Formula sheet

Circuit Analysis and Protection Formula Sheet

Electrical circuit formulas for Ohm's law, Kirchhoff laws, source equivalents, bridges, transients, impedance, insulation, leakage, fault current, and protection energy.

Branch: Electrical Engineering
Content: Formula sheet
Updated: Jun 20, 2026
Revision: v1.1.0 · reviewed

This formula sheet collects first-pass relationships used in electrical circuit analysis, measurement, insulation review, and protection checks. Use it with clear assumptions about DC versus AC, RMS versus peak, source impedance, frequency, grounding, temperature, measurement method, and protective-device ratings.

Protection operating window

A protective device must carry legitimate load and still detect faults that must be cleared. A first screening window is:

I_{load,max}<I_{pickup}<I_{fault,min}

Maximum interrupting-duty margin:

M_{interrupt}=I_{interrupting\ rating}-I_{fault,max}

Minimum sensitivity margin:

M_{sensitivity}=I_{fault,min}-I_{pickup}

These equations are only screens. Final settings require time-current curves, tolerances, CT accuracy, temperature, inrush, ground-fault path, source mode, and coordination margin.

Ohm law

Ohm’s law:

V=IR

Current:

\displaystyle I=\frac{V}{R}

Resistance:

\displaystyle R=\frac{V}{I}

Conductance:

\displaystyle G=\frac{1}{R}

Ohm’s law applies directly to ideal linear ohmic elements. Nonlinear devices require a curve, operating point, or small-signal model.

Kirchhoff laws

Kirchhoff current law:

\displaystyle \sum I=0

Kirchhoff voltage law:

\displaystyle \sum V=0

Nodal analysis form:

Gv=i

where $G$ is conductance matrix, $v$ is node-voltage vector, and $i$ is injected-current vector.

Use consistent current directions and voltage polarities.

Power and Joule heating

Electrical power:

P=VI

Resistive power:

P=I^2R

Alternative resistive form:

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

Energy over time:

E=Pt

Temperature rise depends on heat dissipation, ambient conditions, duty cycle, enclosure, and material limits. Electrical power alone is not a complete thermal design.

Series and parallel resistance

Series resistance:

R_{eq}=\sum_i R_i

Parallel resistance:

\displaystyle \frac{1}{R_{eq}}=\sum_i \frac{1}{R_i}

Two resistors in parallel:

\displaystyle R_{eq}=\frac{R_1R_2}{R_1+R_2}

Lead resistance, contact resistance, and temperature coefficients can matter in low-resistance or high-current circuits.

Voltage and current dividers

Two-resistor voltage divider:

\displaystyle V_{out}=V_{in}\frac{R_2}{R_1+R_2}

Loaded divider with load $R_L$ across $R_2$ :

R_{lower}=R_2\parallel R_L

\displaystyle V_{out}=V_{in}\frac{R_{lower}}{R_1+R_{lower}}

Current divider for two parallel resistors:

\displaystyle I_1=I_{total}\frac{R_2}{R_1+R_2}

Divider formulas assume the source and measurement instrument do not significantly disturb the circuit unless included in the model.

Thevenin equivalent

Open-circuit voltage:

V_{th}=V_{oc}

Equivalent impedance from open-circuit and short-circuit values:

\displaystyle Z_{th}=\frac{V_{oc}}{I_{sc}}

Load current:

\displaystyle I_L=\frac{V_{th}}{Z_{th}+Z_L}

Load voltage:

V_L=I_LZ_L

Maximum power transfer for a purely resistive DC source:

R_L=R_{th}

For AC systems, impedance can be complex and frequency dependent.

Source Transformations and Norton Form

Norton current from Thevenin form:

\displaystyle I_N=\frac{V_{th}}{Z_{th}}

Norton impedance:

Z_N=Z_{th}

Thevenin voltage from Norton form:

V_{th}=I_NZ_N

Load current from Norton form:

\displaystyle I_L=I_N\frac{Z_N}{Z_N+Z_L}

Source transformations preserve terminal behavior for a defined linear network. They do not preserve internal losses, component ratings, or fault-energy paths unless those elements remain in the model.

Capacitance

Capacitor current:

\displaystyle i_C=C\frac{dv}{dt}

Capacitor energy:

\displaystyle E_C=\frac{1}{2}CV^2

Series capacitance:

\displaystyle \frac{1}{C_{eq}}=\sum_i \frac{1}{C_i}

Parallel capacitance:

C_{eq}=\sum_i C_i

RC time constant:

\tau=RC

Ideal capacitor charging:

v_C(t)=V_f+(V_0-V_f)e^{-t/RC}

Check voltage rating, ripple current, leakage, dielectric absorption, temperature, and discharge path.

Inductance

Inductor voltage:

\displaystyle v_L=L\frac{di}{dt}

Inductor energy:

\displaystyle E_L=\frac{1}{2}LI^2

Series inductance:

L_{eq}=\sum_i L_i

Parallel inductance for uncoupled inductors:

\displaystyle \frac{1}{L_{eq}}=\sum_i \frac{1}{L_i}

RL time constant:

\displaystyle \tau=\frac{L}{R}

Ideal current decay:

i(t)=I_0e^{-tR/L}

Interrupting inductive current can create high voltage unless energy is clamped, absorbed, or commutated safely.

AC impedance and admittance

Impedance:

Z=R+jX

Admittance:

\displaystyle Y=\frac{1}{Z}=G+jB

Inductive reactance:

X_L=\omega L=2\pi fL

Capacitive reactance:

\displaystyle X_C=-\frac{1}{\omega C}=-\frac{1}{2\pi fC}

Magnitude of impedance:

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

Phase angle:

\phi=\operatorname{atan2}(X,R)

Use the two-argument arctangent so the phase remains in the correct quadrant. These formulas assume sinusoidal steady-state operation at one frequency.

