Electricity & Magnetism formula solvers

Categories — 73 of them, 1787 solvers in all

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555 Astable Frequency

f = \frac{1}{\ln 2 \, (R_{1} + 2R_{2}) C}

Electrical TradeElectricity & MagnetismOutput frequency of a 555 timer in astable mode; datasheets round the exact 1/ln2 factor in the numerator to 1.44.

Available Short-Circuit Current from Percent Impedance

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

Electricity & MagnetismElectrical TradeFault current a transformer can deliver at its secondary terminals, from its full-load current and nameplate percent impedance, assuming an infinite primary source.

Bandwidth from Q and Centre Frequency

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

Electricity & MagnetismElectrical TradeWidth of a resonant circuit's passband between its half-power points, set entirely by the resonant frequency and the quality factor.

Capacitance (C = Q/V)

C = \frac{Q}{V}

Electricity & MagnetismPhysicsDefines capacitance as the charge stored per volt applied across a capacitor.

Capacitive Reactance (X_C = 1/2πfC)

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

Electricity & MagnetismPhysicsA capacitor's opposition to AC current, falling as frequency rises.

Conductor Resistance Temperature Correction

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

Electrical TradeElectricity & MagnetismCorrects a conductor's resistance from one temperature to another using the material's temperature coefficient of resistance.

Coulomb's Law

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

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

Current Divider

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

Electrical TradeElectricity & MagnetismHow a total current splits between two parallel resistors — each branch takes the share set by the opposite resistance.

DC Motor Back-EMF (Armature Equation)

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

Electricity & MagnetismElectrical TradeVoltage a spinning DC armature generates against its own supply: the terminal voltage less the drop across the armature resistance.

Decibel Power Gain

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

Electrical TradeElectricity & MagnetismGain or loss in decibels between two power levels, the logarithmic ratio used throughout audio, RF and communications work.

Decibel Voltage Gain

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

Electrical TradeElectricity & MagnetismGain or loss in decibels computed from a voltage ratio, which carries a factor of 20 because power varies with the square of voltage.

Delta Line and Phase Current

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

Electrical TradeElectricity & MagnetismIn a delta connection the line current is √3 times the current in each winding, while line and phase voltages are equal.

Delta to Wye Resistance Transformation

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

Electricity & MagnetismElectrical TradeThe wye arm at node A that behaves identically to a delta of three resistors: the product of the two delta legs touching A over the sum of all three.

Electric Charge (Q = It)

Q = I t

Electricity & MagnetismPhysicsTotal charge transferred by a steady current flowing for a given time.

Electric Field of a Point Charge

E = \frac{k_e Q}{r^{2}}

Electricity & MagnetismPhysicsThe radial electric field a distance r from a point charge Q, with kₑ = 8.9875517923×10⁹ N·m²/C².

Electric Field Strength (E = F/q)

E = \frac{F}{q}

Electricity & MagnetismPhysicsDefines the electric field at a point as the force a test charge feels there, divided by that charge.

Electric Potential of a Point Charge

V = \frac{k_e Q}{r}

Electricity & MagnetismPhysicsThe electric potential a distance r from a point charge Q, measured against zero at infinity.

Electrical Energy (E = Pt)

E = P t

Electricity & MagnetismPhysicsEnergy consumed by a device drawing constant power over a period of time.

Electrical Power (P = I²R)

P = I^{2} R

Electricity & MagnetismPhysicsPower dissipated as heat in a resistance carrying a current (Joule heating).

Electrical Power (P = V²/R)

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

Electricity & MagnetismPhysicsPower dissipated in a resistance held at a fixed voltage.

Electrical Power (P = VI)

P = V I

Electricity & MagnetismPhysicsPower delivered to a component as the product of the voltage across it and the current through it.

EMF Induced in a Coupled Coil

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

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

Energy Stored in a Capacitor

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

Electricity & MagnetismPhysicsEnergy banked in a capacitor's electric field from its capacitance and voltage.

