Formula Sheet: One-Page Master Reference

1. DC Fundamentals

DC (direct current) fundamentals are the bedrock of everything else on this sheet: a steady voltage source pushing a steady current through a resistance. The three quantities are linked by Ohm's Law, and together with power and energy they describe almost every low‑frequency circuit you will analyse by hand. Keep the units consistent — volts, amperes, ohms, watts, joules — and convert prefixes before substituting, because a factor‑of‑ten slip is the most common hand‑calculation error.

In pure DC with resistive loads the current is in phase with the voltage and the power is simply the product of the two. The table below collects the core relationships (Ohm's Law in its three forms, power, energy and charge). These are exact for resistive, steady‑state conditions; they break down once you add energy‑storing elements (capacitors, inductors) or time‑varying sources, which is where the AC sections take over.

Related: Power & Energy (§5), Ohm's Law (§2).

2. Ohm's Law

Ohm's Law states that the voltage across a conductor is proportional to the current through it, the constant of proportionality being the resistance. It is the single most‑used relationship in electronics and appears in three equivalent forms depending on which quantity is unknown. The proportionality holds for ohmic (linear, temperature‑stable) devices; real components like diodes, transistors and filaments are non‑ohmic, so the law applies to their small‑signal or resistive parts rather than the whole device.

Use it to size resistors, predict voltage drops, or back out an unknown resistance from a measured voltage and current. All three forms share the same units — volts, amperes, ohms — so the arithmetic is unit‑safe as long as you keep them consistent. It pairs directly with the power formulas, which you reach for the moment a resistor must also stay within its power rating.

Related: Series/Parallel (§3), Power & Energy (§5).

3. Series and Parallel Circuits

In a series circuit the same current flows through every element and the voltages add, so the total resistance is just the sum of the individual resistances. That makes series the natural place for the voltage‑division rule: each resistor takes a share of the total voltage proportional to its resistance. In a parallel circuit the same voltage appears across every branch and the currents add, so it is the conductances (reciprocals of resistance) that sum; for two parallel resistors the product‑over‑sum shortcut R1·R2/(R1+R2) is worth memorising.

These rules let you reduce any resistive network to a single equivalent resistance, after which Ohm's Law and the division rules give every current and voltage. The assumption is linear, resistive, steady‑state behaviour. Current division in a parallel branch is the mirror image of voltage division: a branch carries a share of the total current inversely proportional to its resistance (i.e. proportional to its conductance).

Related: Kirchhoff's Laws (§4), Conductance G = 1/R (§23).

4. Kirchhoff's Laws

Kirchhoff's Current Law (KCL) is a statement of charge conservation: the algebraic sum of currents entering any node is zero, so the current in equals the current out. Kirchhoff's Voltage Law (KVL) is a statement of energy conservation around a loop: the algebraic sum of the voltages around any closed path is zero, so the sum of the rises equals the sum of the drops. Together they let you write enough equations to solve any network, no matter how tangled.

Use KCL at junctions and KVL around independent loops; the number of independent equations equals the number of unknowns (branch currents and node voltages). Sign convention matters — pick a direction for each current and a polarity for each voltage, then let the algebra's signs reveal the true direction. Mesh and nodal analysis are just systematic ways of applying KVL and KCL, and they are the standard tools once a circuit is too irregular for simple series/parallel reduction.

Related: Series/Parallel (§3) is a special case of KVL/KCL.

5. Electrical Power and Energy

Power is the rate at which energy is transferred or converted; in a circuit it is the product of voltage and current. For a resistor the three equivalent forms P = VI = I²R = V²/R let you compute dissipation from whichever pair of quantities you have. Energy is power integrated over time (E = Pt), so the joule is a watt‑second and the practical watt‑hour is 3600 joules. Charge, the quantity a battery is rated in, is current times time (Q = It), with 1 ampere‑hour equal to 3600 coulombs.

Watch the distinction between power (a rate, in watts) and energy (an amount, in joules or watt‑hours) — it is the most common conceptual slip. A component's power rating tells you the maximum it can dissipate continuously; exceeding it raises its temperature and eventually destroys it, which is why resistor wattage is chosen with margin. For AC, real power is reduced by the power factor, covered in its own section.

Related: AC Power (§13), Power Factor (§14).

