Formulas

All formulas

Basics & units

Usual prefixesk = 10³
m = 10⁻³
µ = 10⁻⁶
n = 10⁻⁹

Quantity	Symbol	Unit
Voltage	U	V
Current	I	A
Resistance	R	Ω
Power	P	W
Energy	W	J

Ampere-hour and kilowatt-hour1 Ah = 3 600 C
1 kWh = 3,6 × 10⁶ J

Quantity	Symbol	Unit
Quantity of electricity	Q	C
Energy	W	J

Electrostatics

Coulomb's lawF = k · q · q'd²

Quantity	Symbol	Unit
Force	F	N
Charge	q	C
Distance	d	m
Constant (9 × 10⁹)	k	N·m²/C²

Field between two platesE = Ue

Quantity	Symbol	Unit
Electric field	E	V/m
Voltage	U	V
Plate spacing	e	m

Force on a chargeF = q · E

Quantity	Symbol	Unit
Force	F	N
Charge	q	C
Electric field	E	V/m

Voltage & current

Quantity of electricityQ = I · t

Quantity	Symbol	Unit
Quantity of electricity	Q	C
Current	I	A
Time	t	s

Current densityJ = IS

Quantity	Symbol	Unit
Current density	J	A/mm²
Current	I	A
Cross-section	S	mm²

Cross-section of a round wireS = π · d²4

Quantity	Symbol	Unit
Cross-section	S	mm²
Diameter	d	mm

Ohm's law & resistance

Ohm's lawU = R · I

Quantity	Symbol	Unit
Voltage	U	V
Resistance	R	Ω
Current	I	A

Pouillet's lawR = ρ · lS

Quantity	Symbol	Unit
Resistance	R	Ω
Resistivity	ρ	Ω·m
Length	l	m
Cross-section	S	m²

Copper ρ ≈ 1.7 × 10⁻⁸ Ω·m; aluminium 2.8 × 10⁻⁸; iron 1.0 × 10⁻⁷.

Temperature effectR = R₀ · (1 + α · Δθ)

Quantity	Symbol	Unit
Resistance at 20 °C	R₀	Ω
Temperature coefficient	α	/°C
Temperature change	Δθ	°C

Power, energy, efficiency

Electrical powerP = U · I

Quantity	Symbol	Unit
Power	P	W
Voltage	U	V
Current	I	A

Joule heatingP = R · I² = U²R

Quantity	Symbol	Unit
Power	P	W
Resistance	R	Ω
Current	I	A

EnergyW = P · t

Quantity	Symbol	Unit
Energy	W	J
Power	P	W
Time	t	s

In kWh: power in kW × time in hours.

Efficiencyη = PuPa

Quantity	Symbol	Unit
Efficiency	η	-
Useful power	Pu	W
Absorbed power	Pa	W

Resistor networks

Whatever the network

Ohm's law on each resistorU_n = R_n · I_n
I_n = U_nR_n
R_n = U_nI_n

Quantity	Symbol	Unit
Voltage across resistor n	U_n	V
Resistor n	R_n	Ω
Current through it	I_n	A

Every resistor obeys Ohm's law with its own voltage and current, in series as in parallel.

Equivalent resistanceRéq = UI

Quantity	Symbol	Unit
Equivalent resistance	Réq	Ω
Voltage across the network	U	V
Total current	I	A

Req is the single resistor that, at the same voltage U, would carry the same current I.

Power in each resistorP_n = U_n · I_n
P_n = R_n · I_n²
P_n = U_n²R_n

Quantity	Symbol	Unit
Power dissipated by resistor n	P_n	W

Power balanceP = P₁ + P₂ + ...
P = U · I

Quantity	Symbol	Unit
Total power	P	W

Powers always add up, in series, in parallel or in a mixed network.

In series

Equivalent resistanceRéq = R₁ + R₂ + ...

Quantity	Symbol	Unit
Equivalent resistance	Réq	Ω

Common and split quantitiesI = I₁ = I₂ = ...
U = U₁ + U₂ + ...

Quantity	Symbol	Unit
Current (common)	I	A
Total voltage (split)	U	V

In series the current I is common to every resistor; the voltage U is split between them.

