Circuits & Electrical Power formula sheet: printable

Circuits & Electrical Power-1 Ohm's Law

V = I R

Relates the voltage across a conductor to the current through it and its resistance.

Circuits & Electrical Power-2 Electric Charge (Q = It)

Q = I t

Total charge transferred by a steady current flowing for a given time.

Circuits & Electrical Power-3 Electrical Power (P = VI)

P = V I

Power delivered to a component as the product of the voltage across it and the current through it.

Circuits & Electrical Power-4 Electrical Power (P = I²R)

P = I^{2} R

Power dissipated as heat in a resistance carrying a current (Joule heating).

Circuits & Electrical Power-5 Electrical Power (P = V²/R)

P = \frac{V^{2}}{R}

Power dissipated in a resistance held at a fixed voltage.

Circuits & Electrical Power-6 Two Resistors in Series

R_{t} = R_{1} + R_{2}

Total resistance of two resistors connected end to end is simply their sum.

Circuits & Electrical Power-7 Two Resistors in Parallel

R_{t} = \frac{R_{1} R_{2}}{R_{1} + R_{2}}

Total resistance of two resistors connected side by side: product over sum, always less than either…

Circuits & Electrical Power-8 Voltage Divider

V_{out} = V_{in} \frac{R_{2}}{R_{1} + R_{2}}

Output of two resistors in series across a source: the input voltage split in proportion to the…

Circuits & Electrical Power-9 Current Divider

I_{1} = I_{t} \frac{R_{2}}{R_{1} + R_{2}}

How a total current splits between two parallel resistors, each branch takes the share set by the…

Circuits & Electrical Power-10 Resistance of a Wire (R = ρL/A)

R = \frac{\rho L}{A}

Resistance from material resistivity, length, and cross-sectional area.

Circuits & Electrical Power-11 Conductor Resistance Temperature Correction

R_{2} = R_{1} \left[ 1 + \alpha (T_{2} - T_{1}) \right]

Corrects a conductor's resistance from one temperature to another using the material's temperature…

Circuits & Electrical Power-12 Kirchhoff's Voltage Law (Three-Element Loop)

V_{s} = V_{1} + V_{2} + V_{3}

Around any closed loop the source voltage equals the sum of the drops: here a source feeding three…

Circuits & Electrical Power-13 Kirchhoff's Current Law (Node with Three Branches)

I_{in} = I_{1} + I_{2} + I_{3}

Charge cannot pile up at a junction, so the current arriving at a node equals the sum of the…

Circuits & Electrical Power-14 Thevenin Resistance from an Open-Circuit and Loaded Measurement

R_{Th} = R_{L} \left( \frac{V_{OC}}{V_{L}} - 1 \right)

Internal (Thevenin) resistance of any source, found by measuring its open-circuit voltage and then…

Circuits & Electrical Power-15 Norton Current from the Thevenin Equivalent

I_{N} = \frac{V_{Th}}{R_{Th}}

Every Thevenin source has an identical Norton twin: the same resistance in parallel, driven by the…

Circuits & Electrical Power-16 Maximum Power Transfer to a Matched Load

P_{max} = \frac{V_{Th}^{2}}{4 R_{Th}}

The most power a source can hand to a load, which happens when the load resistance is made equal to…

Circuits & Electrical Power-17 Delta to Wye Resistance Transformation

R_{A} = \frac{R_{ab} R_{ca}}{R_{ab} + R_{bc} + R_{ca}}

The wye arm at node A that behaves identically to a delta of three resistors: the product of the…

Circuits & Electrical Power-18 Wye to Delta Resistance Transformation

R_{ab} = \frac{R_{A} R_{B} + R_{B} R_{C} + R_{C} R_{A}}{R_{C}}

The delta leg between nodes A and B equivalent to a three-arm wye: the sum of the pairwise arm…

Circuits & Electrical Power-19 LED Series Resistor

R = \frac{V_{s} - V_{f}}{I}

Resistor needed to run an LED at a chosen current from a given supply, absorbing the difference…

Circuits & Electrical Power-20 Capacitance (C = Q/V)

C = \frac{Q}{V}

Defines capacitance as the charge stored per volt applied across a capacitor.

Circuits & Electrical Power-21 Energy Stored in a Capacitor

E = \tfrac{1}{2} C V^{2}

Energy banked in a capacitor's electric field from its capacitance and voltage.

Circuits & Electrical Power-22 Two Capacitors in Series

C_{t} = \frac{C_{1} C_{2}}{C_{1} + C_{2}}

Combines two series capacitors into a total that is smaller than either one.