6. AC Fundamentals

AC (alternating current) quantities vary with time, and the sinusoid is the workhorse because a sine wave's derivative and integral are also sine waves, making the analysis of reactive components clean. A voltage is written v(t) = Vp·sin(ωt), where Vp is the peak amplitude and ω = 2πf is the angular frequency; a current may carry a phase offset φ, i(t) = Ip·sin(ωt ± φ). The phase difference between voltage and current is the heart of reactive power and power factor.

Because instantaneous values oscillate between positive and negative, engineers describe AC with RMS (effective) values and peak values rather than a single number. The RMS value of a sine is its peak divided by √2, and it is the number a multimeter reports and the number that makes P = VI work for power. All the reactance, impedance and resonance formulas that follow are written in terms of angular frequency ω = 2πf, so converting between f and ω is a first step in almost every AC calculation.

Related: RMS (§7), Frequency (§8), Reactance (§11).

7. RMS and Peak Values

RMS (root‑mean‑square) is the "heating‑equivalent" value of a varying quantity: the DC value that would produce the same average power in a resistor. For a pure sine, Vrms = Vp/√2 ≈ 0.707·Vp, so Vp = √2·Vrms ≈ 1.414·Vrms, and the peak‑to‑peak value is twice the peak, Vpp = 2·Vp. The full‑wave average of a sine is 0.637·Vp, but it is RMS — not average — that you use for power and for the AC values quoted on supplies and meters.

These relationships assume a pure sinusoid. Other waveforms have different form factors: a square wave has Vrms = Vp, a triangle wave Vrms = Vp/√3, and a rectified signal has yet again different values. Confusing peak with RMS is a classic mistake — a 17 V "peak" supply is a 12 V RMS supply, and a 30 V peak output can exceed a part's rating even though its RMS looks harmless. Always state which value you mean.

Related: AC Fundamentals (§6), AC Power (§13).

8. Frequency, Period, and Angular Frequency

Frequency f (hertz) is cycles per second; its reciprocal is the period T (seconds), so f = 1/T. Angular frequency ω (radians per second) is the rate at which the phase angle advances, ω = 2πf, and it is the quantity that appears naturally in the reactance, impedance and resonance formulas because it is the derivative of the phase ωt. Wavelength λ ties frequency to propagation speed: λ = v/f, which in free space (v ≈ c = 3×10⁸ m/s) gives the familiar λ ≈ 300/f(MHz) metres rule of thumb.

The Hz‑versus‑rad/s distinction is easy to get wrong: 1 Hz equals 2π rad/s ≈ 6.283 rad/s, so a 60 Hz mains frequency is 377 rad/s. If a formula contains ω, feed it radians per second; if it contains f, feed it hertz. Period and frequency are reciprocals (a 1 kHz signal has a 1 ms period), and these three — f, T, ω — plus wavelength cover essentially all timing and RF conversion work.

Related: Frequency Response (§22), Resonance (§15).

9. Capacitors

A capacitor stores energy in an electric field, with capacitance defined as charge per volt, C = Q/V. The charge on a capacitor is Q = CV, and the energy stored is E = ½CV², which is why a large capacitor charged to a high voltage can hold a dangerous amount of energy even after the supply is removed. A capacitor's opposition to AC is its capacitive reactance Xc = 1/(2πfC) = 1/(ωC), which falls as frequency rises — a capacitor looks like an open circuit at DC and like a short circuit at very high frequency.

In an AC circuit the capacitor current leads the voltage by 90°, and its impedance is Zc = 1/(jωC) = −jXc. Capacitances add in parallel (CT = C1 + C2 + …) but combine like resistances in series (1/CT = 1/C1 + 1/C2 + …) — the opposite of resistors. The RC time constant τ = RC sets how fast a capacitor charges and discharges and defines the cutoff of an RC filter, both covered in the RC section.

Related: Reactance (§11), RC Circuits (§16), Resonance (§15).

10. Inductors

An inductor stores energy in a magnetic field and opposes changes in current; its voltage is proportional to the rate of change of current, v = L·(di/dt). Inductance is defined through flux linkage, λ = LI, and the energy stored is E = ½LI², which is why opening the circuit of a current‑carrying inductor can generate a large voltage spike. Its opposition to AC is inductive reactance Xl = 2πfL = ωL, which rises with frequency — an inductor looks like a short at DC and an open at high frequency, the opposite of a capacitor.