Voltage across each resistorU = Réq · I
U₁ = R₁ · I
U_n = R_n · I

Quantity	Symbol	Unit
Voltage across resistor n	U_n	V
Resistor n	R_n	Ω
Current	I	A

CurrentI = URéq
I₁ = U₁R₁
I_n = U_nR_n

Quantity	Symbol	Unit
Current	I	A
Total voltage	U	V
Equivalent resistance	Réq	Ω

Each resistanceRéq = UI
R₁ = U₁I
R_n = U_nI

Quantity	Symbol	Unit
Resistor n	R_n	Ω
Voltage across resistor n	U_n	V
Current	I	A

Voltage dividerU₁ = U · R₁R₁ + R₂

Quantity	Symbol	Unit
Voltage on R₁	U₁	V
Total voltage	U	V

In parallel

Equivalent resistance1Réq = 1R₁ + 1R₂ + ...

Quantity	Symbol	Unit
Equivalent resistance	Réq	Ω

Two resistors: Req = R₁·R₂ / (R₁ + R₂).

Common and split quantitiesU = U₁ = U₂ = ...
I = I₁ + I₂ + ...

Quantity	Symbol	Unit
Voltage (common)	U	V
Total current (split)	I	A

In parallel the voltage U is common to every branch; the current I is split between them.

Current in each branchI = URéq
I₁ = UR₁
I_n = UR_n

Quantity	Symbol	Unit
Current in branch n	I_n	A
Voltage (common)	U	V
Resistance of branch n	R_n	Ω

The branch with the smallest resistance carries the largest current.

VoltageU = Réq · I
U = R₁ · I₁ = R₂ · I₂ = ... = R_n · I_n

Quantity	Symbol	Unit
Voltage (common)	U	V
Total current	I	A
Equivalent resistance	Réq	Ω

Resistance of each branchRéq = UI
R₁ = UI₁
R_n = UI_n

Quantity	Symbol	Unit
Resistance of branch n	R_n	Ω
Voltage (common)	U	V
Current in branch n	I_n	A

Current dividerI₁ = I · R₂R₁ + R₂

Quantity	Symbol	Unit
Current in R₁	I₁	A
Total current	I	A

Two branches: the current splits inversely to the resistances (I₁ gets R₂'s share).

n identical resistorsRéq = Rn

Quantity	Symbol	Unit
Equivalent resistance	Réq	Ω
Resistance of each branch	R	Ω
Number of resistors	n	—

Mixed network

Step-by-step reductionR₂₃ = R₂ · R₃R₂ + R₃
Réq = R₁ + R₂₃

Quantity	Symbol	Unit
Equivalent resistance of R₂ ∥ R₃	R₂₃	Ω
Total equivalent resistance	Réq	Ω

Example: R₁ in series with R₂ ∥ R₃. Replace each series or parallel group by its equivalent resistance until a single resistor remains.

Other case: parallel then seriesR₂₃ = R₂ + R₃
Réq = R₁ · R₂₃R₁ + R₂₃

Quantity	Symbol	Unit
Equivalent resistance of R₂ + R₃	R₂₃	Ω
Total equivalent resistance	Réq	Ω

R₁ in parallel with the R₂ + R₃ branch: reduce the series branch first, then the parallel.

Total currentI = URéq

Quantity	Symbol	Unit
Current delivered by the generator	I	A
Generator voltage	U	V

Back down to each resistorU₁ = R₁ · I
U₂₃ = R₂₃ · I
I₂ = U₂₃R₂
I₃ = U₂₃R₃

Quantity	Symbol	Unit
Voltage on R₁	U₁	V
Voltage shared by R₂ and R₃	U₂₃	V
Current in R₂	I₂	A

Walk back the other way: a series group gets the common current (U = R · I), a parallel group gets the common voltage (I = U / R). Check: U₁ + U₂₃ = U and I₂ + I₃ = I.