Circuits & Electrical Power-23 Two Capacitors in Parallel

C_{t} = C_{1} + C_{2}

Combines two parallel capacitors by simply adding their capacitances.

Circuits & Electrical Power-24 RC Time Constant

\tau = R C

Characteristic charging/discharging time of a resistor-capacitor circuit.

Circuits & Electrical Power-25 RC Capacitor Discharge

V = V_{0} \, e^{-t/\tau}

Exponential decay of the voltage on a capacitor discharging through a resistor.

Circuits & Electrical Power-26 RC Capacitor Charging

V = V_{s}\left(1 - e^{-t/\tau}\right)

The voltage on a capacitor charging toward a supply through a resistor, rising as 1 − e^(−t/τ).

Circuits & Electrical Power-27 RL Time Constant (τ = L/R)

\tau = \frac{L}{R}

How quickly current builds or decays in an inductor-resistor circuit.

Circuits & Electrical Power-28 Energy Stored in an Inductor

E = \tfrac{1}{2} L I^{2}

Energy held in an inductor's magnetic field: half the inductance times current squared.

Circuits & Electrical Power-29 Two Inductors in Series

L_{t} = L_{1} + L_{2}

Combines two uncoupled series inductors by adding their inductances, exactly as series resistors…

Circuits & Electrical Power-30 Two Inductors in Parallel

L_{t} = \frac{L_{1} L_{2}}{L_{1} + L_{2}}

Combines two uncoupled parallel inductors into a total smaller than either one, by product over sum.

Circuits & Electrical Power-31 RMS and Peak Voltage

V_{rms} = \frac{V_{peak}}{\sqrt{2}}

For a sine wave, the effective (heating-equivalent) voltage is the peak divided by √2.

Circuits & Electrical Power-32 Inductive Reactance (X_L = 2πfL)

X_L = 2\pi f L

An inductor's opposition to AC current, rising in proportion to frequency.

Circuits & Electrical Power-33 Capacitive Reactance (X_C = 1/2πfC)

X_C = \frac{1}{2\pi f C}

A capacitor's opposition to AC current, falling as frequency rises.

Circuits & Electrical Power-34 Series RL or RC Impedance

Z = \sqrt{R^{2} + X^{2}}

Impedance magnitude of a resistance in series with a single reactance, whether that reactance is…

Circuits & Electrical Power-35 Series RLC Impedance

Z = \sqrt{R^{2} + (X_{L} - X_{C})^{2}}

Magnitude of impedance in a series RLC circuit, combining resistance with the net reactance left…

Circuits & Electrical Power-36 Phase Angle from Power Factor

\varphi = \arccos(\text{PF})

The angle by which current lags or leads voltage, found from the power factor: the link between the…

Circuits & Electrical Power-37 Phasor Sum of Two Series Impedances

Z = \sqrt{Z_{1}^{2} + Z_{2}^{2} + 2 Z_{1} Z_{2} \cos \Delta\theta}

Magnitude of two series impedances added as phasors: the law of cosines on the impedance triangle,…

Circuits & Electrical Power-38 LC Resonant Frequency

f = \frac{1}{2\pi\sqrt{LC}}

The natural oscillation frequency of an inductor-capacitor pair.

Circuits & Electrical Power-39 Q Factor of a Series Resonant Circuit

Q = \frac{1}{R} \sqrt{\frac{L}{C}}

Quality factor of a series RLC circuit, measuring how sharply it resonates and how much it…

Circuits & Electrical Power-40 Bandwidth from Q and Centre Frequency

BW = \frac{f_{0}}{Q}

Width of a resonant circuit's passband between its half-power points, set entirely by the resonant…

Circuits & Electrical Power-41 RC Cutoff Frequency

f_{c} = \frac{1}{2\pi R C}

The −3 dB corner of a resistor-capacitor filter, where the output has fallen to 0.707 of the input…

Circuits & Electrical Power-42 RL Cutoff Frequency

f_{c} = \frac{R}{2\pi L}

Corner frequency of a resistor-inductor filter, reached when the inductive reactance 2πfL has grown…

Circuits & Electrical Power-43 Decibel Voltage Gain

G_{dB} = 20 \log_{10}\!\left(\frac{V_{2}}{V_{1}}\right)

Gain or loss in decibels computed from a voltage ratio, which carries a factor of 20 because power…

Circuits & Electrical Power-44 Decibel Power Gain

G_{dB} = 10 \log_{10}\!\left(\frac{P_{2}}{P_{1}}\right)