Energy Stored in an Inductor

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

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

Faraday's Law of Induction

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

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

Force Between Parallel Wires

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

Electricity & MagnetismPhysicsMagnetic force between two parallel current-carrying wires separated by distance d.

Induction Motor Slip

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

Electrical TradeElectricity & MagnetismPercent slip of an induction motor: how far the rotor falls behind the rotating magnetic field, as a share of synchronous speed.

Inductive Reactance (X_L = 2πfL)

X_L = 2\pi f L

Electricity & MagnetismPhysicsAn inductor's opposition to AC current, rising in proportion to frequency.

Inverting Op-Amp Gain

G = -\frac{R_{f}}{R_{in}}

Electrical TradeElectricity & MagnetismClosed-loop gain of an inverting operational amplifier, set purely by the ratio of feedback resistor to input resistor.

Kirchhoff's Current Law (Node with Three Branches)

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

Electricity & MagnetismElectrical TradeCharge cannot pile up at a junction, so the current arriving at a node equals the sum of the currents leaving it.

Kirchhoff's Voltage Law (Three-Element Loop)

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

Electricity & MagnetismElectrical TradeAround any closed loop the source voltage equals the sum of the drops — here a source feeding three series elements.

LC Resonant Frequency

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

Electricity & MagnetismWaves & OscillationsPhysicsThe natural oscillation frequency of an inductor-capacitor pair.

LED Series Resistor

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

Electrical TradeElectricity & MagnetismResistor needed to run an LED at a chosen current from a given supply, absorbing the difference above the forward voltage.

Magnetic Field of a Long Straight Wire

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

Electricity & MagnetismPhysicsThe magnetic field circling a long straight conductor, falling off as 1/r with distance from its axis.

Magnetic Field of a Solenoid

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

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

Magnetic Flux (Φ = BA cos θ)

\Phi = B A \cos\theta

Electricity & MagnetismPhysicsThe magnetic field threading a surface: field times area times the cosine of the tilt angle.

Magnetic Force on a Current-Carrying Wire

F = B I L \sin\theta

Electricity & MagnetismPhysicsMagnitude of the force on a straight current-carrying wire in a uniform magnetic field.

Magnetic Force on a Moving Charge

F = q v B \sin\theta

Electricity & MagnetismPhysicsMagnitude of the magnetic (Lorentz) force on a charge moving through a magnetic field.

Maximum Power Transfer to a Matched Load

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

Electricity & MagnetismElectrical TradeThe most power a source can hand to a load, which happens when the load resistance is made equal to the source's own Thevenin resistance.

Motional EMF (ε = BLv)

\varepsilon = B L v

Electricity & MagnetismPhysicsVoltage generated across a conductor of length L moving at speed v through a field B.

Motor Efficiency

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

Electrical TradeElectricity & MagnetismPercentage efficiency of a motor or drive as mechanical output power divided by electrical input power, times one hundred.

Motor Torque from Power and Speed

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

Electrical TradeElectricity & MagnetismShaft torque a motor develops at a given output power and speed — the trade's 5252·HP/RPM rule written in SI, with N in revolutions per second.

Mutual Inductance of Coupled Coils

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

Electricity & MagnetismPhysicsThe mutual inductance of two magnetically coupled coils, from their self-inductances and the coupling coefficient k.

Non-Inverting Op-Amp Gain

G = 1 + \frac{R_{f}}{R_{in}}

Electrical TradeElectricity & MagnetismClosed-loop gain of a non-inverting operational amplifier, always one more than the feedback-to-ground resistor ratio.

Norton Current from the Thevenin Equivalent

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

Electricity & MagnetismElectrical TradeEvery Thevenin source has an identical Norton twin: the same resistance in parallel, driven by the current the source would deliver into a dead short.

Number of Elementary Charges (N = Q/e)

N = \frac{Q}{e}

Electricity & MagnetismPhysicsHow many elementary charges make up a given charge, with e = 1.602176634×10⁻¹⁹ C exactly.

Ohm's Law

V = I R

Electricity & MagnetismPhysicsRelates the voltage across a conductor to the current through it and its resistance.