Power line

Line voltage dropΔU = 2 · R_l · I

Quantity	Symbol	Unit
Voltage drop	ΔU	V
Resistance of one wire	R_l	Ω
Current	I	A

Real generators & loads

Real generatorU = E − r · I

Quantity	Symbol	Unit
EMF	E	V
Internal resistance	r	Ω
Terminal voltage	U	V

Active loadU = E' + r' · I

Quantity	Symbol	Unit
Back-EMF	E'	V
Internal resistance	r'	Ω

Generalised Ohm's lawI = ΣE − ΣE'ΣR

Quantity	Symbol	Unit
Current	I	A
Sum of loop resistances	ΣR	Ω

CharacteristicU = f(I) : droite de pente −r, ordonnée à l'origine E

Kirchhoff's laws

Junction ruleΣ I entrants = Σ I sortants

Quantity	Symbol	Unit
Current	I	A

Loop ruleΣ E − Σ R · I = 0

Quantity	Symbol	Unit
EMF	E	V
Voltage drop	R·I	V

Capacitors

CapacitanceQ = C · U

Quantity	Symbol	Unit
Charge	Q	C
Capacitance	C	F
Voltage	U	V

Stored energyW = ½ · C · U²

Quantity	Symbol	Unit
Energy	W	J

Combinationsparallèle : Céq = C₁ + C₂
série : 1Céq = 1C₁ + 1C₂

Quantity	Symbol	Unit
Equivalent capacitance	Céq	F

RC time constantτ = R · C

Quantity	Symbol	Unit
Time constant	τ	s
Resistance	R	Ω
Capacitance	C	F

3-digit marking: two digits then number of zeros, in pF (104 = 100 000 pF = 100 nF).

Magnetism & electromagnetism

Magnetic fluxΦ = B · S

Quantity	Symbol	Unit
Flux	Φ	Wb
Magnetic flux density	B	T
Area	S	m²

Field of a long coilB = µ₀ · N · Il

Quantity	Symbol	Unit
Vacuum permeability (4π × 10⁻⁷)	µ₀	T·m/A
Number of turns	N	-
Length	l	m

Laplace forceF = B · I · l

Quantity	Symbol	Unit
Force	F	N
Current	I	A
Conductor length	l	m

Induction & self-induction

Faraday's lawE = −N · ΔΦΔt

Quantity	Symbol	Unit
Induced EMF	E	V
Number of turns	N	-
Flux	Φ	Wb
Time	t	s

Cut fluxE = B · l · v

Quantity	Symbol	Unit
Magnetic flux density	B	T
Length	l	m
Speed	v	m/s

Self-inductionE = −L · ΔIΔt

Quantity	Symbol	Unit
Inductance	L	H
Current	I	A

Energy of a coilW = ½ · L · I²

Quantity	Symbol	Unit
Energy	W	J

RL time constantτ = LR

Quantity	Symbol	Unit
Time constant	τ	s

Alternating current

Sinusoidal quantityu(t) = Umax · sin(ω · t + φ)

Quantity	Symbol	Unit
Peak value	Umax	V
Angular frequency	ω	rad/s
Initial phase	φ	rad

Angular frequency, frequency, periodω = 2 · π · f
f = 1T

Quantity	Symbol	Unit
Frequency	f	Hz
Period	T	s

RMS valueUeff = Umax√2

Quantity	Symbol	Unit
RMS value	Ueff	V
Peak value	Umax	V

RLC circuits

ReactancesX_L = L · ω
X_C = 1C · ω

Quantity	Symbol	Unit
Reactance	X	Ω
Inductance	L	H
Capacitance	C	F
Angular frequency	ω	rad/s

Series impedanceZ = √(R² + (X_L − X_C)²)

Quantity	Symbol	Unit
Impedance	Z	Ω
Resistance	R	Ω

Ohm's law in ACU = Z · I

Quantity	Symbol	Unit
RMS voltage	U	V
RMS current	I	A

Phase shifttan φ = X_L − X_CR
cos φ = RZ

Quantity	Symbol	Unit
Phase shift	φ	rad

AC power

The three powersS = U · I
P = U · I · cos φ
Q = U · I · sin φ

Quantity	Symbol	Unit
Apparent power	S	VA
Active power	P	W
Reactive power	Q	var

Power triangleS² = P² + Q²
cos φ = PS

Quantity	Symbol	Unit
Power factor	cos φ	-

Power factor correctionC = P · (tan φ₁ − tan φ₂)U² · ω

Quantity	Symbol	Unit
Capacitance to add	C	F
Phase shifts before / after	φ₁, φ₂	rad

Measurement & instruments

Reading a dialvaleur = divisions luesdivisions totales × calibre

Ammeter shuntR_s = r_A · iI − i

Quantity	Symbol	Unit
Shunt resistance	R_s	Ω
Ammeter resistance	r_A	Ω
Full-scale current	i	A
Total current to measure	I	A

Series multiplier (voltmeter)R_add = Ui − r_V

Quantity	Symbol	Unit
Series multiplier	R_add	Ω
Voltmeter resistance	r_V	Ω

Volt-ammeter methodR = UI

Quantity	Symbol	Unit
Measured resistance	R	Ω

Downstream: the voltmeter is across R alone (error from the current shunted by V); upstream: the voltmeter spans A (error from the drop in A).

Simulated time00:00:00

Q = I·t-

W = P·t-