Gain or loss in decibels between two power levels, the logarithmic ratio used throughout audio, RF…

Circuits & Electrical Power-45 Single-Phase Real Power with Power Factor

P = V I \, \text{PF}

True power of a single-phase AC load: volts times amps times power factor, the fraction of the…

Circuits & Electrical Power-46 Power Factor from Real and Apparent Power

\text{PF} = \frac{P}{S}

Power factor as the ratio of real power in watts to apparent power in volt-amperes, the share of…

Circuits & Electrical Power-47 Reactive Power (Power Triangle)

Q = \sqrt{S^{2} - P^{2}}

Reactive power in vars from the power triangle, where apparent power is the hypotenuse over real…

Circuits & Electrical Power-48 Three-Phase Real Power

P = \sqrt{3} \, V_{L} I_{L} \, \text{PF}

Real power drawn by a balanced three-phase load from its line-to-line voltage, line current, and…

Circuits & Electrical Power-49 Three-Phase Apparent Power

S = \sqrt{3} \, V_{L} I_{L}

Apparent power in volt-amperes for a balanced three-phase load: the quantity that sizes…

Circuits & Electrical Power-50 Wye Line and Phase Voltage

V_{L} = \sqrt{3} \, V_{\varphi}

In a wye (star) connection the line-to-line voltage is √3 times the line-to-neutral phase voltage…

Circuits & Electrical Power-51 Delta Line and Phase Current

I_{L} = \sqrt{3} \, I_{\varphi}

In a delta connection the line current is √3 times the current in each winding, while line and…

Circuits & Electrical Power-52 Power-Factor Correction kvar

Q_{c} = P \left( \tan\varphi_{1} - \tan\varphi_{2} \right)

Size of the capacitor bank in vars needed to lift a load from its existing power factor up to a…

Circuits & Electrical Power-53 Power-Factor Correction Capacitance

C = \frac{Q_{c}}{2\pi f V^{2}}

Capacitance needed across a single-phase load to supply a given number of vars at the supply…

Circuits & Electrical Power-54 Electrical Energy (E = Pt)

E = P t

Energy consumed by a device drawing constant power over a period of time.

Circuits & Electrical Power-55 Energy Cost from a Utility Rate

C_e = E \, p_e

Cost of the energy a system consumes: kilowatt-hours or fuel BTUs times the utility rate, for tower…

Circuits & Electrical Power-56 Utility Demand Charge

C_{d} = P_{d} \, r_{d}

The part of a commercial electricity bill charged on the highest average power drawn in any…

Circuits & Electrical Power-57 Voltage Drop, Single Phase

V_{d} = \frac{2 \rho L I}{A}

Voltage lost in a single-phase run from conductor resistivity, one-way length, current and area:…

Circuits & Electrical Power-58 Voltage Drop, Three Phase

V_{d} = \frac{\sqrt{3} \, \rho L I}{A}

Line-to-line voltage drop on a balanced three-phase run, using √3 rather than 2 because there is no…

Circuits & Electrical Power-59 Percent Voltage Drop

\%V_{d} = \frac{100 \, V_{d}}{V_{s}}

Voltage drop expressed as a percentage of the supply voltage, the form codes use for the 3% branch…

Circuits & Electrical Power-60 Peukert's Law (Battery Runtime)

t = H \left( \frac{C}{I H} \right)^{k}

How long a battery really lasts at a chosen current, given its rated capacity, the discharge time…

Circuits & Electrical Power-61 Synchronous Speed from Frequency and Poles

N_{s} = \frac{2f}{p}

Speed of an AC machine's rotating field from supply frequency and pole count: the familiar 120f/p…

Circuits & Electrical Power-62 Induction Motor Slip

s = \frac{100 \, (N_{s} - N_{r})}{N_{s}}

Percent slip of an induction motor: how far the rotor falls behind the rotating magnetic field, as…

Circuits & Electrical Power-63 Motor Torque from Power and Speed

T = \frac{P}{2\pi N}

Shaft torque a motor develops at a given output power and speed: the trade's 5252·HP/RPM rule…

Circuits & Electrical Power-64 DC Motor Back-EMF (Armature Equation)

E_{b} = V - I_{a} R_{a}

Voltage a spinning DC armature generates against its own supply: the terminal voltage less the drop…

Circuits & Electrical Power-65 Motor Efficiency

\eta = \frac{100 \, P_{out}}{P_{in}}

Percentage efficiency of a motor or drive as mechanical output power divided by electrical input…