Parallel RLC Admittance

Y = \sqrt{G^{2} + (B_{C} - B_{L})^{2}}

Electricity & MagnetismElectrical TradeTotal admittance of a parallel RLC branch in siemens, where the capacitive and inductive susceptances cancel before joining the conductance.

Per-Unit Base Impedance

Z_{base} = \frac{V_{base}^{2}}{S_{base}}

Electricity & MagnetismElectrical TradeThe ohms that count as 1.0 per unit on a chosen voltage and apparent-power base — the yardstick every per-unit fault study is measured against.

Percent Voltage Drop

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

Electrical TradeElectricity & MagnetismVoltage drop expressed as a percentage of the supply voltage, the form codes use for the 3% branch and 5% total limits.

Peukert's Law (Battery Runtime)

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

Electricity & MagnetismElectrical TradeHow long a battery really lasts at a chosen current, given its rated capacity, the discharge time that rating was measured over, and its Peukert exponent.

Phase Angle from Power Factor

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

Electrical TradeElectricity & MagnetismThe angle by which current lags or leads voltage, found from the power factor — the link between the ratio and the triangle.

Phasor Sum of Two Series Impedances

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

Electricity & MagnetismElectrical TradeMagnitude of two series impedances added as phasors — the law of cosines on the impedance triangle, where Δθ is the angle between the two.

Power Factor from Real and Apparent Power

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

Electrical TradeElectricity & MagnetismPower factor as the ratio of real power in watts to apparent power in volt-amperes, the share of supplied capacity doing work.

Power-Factor Correction Capacitance

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

Electrical TradeElectricity & MagnetismCapacitance needed across a single-phase load to supply a given number of vars at the supply voltage and frequency.

Power-Factor Correction kvar

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

Electrical TradeElectricity & MagnetismSize of the capacitor bank in vars needed to lift a load from its existing power factor up to a chosen target power factor.

Q Factor of a Series Resonant Circuit

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

Electrical TradeElectricity & MagnetismQuality factor of a series RLC circuit, measuring how sharply it resonates and how much it magnifies voltage at resonance.

RC Capacitor Charging

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

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

RC Capacitor Discharge

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

Electricity & MagnetismPhysicsExponential decay of the voltage on a capacitor discharging through a resistor.

RC Cutoff Frequency

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

Electricity & MagnetismElectrical TradeThe −3 dB corner of a resistor-capacitor filter, where the output has fallen to 0.707 of the input and the phase shift is 45°.

RC Time Constant

\tau = R C

Electricity & MagnetismPhysicsCharacteristic charging/discharging time of a resistor-capacitor circuit.

Reactive Power (Power Triangle)

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

Electrical TradeElectricity & MagnetismReactive power in vars from the power triangle, where apparent power is the hypotenuse over real and reactive legs.

Resistance of a Wire (R = ρL/A)

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

Electricity & MagnetismPhysicsResistance from material resistivity, length, and cross-sectional area.

RL Cutoff Frequency

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

Electricity & MagnetismElectrical TradeCorner frequency of a resistor-inductor filter, reached when the inductive reactance 2πfL has grown to equal the resistance.

RL Time Constant (τ = L/R)

\tau = \frac{L}{R}

Electricity & MagnetismPhysicsHow quickly current builds or decays in an inductor-resistor circuit.

RMS and Peak Voltage

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

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

Series RL or RC Impedance

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

Electrical TradeElectricity & MagnetismImpedance magnitude of a resistance in series with a single reactance, whether that reactance is inductive or capacitive.

Series RLC Impedance

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

Electrical TradeElectricity & MagnetismMagnitude of impedance in a series RLC circuit, combining resistance with the net reactance left after XL and XC cancel.

Single-Phase Real Power with Power Factor

P = V I \, \text{PF}

Electrical TradeElectricity & MagnetismTrue power of a single-phase AC load: volts times amps times power factor, the fraction of the current doing real work.

Skin Depth

\delta = \sqrt{\frac{2\rho}{\omega\mu}}

Electricity & MagnetismElectrical TradeDepth at which alternating current has fallen to 1/e of its surface value — why fat conductors and high frequencies waste copper.