In an AC circuit the inductor voltage leads the current by 90°, and its impedance is Zl = jωL = jXl. Inductances add in series (LT = L1 + L2 + …) and combine like resistances in parallel (1/LT = 1/L1 + 1/L2 + …). The ideal inductance of a solenoid is L = μ·N²·A/l (permeability × turns² × area / magnetic path length). The RL time constant τ = L/R governs current rise and decay, handled in the RL section.

Related: Reactance (§11), RL Circuits (§17), Transformers (§19).

11. Reactance

Reactance is the opposition that a capacitor or inductor offers to AC, measured in ohms just like resistance but frequency‑dependent and — unlike resistance — not dissipating power on average. Capacitive reactance Xc = 1/(ωC) decreases as frequency increases, while inductive reactance Xl = ωL increases with frequency; at a single frequency they can be equal in magnitude, which is the condition for resonance. The negative sign in the capacitive impedance (−jXc) encodes the 90° phase shift.

Because reactance scales with frequency, a component that is negligible at audio frequency can dominate at RF — this is why high‑frequency behaviour often surprises. Reactance combines with resistance into a complex impedance Z = R + jX, and the net reactance of a series L and C is X = Xl − Xc. Understanding reactance is the gateway to impedance, AC power and resonance, all of which build on these two formulas.

Related: Impedance (§12), Resonance (§15).

12. Impedance

Impedance Z generalises resistance to AC: a complex quantity Z = R + jX whose real part is resistance (dissipates power) and whose imaginary part is reactance (stores and returns power). Its magnitude is |Z| = √(R² + X²) and its phase angle θ = arctan(X/R) tells you how far the current leads or lags the voltage. Admittance Y = 1/Z = G + jB is the reciprocal, convenient for parallel combinations.

For a series RLC the impedance is Z = R + j(ωL − 1/(ωC)), so the inductive and capacitive reactances partially cancel; at resonance they cancel completely and |Z| = R. For parallel elements it is usually easier to add admittances (1/Z = 1/R + 1/(jωL) + jωC) and invert. Once you have Z, the AC forms of Ohm's Law (V = IZ) and power (P = I²R, S = VI*) follow directly.

Related: Reactance (§11), AC Power (§13), Resonance (§15).

13. AC Power

In an AC circuit the instantaneous power p(t) = v(t)·i(t) oscillates, so the quantity that matters is the average (real) power P, which is Vrms·Irms·cos φ, where φ is the phase angle between voltage and current. Apparent power S = Vrms·Irms (in volt‑amperes) is the magnitude of the complex power, and reactive power Q = Vrms·Irms·sin φ (in volt‑amperes reactive) is the power that sloshes back and forth with the reactive elements. They obey |S|² = P² + Q², and the complex power is S = P + jQ = V·I*.

Real power is the only part actually consumed (turned into heat, light, motion); reactive power is exchanged with inductors and capacitors but not consumed, which is why it can be reduced by correction. For a purely resistive load φ = 0 so S = P and Q = 0; for a purely reactive load φ = 90° so P = 0 and S = Q. Only the resistive part of the impedance dissipates power, which is why P = Irms²·R remains valid in AC.

Related: Power Factor (§14), RMS (§7).

14. Power Factor

Power factor is the ratio of real power to apparent power, pf = P/S = cos φ, and it lies between 0 and 1 (or 0–100%). It tells you how effectively the current is being converted into useful work: a pf of 1 means all the apparent power is real, while a low pf means a large circulating reactive current that stresses wiring and sources without doing useful work. Inductive loads (motors, transformers, inductive ballasts) draw lagging current, while capacitive loads draw leading current.

Suppliers charge for apparent power (or penalise low pf), because the extra current for a given real power increases I²R losses in cables and transformers. Correction is usually done by adding capacitance in parallel with an inductive load, which supplies the reactive power locally and raises pf toward unity. Note that "leading" or "lagging" must be stated with the pf value, since both have the same magnitude but opposite sign of Q.

Related: AC Power (§13), Capacitors (§9).

