8.1 Alternating current

Fundamentals of alternating current

Alternating current (AC) reverses direction periodically, typically in a sinusoidal pattern. This periodic reversal is what makes AC so useful: it allows voltage levels to be easily stepped up or down using transformers, which is why AC dominates modern power distribution. Direct current (DC) can't do this without complex electronic converters.

Definition and characteristics

A few key properties define any AC signal:

• Frequency is the number of complete cycles per second, measured in hertz (Hz). Power systems use 60 Hz (North America) or 50 Hz (most of the rest of the world).
• Amplitude is the maximum value the voltage or current reaches during each cycle.
• Phase angle describes where the waveform sits relative to some reference point in time.
• RMS (Root Mean Square) value is the effective DC-equivalent value used for power calculations. For a sinusoidal wave, $V_{RMS} = \frac{V_{peak}}{\sqrt{2}} \approx 0.707 \cdot V_{peak}$. When someone says "120 V outlet," that's the RMS voltage; the actual peak voltage is about 170 V.

AC vs DC comparison

AC naturally comes out of generators because a coil rotating in a magnetic field produces a sinusoidal voltage. DC is preferred where constant polarity matters, like in microchips and battery charging.

Sinusoidal waveform properties

The voltage of a sinusoidal AC source at any instant is described by:

$v(t) = V_m \sin(\omega t + \phi)$

where $V_m$ is the peak voltage, $\omega$ is the angular frequency, $t$ is time, and $\phi$ is the phase angle.

• Period ($T$) is the time for one full cycle: $T = \frac{1}{f}$
• Angular frequency: $\omega = 2\pi f$, measured in rad/s
• The waveform reaches its positive peak at 90°, crosses zero at 180°, hits its negative peak at 270°, and completes the cycle at 360°

AC circuit components

Resistors, capacitors, and inductors each respond differently to AC. Resistors behave the same as in DC circuits, but capacitors and inductors are reactive components that cause phase shifts between voltage and current. This is the central challenge of AC circuit analysis.

Resistors in AC circuits

Resistors in AC circuits follow Ohm's law just like in DC circuits. Voltage and current stay perfectly in phase (no phase shift). Their impedance doesn't change with frequency.

• Power dissipation uses RMS values: $P = I^2_{RMS} R$
• Resistors are used for current limiting and voltage division in AC, same as in DC

Capacitors in AC circuits

Capacitors store energy in electric fields. In AC circuits, current leads voltage by 90°. Think of it this way: current must flow into the capacitor before voltage can build up across it.

• Capacitive reactance measures how much a capacitor opposes AC: $X_C = \frac{1}{2\pi fC}$
• Notice that $X_C$ decreases as frequency increases. At very high frequencies, a capacitor acts almost like a short circuit. At DC ($f = 0$), it acts like an open circuit.
• This frequency-dependent behavior makes capacitors useful as high-pass filters, blocking low frequencies while passing high ones.
• Capacitors also provide power factor correction in industrial settings by offsetting the effects of inductive loads.

Inductors in AC circuits

Inductors store energy in magnetic fields. In AC circuits, current lags voltage by 90°. The inductor resists changes in current, so the current responds after the voltage drives it.

• Inductive reactance: $X_L = 2\pi fL$
• $X_L$ increases with frequency. At high frequencies, an inductor blocks current. At DC, it acts like a plain wire.
• This makes inductors useful as low-pass filters, passing low frequencies and blocking high ones.
• Inductors appear in transformers, power supplies, and EMI suppression circuits.

AC circuit analysis

Analyzing AC circuits requires tracking both magnitude and phase. Phasors and complex impedance are the tools that make this manageable.

Phasor diagrams

Phasors represent AC quantities as rotating vectors on a 2D plane. The length of the phasor corresponds to the amplitude (or RMS value), and the angle represents the phase.

• You can add and subtract AC voltages or currents geometrically using phasors, which is much easier than working with sine functions directly.
• Phasor diagrams give you a visual picture of phase relationships. For example, in a series RL circuit, you'd draw the resistor voltage along the horizontal axis and the inductor voltage pointing 90° upward, then find the total voltage as the vector sum.

Complex impedance

Complex impedance combines resistance and reactance into one quantity:

$Z = R + jX$

where $R$ is resistance, $X$ is net reactance ($X_L - X_C$), and $j$ is the imaginary unit.

