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555 Astable Frequency

f=1ln⁡2 (R1+2R2)Cf = \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

ISC=100 IFL%ZI_{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=f0QBW = \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.

Conductor Resistance Temperature Correction

R2=R1[1+α(T2−T1)]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.

Current Divider

I1=ItR2R1+R2I_{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.

Decibel Power Gain

GdB=10log⁡10 ⁣(P2P1)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

GdB=20log⁡10 ⁣(V2V1)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

IL=3 IφI_{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

RA=RabRcaRab+Rbc+RcaR_{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.

Induction Motor Slip

s=100 (Ns−Nr)Nss = \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.

Inverting Op-Amp Gain

G=−RfRinG = -\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)

Iin=I1+I2+I3I_{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)

Vs=V1+V2+V3V_{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.

LED Series Resistor

R=Vs−VfIR = \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.

Maximum Power Transfer to a Matched Load

Pmax=VTh24RThP_{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.

Motor Efficiency

η=100 PoutPin\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=P2πNT = \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.

Non-Inverting Op-Amp Gain

G=1+RfRinG = 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

IN=VThRThI_{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.

Parallel RLC Admittance

Y=G2+(BC−BL)2Y = \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

Zbase=Vbase2SbaseZ_{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

%Vd=100 VdVs\%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(CIH)kt = 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

φ=arccos⁡(PF)\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.

Power Factor from Real and Apparent Power

PF=PS\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=Qc2πfV2C = \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

Qc=P(tan⁡φ1−tan⁡φ2)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=1RLCQ = \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 Cutoff Frequency

fc=12πRCf_{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°.

Reactive Power (Power Triangle)

Q=S2−P2Q = \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.

RL Cutoff Frequency

fc=R2πLf_{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.

Series RL or RC Impedance

Z=R2+X2Z = \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=R2+(XL−XC)2Z = \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=VI PFP = 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

δ=2ρωμ\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

Ns=2fpN_{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

RTh=RL(VOCVL−1)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=3 VLILS = \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=Pout3 V PF ηI = \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=3 VLIL PFP = \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 Percent-Impedance Voltage Drop

Vd=%Z100⋅SLSR⋅VRV_{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.

Voltage Divider

Vout=VinR2R1+R2V_{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

Vd=2ρLIAV_{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

Vd=3 ρLIAV_{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=100 (Vnl−Vfl)Vfl\%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.

Wye Line and Phase Voltage

VL=3 VφV_{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

Rab=RARB+RBRC+RCRARCR_{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.