15. Resonance

Resonance occurs when the inductive and capacitive reactances in a circuit are equal and opposite, Xl = Xc, so their imaginary parts cancel. This happens at the resonant frequency f0 = 1/(2π√(LC)), equivalently ω0 = 1/√(LC). In a series RLC, |Z| is minimum (equal to R) and the current is maximum at resonance; in an ideal parallel RLC, |Z| is maximum. Resonance is the principle behind tuning, filtering and energy storage in oscillators.

The sharpness of resonance is set by the quality factor Q = ω0·L/R = 1/(ω0·R·C) = (1/R)·√(L/C), and the bandwidth (the frequency span between the half‑power points) is BW = f0/Q. A high Q means a narrow, selective peak; a low Q means a broad, flat response. These three numbers — f0, Q and BW — summarise any resonant circuit and link directly to filter design.

Related: RLC Circuits (§18), Frequency Response (§22).

16. RC Circuits

A resistor and capacitor together set a time constant τ = RC that governs how quickly the capacitor voltage changes: during charging, Vc(t) = V(1 − e^(−t/τ)), and during discharging, Vc(t) = V0·e^(−t/τ). After one time constant the voltage has reached (or fallen to) about 63%, and after about 5τ it is essentially settled — the rule of thumb for "fully charged" in practice. The same τ, expressed in the frequency domain, gives the RC low‑pass cutoff fc = 1/(2πRC).

At the cutoff frequency the output of a first‑order RC filter is 1/√2 ≈ 0.707 of its low‑frequency value, i.e. −3 dB, and the phase shift is −45°; the magnitude then rolls off at 20 dB per decade and the phase approaches −90°. The general phase is θ = −arctan(f/fc) = −arctan(ωRC) for a low‑pass. Because the response is first order, RC sections are the building blocks of smoothing, coupling, timing and most audio filters.

Related: Capacitors (§9), Frequency Response (§22).

17. RL Circuits

A resistor and inductor set a time constant τ = L/R that governs how the current builds and decays: on energising, i(t) = (V/R)(1 − e^(−tR/L)), and on removal, i(t) = I0·e^(−tR/L). As with RC, about five time constants is treated as settled. In the frequency domain an RL low‑pass has cutoff fc = R/(2πL), the mirror image of the RC case, and an RL high‑pass is the complementary arrangement.

At the RL low‑pass cutoff the output is again −3 dB (1/√2 of the low‑frequency value) with a −45° phase, rolling off at 20 dB/decade toward −90°. The phase is θ = −arctan(ωL/R). RL filters appear in smoothing (with rectifiers), timing, and as the inductive half of many second‑order filters; pairing an RL and an RC section is how practical two‑pole responses are built.

Related: Inductors (§10), Frequency Response (§22).

18. RLC Circuits

A resistor, inductor and capacitor together produce the full resonance behaviour: the series impedance is Z = R + j(ωL − 1/(ωC)), so the response peaks when ωL = 1/(ωC), i.e. at f0 = 1/(2π√(LC)). Series and parallel RLC circuits both share this natural frequency (for ideal L and C) but differ in whether the impedance is minimum (series) or maximum (parallel) at resonance. The quality factor and bandwidth, Q = (1/R)·√(L/C) and BW = f0/Q, set selectivity as in the resonance section.

The damping of an RLC circuit is controlled by R relative to √(L/C): low damping (high Q) gives an under‑damped, ringing response, critical damping gives the fastest settle without overshoot, and heavy damping is over‑damped and sluggish. These second‑order circuits underpin band‑pass and band‑stop filters, oscillators and tuned amplifiers, and their time‑domain step response (rise, overshoot, settling) is the same mathematics as control‑system stability.

Related: Resonance (§15), Impedance (§12).

19. Transformers

A transformer couples two windings through a shared magnetic flux, changing voltage and current in the ratio of their turns. The turns ratio is a = Np/Ns, and the ideal voltage and current relationships are Vp/Vs = Np/Ns and Ip/Is = Ns/Np, so a step‑down in voltage is a step‑up in current by the same factor. Because an ideal transformer conserves power, Pp = Ps = Vp·Ip = Vs·Is, and the VA rating tells you the maximum product it can pass.