• Magnitude: $|Z| = \sqrt{R^2 + X^2}$
• Phase angle: $\theta = \tan^{-1}\left(\frac{X}{R}\right)$
• Ohm's law extends to AC: $V = IZ$, where $V$, $I$, and $Z$ are all complex quantities
• Series impedances add directly. Parallel impedances combine using the reciprocal formula, just like resistors in parallel but with complex arithmetic.

Power factor

Not all the power delivered to an AC circuit does useful work. Power factor quantifies how efficiently power is being used:

$PF = \frac{P}{S} = \cos\theta$

where $P$ is real power (watts), $S$ is apparent power (volt-amps), and $\theta$ is the phase angle between voltage and current.

• $PF = 1$ (unity) means all power is doing useful work. This happens in a purely resistive circuit.
• A lagging power factor (current lags voltage) comes from inductive loads like motors.
• A leading power factor (current leads voltage) comes from capacitive loads.
• Utilities charge industrial customers penalties for low power factor because it wastes grid capacity. Adding capacitors to offset inductive loads is a common fix.

RLC circuits

RLC circuits combine all three passive components and exhibit frequency-dependent behavior, including the important phenomenon of resonance.

Series RLC circuits

In a series RLC circuit, the same current flows through every component. The total impedance is:

$Z = R + j(X_L - X_C)$

• When $X_L > X_C$, the circuit is net inductive (current lags voltage).
• When $X_C > X_L$, the circuit is net capacitive (current leads voltage).
• When $X_L = X_C$, the reactive parts cancel and the circuit is purely resistive. That's resonance.

Parallel RLC circuits

In a parallel RLC circuit, all components share the same voltage. Analysis is easier using admittance (the reciprocal of impedance):

$Y = \frac{1}{R} + j\left(\frac{1}{X_C} - \frac{1}{X_L}\right)$

Note the sign convention flips compared to series. At resonance, a parallel RLC circuit exhibits maximum impedance and minimum current drawn from the source. This is the opposite of series resonance.

Parallel RLC circuits are used in bandstop filters and impedance matching networks.

Resonance in RLC circuits

Resonance occurs when inductive and capacitive reactances are equal ($X_L = X_C$). Solving for frequency gives:

$f_r = \frac{1}{2\pi\sqrt{LC}}$

This frequency depends only on $L$ and $C$, not on $R$.

The quality factor (Q-factor) describes how sharp the resonance peak is:

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

A high Q means a narrow, sharp peak (selective). A low Q means a broad peak (less selective). This matters for tuning circuits in radios and communication systems.

AC power transmission

The ability to step voltage up and down with transformers is the main reason AC won out over DC for power grids. Higher voltage means lower current for the same power, and lower current means less $I^2R$ loss in the transmission lines.

Transformers and voltage conversion

Transformers work through electromagnetic induction between two coils (windings) wrapped around a shared iron core. The voltage ratio equals the turns ratio:

$\frac{V_p}{V_s} = \frac{N_p}{N_s}$

where $V_p$ and $V_s$ are primary and secondary voltages, and $N_p$ and $N_s$ are the number of turns.

• Step-up transformers increase voltage for long-distance transmission (e.g., from 20 kV at a power plant to 500 kV on transmission lines).
• Step-down transformers decrease voltage for distribution and end use (e.g., from 500 kV down to 240/120 V at your house).
• Transformers also provide electrical isolation between primary and secondary circuits.

Three-phase AC systems

Most power grids use three-phase AC: three sinusoidal voltages offset by 120° from each other. Compared to single-phase:

• Power delivery is smoother and more constant
• Motors run more efficiently and can be simpler in design
• Less conductor material is needed for the same power capacity

Two common configurations exist: wye (Y) and delta (Δ). In a wye system, the line-to-line voltage is $\sqrt{3}$ times the phase voltage.

Power grid infrastructure

The grid moves power from generation to consumption in stages:

1. Generation: Power plants produce AC (typically at 10-25 kV)
2. Transmission: Step-up transformers boost voltage to 100-500+ kV for long-distance lines, minimizing $I^2R$ losses
3. Substations: Step-down transformers reduce voltage for regional distribution
4. Distribution: Power reaches homes and businesses at standard voltages (120/240 V in North America)

Modern "smart grids" add digital monitoring and control to improve reliability and integrate renewable energy sources.

AC measurements and instruments

Measuring AC requires instruments that can handle time-varying signals. The key difference from DC measurement is that AC values are constantly changing, so you need to know which value you're reading.

