Learn DC and AC circuit analysis, power factor and three-phase basics, safe measurement on battery circuits, simulation, transformers, motors and systematic troubleshooting.

Electrical Circuits

## Why it matters

The power from a wall outlet, a generator or a utility transformer is **alternating current (AC)**: the voltage swings positive and negative many times a second. Batteries, solar panels and the inside of most electronics use **direct current (DC)**, which stays in one direction. To read a nameplate, choose a meter setting or compare a 120 V supply with a 170 V peak on an oscilloscope, you need the vocabulary of AC waveforms: peak, peak-to-peak, RMS, frequency and period.

## DC and AC

- **DC** has a steady polarity. A 9 V battery's + terminal is always about 9 V above its − terminal.
- **AC** reverses polarity in a repeating pattern. The usual shape is a **sine wave**, because that is what a rotating generator naturally produces.

AC won for power distribution mainly because transformers can step AC voltage up for efficient long-distance transmission and back down for safe use. You will see why in the transformers lesson. DC is making a comeback in solar, batteries, data centres and long high-voltage links, but buildings are still wired for AC.

## Describing a sine wave

A sine-wave voltage can be written as:

```text
v(t) = Vpeak × sin(2π × f × t)
```

- **Peak (Vpeak):** the highest value above zero.
- **Peak-to-peak (Vpp):** from the top of the wave to the bottom, `Vpp = 2 × Vpeak`.
- **Frequency (f, hertz):** complete cycles per second. North American power is 60 Hz; most of the rest of the world uses 50 Hz.
- **Period (T, seconds):** time for one cycle, `T = 1 / f`. At 60 Hz, T = 1/60 s ≈ 16.7 ms. At 50 Hz, T = 20 ms.

## RMS: the "effective" value

If you connect a heater to AC, the voltage spends most of its time below the peak, so it does not heat as much as a DC supply at the peak voltage would. The **RMS (root mean square)** value is the DC voltage that would produce the **same heating** in a resistor.

For a sine wave:

```text
Vrms = Vpeak / √2 ≈ 0.707 × Vpeak
Vpeak = Vrms × √2 ≈ 1.414 × Vrms
```

The same ratio applies to current. **When someone says "120 V AC" or "230 V AC", they mean the RMS value.** Power formulas like `P = Vrms × Irms` (for a resistor) work with RMS values exactly as they do for DC.

The √2 ratio is only for sine waves. A square wave's RMS equals its peak; a triangle wave's RMS is `Vpeak / √3`. Many cheap meters assume a sine wave; **true-RMS** meters measure correctly on distorted waveforms, which are common around variable-speed drives and electronic loads.

## Worked examples

**Example 1.** A nameplate says 120 V, 60 Hz. What are the peak, peak-to-peak and period?

```text
Vpeak = 120 × 1.414 = 169.7 V
Vpp   = 2 × 169.7  = 339.4 V
T     = 1 / 60     = 0.0167 s = 16.7 ms
```

**Example 2.** An oscilloscope (in a training lab or a simulator) shows a sine wave with a peak of 10 V and one cycle every 2 ms.

```text
f    = 1 / 0.002 = 500 Hz
Vrms = 10 / 1.414 = 7.07 V
```

**Example 3.** A 230 V RMS heater has a resistance of 26.45 Ω. What power does it use?

```text
P = Vrms² / R = 230² / 26.45 = 2,000 W
```

## Reading the waveform

- **Phase** describes where a wave is in its cycle, in degrees (360° per cycle). Two waves at the same frequency can be shifted; for example, current lags voltage in a motor. Phase shift becomes important in the next two lessons.
- **Angular frequency** `ω = 2πf` (radians per second) appears in reactance formulas. At 60 Hz, ω ≈ 377 rad/s.

## Common mistakes

- **Treating 120 V as the peak.** It is the RMS value; the peak is about 170 V. Insulation and component voltage ratings must cover the peak.
- **Using the √2 ratio on non-sine waves.** Use a true-RMS meter or the right formula for the waveform.
- **Mixing up frequency and period.** Hertz is cycles per second; period is seconds per cycle.
- **Confusing peak with peak-to-peak.** An oscilloscope often shows Vpp; halve it to get the peak.

## Safety note

Mains AC is dangerous at any of these values and can cause a fatal shock or an arc flash. In this course you look at AC waveforms only in simulators or videos. Measuring live mains circuits is done only after employer or school training, with rated meters and protective equipment, under supervision. See NFPA 70E (https://www.nfpa.org/codes-and-standards/nfpa-70e-standard-development/70e) and OSHA's electrical safety pages (https://www.osha.gov/electrical). Checked on 2026-10-23.

## Recap

- DC holds one polarity; AC reverses, usually as a sine wave.
- `T = 1 / f`; North America uses 60 Hz, most other places 50 Hz.
- `Vrms = Vpeak / √2` for a sine wave; quoted mains voltages are RMS.

## Try it

Open the **Do** tab and write the RMS formula in a small AC waveform calculator.


