W = Pt = V²t / R = VIt = I²tR

Similarly, the energy W consumed over time t due to power P dissipated in a conductance G carrying a current I with a voltage drop V is:
W = Pt = V²tG = VIt = I²t / G

The energy W stored in a capacitance C holding voltage V with charge Q is:
W = CV² / 2 = QV / 2 = Q² / 2C

The energy W stored in an inductance L carrying current I with magnetic linkage Y is:
W = LI² / 2 = YI / 2 = Y² / 2L

Batteries

If a battery of open-circuit voltage E_B has a loaded voltage V_L when supplying load current I_L, the battery internal resistance R_B is:
R_B = (E_B - V_L) / I_L

The load voltage V_L and load current I_L for a load resistance R_L are:
V_L = I_LR_L = E_B - I_LR_B = E_BR_L / (R_B + R_L)
I_L = V_L / R_L = (E_B - V_L) / R_B = E_B / (R_B + R_L)

The battery short-circuit current I_sc is:
I_sc = E_B / R_B = E_BI_L / (E_B - V_L)

Voltmeter Multiplier

The resistance R_S to be connected in series with a voltmeter of full scale voltage V_V and full scale current drain I_V to increase the full scale voltage to V is:
R_S = (V - V_V) / I_V

The power P dissipated by the resistance R_S with voltage drop (V - V_V) carrying current I_V is:
P = (V - V_V)² / R_S = (V - V_V)I_V = I_V²R_S

Ammeter Shunt

The resistance R_P to be connected in parallel with an ammeter of full scale current I_A and full scale voltage drop V_A to increase the full scale current to I is:
R_P = V_A / (I - I_A)

The power P dissipated by the resistance R_P with voltage drop V_A carrying current (I - I_A) is:
P = V_A² / R_P = V_A(I - I_A) = (I - I_A)²R_P

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