Oscilloscopes for AC analysis

An oscilloscope displays voltage as a function of time, letting you see the actual waveform shape. You can measure:

• Amplitude (peak voltage)
• Frequency and period
• Phase relationships between two signals (using two channels)

Bandwidth determines the highest frequency the scope can accurately capture. Probe selection matters too: a 10X probe reduces loading on the circuit but divides the signal by 10 (the scope compensates automatically).

RMS vs peak values

This distinction trips up a lot of students. For a sinusoidal waveform:

• Peak voltage ($V_{peak}$): the maximum value the waveform reaches
• Peak-to-peak voltage ($V_{pp}$): the full swing from negative peak to positive peak, so $V_{pp} = 2V_{peak}$
• RMS voltage: $V_{RMS} = \frac{V_{peak}}{\sqrt{2}}$

RMS is the value that matters for power calculations because it gives the equivalent DC voltage that would deliver the same power to a resistor. When you plug numbers into $P = \frac{V^2}{R}$, use RMS values.

The crest factor (peak divided by RMS) equals $\sqrt{2} \approx 1.414$ for a pure sine wave. Non-sinusoidal waveforms have different crest factors, which is why "true RMS" meters exist for accurate measurements of distorted waveforms.

Power meters and wattmeters

• Wattmeters measure real power (watts) by accounting for the phase angle between voltage and current.
• Digital power analyzers provide detailed breakdowns of real power, reactive power, apparent power, and power factor.
• Clamp-on ammeters measure current without breaking the circuit by sensing the magnetic field around a conductor.
• Power factor meters help identify inefficient loads in industrial settings.

Applications of AC

Electric motors and generators

• Induction motors are the workhorses of industry. A rotating magnetic field from the stator induces current in the rotor, producing torque. No electrical connection to the rotor is needed.
• Synchronous generators in power plants convert mechanical energy to AC electrical energy.
• Variable frequency drives (VFDs) control motor speed by adjusting the frequency of the AC supplied to the motor, saving significant energy in pumps and fans.

Household appliances

Nearly every appliance in your home runs on AC or converts AC to DC internally. Refrigerators and air conditioners use AC-powered compressors. Microwave ovens convert AC to high-voltage DC for the magnetron. LED lights use small AC-to-DC converters. Washing machines use AC motors with electronic speed control.

Industrial applications

• Induction heating uses high-frequency AC to heat metals without contact
• Electric arc furnaces use high-power AC for steel production
• Variable speed pumps and compressors improve energy efficiency
• Welding equipment uses AC or rectified AC depending on the process

AC safety considerations

AC is particularly dangerous because the alternating nature can cause muscles to contract and "lock on," making it hard to let go of a live conductor. Frequencies around 50-60 Hz are especially hazardous to the heart.

Grounding and isolation

• Grounding provides a low-impedance path for fault currents to flow safely to earth, triggering protective devices.
• GFCIs (Ground Fault Circuit Interrupters) detect tiny imbalances between the current flowing out on the hot wire and returning on the neutral. If even 4-6 mA goes somewhere else (like through you), the GFCI trips in milliseconds.
• Isolation transformers separate circuits so that touching one conductor doesn't create a path through your body to ground.
• Double insulation (marked with a square-within-a-square symbol) eliminates the need for a ground wire in some tools and appliances.

Circuit breakers and fuses

These devices protect against overcurrent, which can cause fires or equipment damage.

• Fuses contain a metal element that melts when current exceeds the rating, breaking the circuit. They must be replaced after tripping.
• Circuit breakers use thermal (for sustained overloads) and magnetic (for short circuits) mechanisms to trip. They can be reset.
• Time-delay fuses tolerate brief current surges, like the inrush current when a motor starts.
• Residual current devices (RCDs) work similarly to GFCIs, detecting current imbalances between live and neutral conductors.

Proper coordination between protective devices ensures that only the device closest to the fault trips, keeping the rest of the system running.

Electrical shock prevention

• Insulate all live parts to prevent accidental contact
• Connect exposed metal enclosures to ground (protective earthing)
• Use extra-low voltage (ELV) systems in wet environments like bathrooms
• Follow lockout/tagout procedures during maintenance: physically disconnect and lock out the power source before working on equipment
• Test insulation resistance and ground continuity regularly to catch degradation before it becomes dangerous