Wye-Delta Conversion

For delta resistances or impedances $Z_{ab}$ , $Z_{bc}$ , and $Z_{ca}$ , the equivalent wye branch to node $a$ is:

\displaystyle Z_a=\frac{Z_{ab}Z_{ca}}{Z_{ab}+Z_{bc}+Z_{ca}}

Similarly:

\displaystyle Z_b=\frac{Z_{ab}Z_{bc}}{Z_{ab}+Z_{bc}+Z_{ca}}

\displaystyle Z_c=\frac{Z_{bc}Z_{ca}}{Z_{ab}+Z_{bc}+Z_{ca}}

For wye impedances $Z_a$ , $Z_b$ , and $Z_c$ , the equivalent delta branch between nodes $a$ and $b$ is:

\displaystyle Z_{ab}=Z_a+Z_b+\frac{Z_aZ_b}{Z_c}

Use the same frequency and operating condition for all impedances. Mutual coupling, nonlinear loads, and unbalanced sources require a more explicit model.

RMS and peak values

For an ideal sine wave:

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

\displaystyle I_{rms}=\frac{I_{peak}}{\sqrt{2}}

Peak-to-peak voltage:

V_{pp}=2V_{peak}

Use RMS values for heating and most AC power calculations. Use peak values for insulation, rectifier, clamping, and transient checks.

Bridge and low-resistance measurement

Wheatstone bridge balance:

\displaystyle \frac{R_1}{R_2}=\frac{R_3}{R_x}

Unknown resistance at balance:

\displaystyle R_x=R_3\frac{R_2}{R_1}

Four-wire voltage measurement:

\displaystyle R_x=\frac{V_{sense}}{I_{test}}

Four-wire methods reduce lead and contact resistance error in low-resistance measurements.

Insulation and leakage

Insulation resistance:

\displaystyle R_{ins}=\frac{V_{test}}{I_{leak}}

Leakage current estimate:

\displaystyle I_{leak}=\frac{V}{R_{ins}}

Capacitive leakage current:

I_C=2\pi fCV

Insulation tests must consider connected electronics, test voltage, temperature, humidity, discharge time, and safety procedures.

Fault current approximation

Simple bolted-fault estimate:

\displaystyle I_{fault}\approx \frac{V}{Z_{source}+Z_{path}}

Short-circuit current from Thevenin equivalent:

\displaystyle I_{sc}=\frac{V_{th}}{Z_{th}}

Prospective fault current margin:

M_I=I_{interrupting\ rating}-I_{available}

Protective devices must have suitable voltage rating, interrupting rating, trip curve, withstand rating, and coordination.

Mini example: Thevenin fault current

For a 24 V source with equivalent source resistance:

R_{th}=0.08\ \Omega

and path resistance:

R_{path}=0.04\ \Omega

estimated branch-end fault current is:

\displaystyle I_{fault}\approx\frac{24}{0.08+0.04}=200\ \text{A}

If the protective device pickup is set at 40 A, the minimum sensitivity margin for this case is:

M_{sensitivity}=200-40=160\ \text{A}

The same method scales to feeder studies when impedance is represented consistently. A complete protection study also checks maximum fault current, clearing time, interrupting rating, conductor withstand, grounding, and alternate source modes.

Thermal withstand and protection energy

Approximate fault energy metric:

I^2t

Conductor heating screen:

Q\propto I^2Rt

Protection selectivity condition, conceptually:

t_{downstream}<t_{upstream}

for the same downstream fault current, with coordination margin included.

Current rating alone is not enough. Verify fault level, clearing time, let-through energy, conductor withstand, arc hazard, and equipment duty.

Rating and Derating Margins

Voltage margin:

M_V=V_{rated}-V_{max}

Current margin:

M_I=I_{rated}-I_{max}

Power margin:

M_P=P_{rated}-P_{dissipated}

Percent loading:

\displaystyle Loading_{\%}=100\frac{x_{operating}}{x_{rated}}

Temperature-adjusted resistance, first-order approximation:

R(T)=R_0[1+\alpha(T-T_0)]

Ratings depend on enclosure, ambient temperature, conductor grouping, duty cycle, cooling, altitude, pollution degree, and installation category.

Ground fault and residual current

Residual current in a balanced circuit:

I_{\Delta}=|I_{line}-I_{neutral}|

For three-phase systems:

I_{\Delta}=|I_a+I_b+I_c+I_n|

A nonzero residual can indicate ground leakage or fault current, but normal capacitive leakage and filters can also contribute.

Ground-fault protection settings must be coordinated with grounding method, leakage current, touch-voltage risk, and nuisance-trip tolerance.

Measurement checks

Meter loading current:

\displaystyle I_{meter}=\frac{V}{R_{in}}

Burden voltage for current measurement:

V_{burden}=I R_{burden}

Measurement relative error:

\displaystyle e_{rel}=\frac{|x_{measured}-x_{reference}|}{|x_{reference}|}

Before measuring, verify category rating, maximum voltage, maximum current, fuse condition, lead rating, range, bandwidth, and whether the circuit can deliver dangerous energy.

Validation record

For protection calculations, record:

the circuit state, source mode, and grounding method;
whether values are DC, RMS AC, peak, or phasor quantities;
source impedance, path impedance, and temperature assumptions;
maximum load current, minimum fault current, and maximum fault current;
protective-device pickup, time-current curve, interrupting rating, and coordination margin;
measurement instrument range, category rating, calibration state, and connection method;
acceptance criteria and the action required when a margin is negative.

The formula result is useful only when it is traceable to a device rating, test result, operating limit, or protection setting.

REF

Disciplines