Reflected (transformed) impedance is one of the most useful results: the impedance seen at the primary is Zp = a²·Zs = (Np/Ns)²·Zs, which is how transformers match a source to a load (and how antenna and speaker matching networks work). Real transformers deviate from ideal — copper (I²R) losses, core (hysteresis and eddy‑current) losses, leakage inductance and finite magnetising current — so efficiency η = Pout/Pin is quoted, typically 90–99% for power transformers.

Related: Inductors (§10), Impedance (§12), AC Power (§13).

20. Decibels (dB)

The decibel expresses a ratio on a logarithmic scale. For power ratios it is dB = 10·log10(P2/P1); for voltage or current ratios it is dB = 20·log10(V2/V1) because power scales with the square of voltage at a fixed impedance. The factor of 20 is therefore not a different definition but the same ratio applied to a squared quantity. Key anchors: 0 dB is unity (no change), +3 dB is roughly double the power (×1.414 voltage), −3 dB is half power, +10 dB is ×10 power, and +20 dB is ×10 voltage.

The dBm scale references power to 1 mW: dBm = 10·log10(P/1 mW), so 0 dBm = 1 mW, +10 dBm = 10 mW, +30 dBm = 1 W. dBu references voltage to 0.775 V (the voltage that produces 1 mW into 600 Ω). Use ratios (plain dB) when comparing two levels and absolute scales (dBm, dBu) when you need a definite value. Because dB is a ratio, you add dB gains and losses around a signal chain, which is why it is the natural unit for amplifiers, attenuators, antennas and noise.

Related: Voltage/Current Gain (§21), Frequency Response (§22).

21. Voltage / Current Gain

Gain is the ratio of an output quantity to its input. Voltage gain Av = Vout/Vin, current gain Ai = Iout/In, and power gain Ap = Pout/Pin = Av·Ai. Each can be expressed as a plain ratio or in decibels (20·log10 for voltage/current, 10·log10 for power). For a simple two‑port, knowing the source and load impedances tells you how much of the available power is actually delivered, so "gain" is always stated with the reference conditions in mind.

Operational amplifiers give the most common concrete gains: the inverting configuration has gain −Rf/Rin (a 180° phase inversion), and the non‑inverting configuration has gain 1 + Rf/Rin. These ideal results assume a high open‑loop gain and the virtual‑short condition at the inverting input; real op‑amp gain is limited by the gain‑bandwidth product, so the closed‑loop gain falls as frequency rises. The same dB arithmetic applies, which is why amplifier specs are given in dB.

Related: Decibels (§20), Frequency Response (§22).

22. Frequency Response

The frequency response of a circuit is how its gain (magnitude) and phase vary with frequency, usually plotted in dB on a log‑frequency axis (Bode plot). The −3 dB points mark where the power has fallen to half (gain to 1/√2 ≈ 0.707), and the bandwidth is the difference between the upper and lower −3 dB frequencies, BW = f_high − f_low. Frequency is often discussed in decades (×10) and octaves (×2), the natural units of a log axis.

Slope is quoted in dB per decade or dB per octave: a single reactive pole (one RC or RL section) gives 20 dB/decade (≈ 6 dB/octave), and each additional pole adds another 20 dB/decade, which is how filter order sets roll‑off. The phase shift accumulates with each pole (−90° per pole for a low‑pass). Reading gain, bandwidth and roll‑off from a Bode plot is a core skill for amplifiers, filters and communication links.

Related: Resonance (§15), Decibels (§20), Gain (§21).

23. Additional Essential Formulas

This final collection gathers the high‑utility relationships that do not fit a single section: conductance G = 1/R and admittance Y = 1/Z; resistivity and wire sizing, R = ρ·l/A with the circular cross‑section A = πd²/4; temperature dependence, R(T) = R0(1 + α·ΔT); the parallel‑plate capacitor C = ε0·εr·A/d; and the solenoid inductance L = μ·N²·A/l. Magnetic relationships close the loop: B = μH, flux Φ = BA, and Faraday's law V = −N·(dΦ/dt).

Also worth keeping to hand: the speed of light c ≈ 3×10⁸ m/s and the free‑space wavelength λ = c/f; the electric field E = V/d between parallel plates; and the energy storage formulas already met (½CV², ½LI²) which reappear in switching and RF. These formulas are the "glue" between the component‑level sections and the system‑level calculations, and most of them reduce to a constant times a geometry or a material property.