Synchronous Speed from Frequency and Poles

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

Electrical TradeElectricity & MagnetismSpeed of an AC machine's rotating field from supply frequency and pole count — the familiar 120f/p when read out in rpm.

Thevenin Resistance from an Open-Circuit and Loaded Measurement

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

Electricity & MagnetismElectrical TradeInternal (Thevenin) resistance of any source, found by measuring its open-circuit voltage and then the voltage it holds under a known load.

Three-Phase Apparent Power

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

Electrical TradeElectricity & MagnetismApparent power in volt-amperes for a balanced three-phase load — the quantity that sizes transformers, cables and breakers.

Three-Phase Motor Full-Load Current

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

Electrical TradeElectricity & MagnetismLine current of a three-phase motor from its shaft output power, voltage, power factor and nameplate efficiency in percent.

Three-Phase Real Power

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

Electrical TradeElectricity & MagnetismReal power drawn by a balanced three-phase load from its line-to-line voltage, line current, and power factor.

Transformer Efficiency from Core and Copper Losses

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

Electricity & MagnetismElectrical TradeEfficiency of a transformer at any fraction of rated load, from the constant core loss and the full-load copper loss that the open- and short-circuit tests measure.

Transformer Percent-Impedance Voltage Drop

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

Electrical TradeElectricity & MagnetismVolts lost inside a transformer's own windings at a given loading, taken straight from its nameplate percent impedance.

Transformer Voltage Ratio

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

Electricity & MagnetismPhysicsRelates the primary and secondary voltages of an ideal transformer to its turns ratio.

Two Capacitors in Parallel

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

Electricity & MagnetismPhysicsCombines two parallel capacitors by simply adding their capacitances.

Two Capacitors in Series

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

Electricity & MagnetismPhysicsCombines two series capacitors into a total that is smaller than either one.

Two Inductors in Parallel

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

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

Two Inductors in Series

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

Electricity & MagnetismPhysicsCombines two uncoupled series inductors by adding their inductances, exactly as series resistors add.

Two Resistors in Parallel

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

Electricity & MagnetismPhysicsTotal resistance of two resistors connected side by side: product over sum, always less than either branch.

Two Resistors in Series

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

Electricity & MagnetismPhysicsTotal resistance of two resistors connected end to end is simply their sum.

Uniform Field Between Parallel Plates (E = V/d)

E = \frac{V}{d}

Electricity & MagnetismPhysicsThe uniform electric field between two parallel plates: the voltage across them divided by the gap.

Utility Demand Charge

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

Electrical TradeElectricity & MagnetismThe part of a commercial electricity bill charged on the highest average power drawn in any interval of the month, rather than on energy used.

Voltage Divider

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

Electrical TradeElectricity & MagnetismOutput of two resistors in series across a source: the input voltage split in proportion to the lower resistor's share.

Voltage Drop, Single Phase

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

Electrical TradeElectricity & MagnetismVoltage lost in a single-phase run from conductor resistivity, one-way length, current and area — the 2 counts both conductors.

Voltage Drop, Three Phase

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

Electrical TradeElectricity & MagnetismLine-to-line voltage drop on a balanced three-phase run, using √3 rather than 2 because there is no return conductor.

Voltage Regulation

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

Electrical TradeElectricity & MagnetismHow far a source's terminal voltage sags between no load and full load, stated as a percentage of the full-load voltage.

Work to Move a Charge (W = qΔV)

W = q \, \Delta V

Electricity & MagnetismPhysicsThe work done on a charge q carried through a potential difference ΔV — the energy it gains or gives up.

Wye Line and Phase Voltage

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

Electrical TradeElectricity & MagnetismIn a wye (star) connection the line-to-line voltage is √3 times the line-to-neutral phase voltage of each winding.

Wye to Delta Resistance Transformation

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

Electricity & MagnetismElectrical TradeThe delta leg between nodes A and B equivalent to a three-arm wye: the sum of the pairwise arm products divided by the arm at the opposite node.