Describe a sine wave with peak, peak-to-peak, RMS, frequency and period, and convert between them.

AC waveforms

Difference between AC and DC Current Explained | AddOhms #5

RMS Value of AC Circuits

## Why it matters

Resistors are not the only things in AC circuits. Motor windings, transformers, fluorescent ballasts and long cables behave like **inductors**; power-factor correction banks, filters and motor start circuits use **capacitors**. Both oppose AC current in a way that changes with frequency and shifts the timing between voltage and current. That opposition is called **reactance**, and combined with resistance it is **impedance**. Without it you cannot size a capacitor, predict a motor's current or understand filters.

## Capacitors and inductors in one paragraph each

A **capacitor** is two conductors separated by an insulator. It stores energy in an electric field and resists changes in **voltage**. Its value is capacitance, C, in farads (F); practical values are microfarads (µF, 10⁻⁶ F) or nanofarads (nF, 10⁻⁹ F). A capacitor blocks steady DC once charged but passes AC, more easily at higher frequencies.

An **inductor** is a coil of wire, often around an iron core. It stores energy in a magnetic field and resists changes in **current**. Its value is inductance, L, in henries (H); practical values are millihenries (mH). An inductor passes DC easily (just the wire resistance) but opposes AC, more strongly at higher frequencies.

## Reactance

Reactance, X, is measured in ohms like resistance:

```text
Inductive reactance:   XL = 2π × f × L
Capacitive reactance:  XC = 1 / (2π × f × C)
```

- XL **rises** with frequency. Double the frequency, double XL.
- XC **falls** with frequency. Double the frequency, halve XC.

**Example 1.** A 100 mH coil at 60 Hz:

```text
XL = 2π × 60 × 0.1 = 37.7 Ω
```

**Example 2.** A 10 µF capacitor at 60 Hz:

```text
XC = 1 / (2π × 60 × 0.000010) = 265 Ω
```

At 600 Hz the same capacitor is 26.5 Ω and the same coil is 377 Ω.

## Phase shift

In a pure resistor, voltage and current peak together. In a pure inductor, current **lags** voltage by 90°. In a pure capacitor, current **leads** voltage by 90°. A memory aid used in many trade schools is **"ELI the ICE man"**: in an inductor (L), voltage E comes before current I; in a capacitor (C), current I comes before E.

## Impedance

Because reactance is 90° out of step with resistance, you cannot simply add ohms. For a resistor, inductor and capacitor in **series**, the net reactance is `X = XL − XC`, and the impedance magnitude is found like the hypotenuse of a right triangle:

```text
|Z| = √( R² + (XL − XC)² )
phase angle θ = arctan( (XL − XC) / R )
```

A positive angle means the circuit is **inductive** (current lags); negative means **capacitive** (current leads). Ohm's law works with impedance: `I = V / |Z|`.

**Example 3.** A series circuit with R = 50 Ω, L = 100 mH and C = 10 µF on 120 V, 60 Hz:

```text
XL = 37.7 Ω,  XC = 265.3 Ω
X  = 37.7 − 265.3 = −227.6 Ω   (capacitive)
|Z| = √(50² + 227.6²) = 233.0 Ω
θ  = arctan(−227.6 / 50) = −77.6°
I  = 120 / 233.0 = 0.515 A
```

## Resonance

When XL equals XC, they cancel and the impedance is just R. This happens at the **resonant frequency**:

```text
f0 = 1 / (2π × √(L × C))
```

For 100 mH and 10 µF, f0 ≈ 159 Hz. Resonance is used in radio tuning and filters, and it is also something power engineers avoid, because it can cause unexpectedly high voltages or currents.

Engineers often write impedance as a complex number, `Z = R + jX`, where j is the imaginary unit. Complex numbers make phase bookkeeping automatic; you will use them in the power and electronics course.

## Common mistakes

- **Forgetting unit prefixes.** 10 µF is 0.000010 F; 100 mH is 0.1 H.
- **Adding R and X directly.** 50 Ω + 227.6 Ω is wrong; use the square-root formula.
- **Mixing up which reactance rises with frequency.** Inductors oppose fast change in current, so XL grows with f.
- **Treating a charged capacitor as safe.** Large capacitors in power supplies and motor circuits can hold a dangerous charge after power is removed. Only trained, supervised workers discharge and verify them.

## Recap

- `XL = 2πfL` rises with frequency; `XC = 1/(2πfC)` falls with frequency.
- Inductor current lags voltage; capacitor current leads.
- Series impedance `|Z| = √(R² + (XL − XC)²)`; then `I = V / |Z|`.

## Try it

Open the **Do** tab and write the capacitive reactance formula in a Python impedance calculator.


Compute inductive and capacitive reactance and the impedance of a series R-L-C circuit at a given frequency.