Circuits & Electrical Power-66 Three-Phase Motor Full-Load Current

I = \frac{P_{out}}{\sqrt{3} \, V \, \text{PF} \, \eta}

Line current of a three-phase motor from its shaft output power, voltage, power factor and…

Circuits & Electrical Power-67 Motor Locked-Rotor Starting Current

I_{LR} = \frac{1000 \, k \, P}{\sqrt{3} \, V}

Inrush a three-phase motor draws at the instant of across-the-line starting, from the locked-rotor…

Circuits & Electrical Power-68 Transformer Voltage Ratio

\frac{V_{s}}{V_{p}} = \frac{N_{s}}{N_{p}}

Relates the primary and secondary voltages of an ideal transformer to its turns ratio.

Circuits & Electrical Power-69 Transformer Full-Load Current

I_{FL} = \frac{S}{k \, V}

Rated current on either side of a transformer from its kVA and that side's voltage, with one factor…

Circuits & Electrical Power-70 Transformer Efficiency from Core and Copper Losses

\eta = \frac{100 \, x S \cos\phi}{x S \cos\phi + P_{c} + x^{2} P_{cu}}

Efficiency of a transformer at any fraction of rated load, from the constant core loss and the…

Circuits & Electrical Power-71 Voltage Regulation

\%VR = \frac{100 \, (V_{nl} - V_{fl})}{V_{fl}}

How far a source's terminal voltage sags between no load and full load, stated as a percentage of…

Circuits & Electrical Power-72 Transformer Percent-Impedance Voltage Drop

V_{d} = \frac{\%Z}{100} \cdot \frac{S_{L}}{S_{R}} \cdot V_{R}

Volts lost inside a transformer's own windings at a given loading, taken straight from its…

Circuits & Electrical Power-73 Available Short-Circuit Current from Percent Impedance

I_{SC} = \frac{100 \, I_{FL}}{\%Z}

Fault current a transformer can deliver at its secondary terminals, from its full-load current and…

Circuits & Electrical Power-74 Generator Sizing from Connected Load

P_{g} = P_{c} \, D_{f} \left( 1 + m \right)

Continuous kilowatts a generator must deliver: connected load reduced by a demand factor, then…

Circuits & Electrical Power-75 Coulomb's Law

F = \frac{k_e \, q_{1} q_{2}}{r^{2}}

Electrostatic force between two point charges, with kₑ = 8.9875517923×10⁹ N·m²/C².

Circuits & Electrical Power-76 Magnetic Force on a Moving Charge

F = q v B \sin\theta

Magnitude of the magnetic (Lorentz) force on a charge moving through a magnetic field.

Circuits & Electrical Power-77 Magnetic Field of a Long Straight Wire

B = \frac{\mu_0 I}{2\pi r}

The magnetic field circling a long straight conductor, falling off as 1/r with distance from its…

Circuits & Electrical Power-78 Magnetic Force on a Current-Carrying Wire

F = B I L \sin\theta

Magnitude of the force on a straight current-carrying wire in a uniform magnetic field.

Circuits & Electrical Power-79 Force Between Parallel Wires

F = \frac{\mu_0 I_1 I_2 \ell}{2\pi d}

Magnetic force between two parallel current-carrying wires separated by distance d.

Circuits & Electrical Power-80 Magnetic Field of a Solenoid

B = \frac{\mu_0 N I}{L}

The uniform field inside a long coil of N turns and length L carrying current I.

Circuits & Electrical Power-81 Magnetic Flux (Φ = BA cos θ)

\Phi = B A \cos\theta

The magnetic field threading a surface: field times area times the cosine of the tilt angle.

Circuits & Electrical Power-82 Faraday's Law of Induction

\varepsilon = N \frac{\Delta\Phi}{\Delta t}

The EMF induced in a coil of N turns by flux changing at ΔΦ/Δt (magnitude form).

Circuits & Electrical Power-83 Mutual Inductance of Coupled Coils

M = k \sqrt{L_{1} L_{2}}

The mutual inductance of two magnetically coupled coils, from their self-inductances and the…

Circuits & Electrical Power-84 EMF Induced in a Coupled Coil

\varepsilon_{2} = M \frac{\Delta I_{1}}{\Delta t}

The voltage a changing current in one coil induces in a second coupled to it (magnitude form).

Circuits & Electrical Power-85 Motional EMF (ε = BLv)

\varepsilon = B L v

Voltage generated across a conductor of length L moving at speed v through a field B.