Inductors, capacitors, impedance

Capacitive Reactance, Impedance, Power Factor, AC Circuits, Physics

Inductive Reactance, Impedance, & Power Factor - AC Circuits - Physics

## Why it matters

Utilities bill large customers for poor power factor, motors run hotter when it is low, and conductors, breakers and transformers must be sized for the **apparent** power, not just the useful power. Factories, hospitals and data centres are supplied with **three-phase** power, and every electrician and electrical engineer needs to calculate three-phase loads. This lesson connects the phase shift from the last lesson to real, reactive and apparent power.

## Three kinds of power

When current is out of phase with voltage, not all of the current does useful work. Engineers split power into three parts:

- **Real (active) power, P, in watts (W):** the power actually converted into heat, light or motion.
- **Reactive power, Q, in volt-amperes reactive (VAR):** power that flows back and forth to build the magnetic fields of motors and transformers (or the electric fields of capacitors). It does no net work but still flows in the wires.
- **Apparent power, S, in volt-amperes (VA):** simply `Vrms × Irms`. This is what the wires, transformer and breaker "feel".

They form a right triangle, the **power triangle**:

```text
S² = P² + Q²
P = S × cos θ
Q = S × sin θ
```

where θ is the phase angle between voltage and current.

## Power factor

```text
power factor (pf) = P / S = cos θ
```

- pf = 1 (θ = 0°): purely resistive, like a heater. All current does work.
- pf below 1: some current is reactive. Induction motors at light load can have a power factor of 0.7 or less.
- **Lagging** pf: inductive loads (motors, transformers), current lags voltage.
- **Leading** pf: capacitive loads.

**Worked example 1.** A single-phase motor draws 10 A at 240 V with a power factor of 0.8 lagging.

```text
S = 240 × 10 = 2,400 VA
P = 2,400 × 0.8 = 1,920 W
Q = √(2,400² − 1,920²) = 1,440 VAR
```

The wires carry 10 A, but only 1,920 W is doing useful work. If the power factor were 1.0, the same 1,920 W would need only 8 A.

## Power-factor correction

Because capacitors draw leading reactive power and motors draw lagging reactive power, a capacitor bank placed near motors can cancel some of the motor's Q. The total current drops, losses fall, and the utility may stop charging a power-factor penalty. Sizing correction capacitors is engineering work; install and maintenance is done by qualified electricians under the site's safety program.

## Three-phase power

A three-phase supply has three AC voltages of the same size, each shifted 120° from the others. Benefits:

- Power delivery is smooth (the total never drops to zero), so motors run with less vibration.
- Three-phase motors start and run without extra start windings.
- For the same power, less copper is needed than with single-phase.

Two voltages matter:

- **Line voltage (VL):** between any two phase conductors, for example 208 V, 480 V or 400 V.
- **Phase voltage (Vph):** between one phase conductor and neutral in a wye (star) system. `VL = √3 × Vph`. That is why 120/208 V and 277/480 V systems come in those pairs: 120 × 1.732 = 208, and 277 × 1.732 = 480.

For a **balanced** three-phase load:

```text
S = √3 × VL × IL
P = √3 × VL × IL × pf
```

**Worked example 2.** A three-phase motor draws 20 A per line at 480 V with a power factor of 0.85.

```text
S = 1.732 × 480 × 20 = 16,628 VA ≈ 16.6 kVA
P = 16,628 × 0.85 = 14,134 W ≈ 14.1 kW
```

**Worked example 3.** What line current does a balanced 30 kW, 0.9 pf load draw at 208 V?

```text
IL = P / (√3 × VL × pf) = 30,000 / (1.732 × 208 × 0.9) = 92.5 A
```

## Common mistakes

- **Using P = V × I on a motor.** That gives VA, not watts, unless you multiply by the power factor.
- **Forgetting √3 in three-phase.** Multiplying by 3 instead of √3 when using line voltage overstates power by 73%.
- **Mixing line and phase voltage.** Check whether a value is phase-to-phase or phase-to-neutral.
- **Sizing equipment on watts.** Transformers and generators are rated in kVA because they must carry the full current.

## Safety and real-world requirements

Three-phase industrial systems carry high fault currents and serious arc-flash hazards. These calculations are paper exercises here. Working on such equipment requires employer training, an energized-work policy based on NFPA 70E (https://www.nfpa.org/codes-and-standards/nfpa-70e-standard-development/70e), lockout/tagout under OSHA rules (https://www.osha.gov/control-hazardous-energy) and, for installation work, a licensed electrician. Checked on 2026-10-23.

## Recap

- Real power P (W) does work; reactive Q (VAR) sustains fields; apparent S (VA) = V × I.
- `pf = P / S = cos θ`; motors are usually lagging.
- Balanced three-phase: `S = √3 × VL × IL`, `P = S × pf`; `VL = √3 × Vph` in a wye system.

## Try it

Open the **Do** tab and write the real-power formula in a single- and three-phase power calculator.


Compute real, reactive and apparent power, power factor, and balanced three-phase loads.

Power factor and three-phase

Power Factor Explained - The basics what is power factor pf

How Three Phase Electricity works - The basics explained

AC waveforms and RMS values, reactance and impedance, power factor and the basics of three-phase power.