Alternating Current

Overview

Alternating Current refers to electric current that varies periodically in magnitude and reverses direction with time. It is the standard form of electrical supply used in homes, industry, and national power grids.

Alternating current is important because:

• voltages can be changed efficiently using transformers
• high-voltage transmission reduces energy loss
• many generators naturally produce alternating emf

This topic builds on ideas from Current Electricity Fundamentals, DC Circuits, and Electromagnetic Induction.

Core Ideas

Alternating-current questions revolve around a small set of ideas:

• current and voltage can vary sinusoidally with time
• mean current over a full cycle is not useful for heating calculations
• rms values are the effective values used in power calculations
• transformers change voltage and current combinations, not total ideal power
• high-voltage transmission reduces cable losses because losses depend on $I^{2}R$
• rectification converts AC into unidirectional, usually pulsating, DC

Core Physical Idea

In a metal conductor, charge carriers drift back and forth as the electric field reverses direction periodically.

Although the charges do not travel continuously in one direction over long distances, energy is still transferred through the circuit.

For sinusoidal AC, current and voltage vary smoothly with time.

AC vs DC

Direct Current

• flows in one direction only
• usually has constant magnitude
• supplied by batteries or DC sources

Alternating Current

• reverses direction periodically
• magnitude changes continuously
• supplied by power stations and AC generators

Comparison Table

Quantity	DC	AC
Direction	constant	reverses
Magnitude	constant or steady	varies
Frequency	0 Hz	non-zero
Easily transformed?	no	yes

Sinusoidal Alternating Current

The most common AC waveform is sinusoidal.

Current

$$I=I_0\sin (\omega t)$$

Voltage

$$V=V_0\sin (\omega t)$$

where:

• $I$ = instantaneous current
• $V$ = instantaneous voltage
• $I_0$ = peak current
• $V_0$ = peak voltage
• $\omega$ = angular frequency
• $t$ = time

Waveform Quantities

Period

Time for one complete cycle:

$$T=\frac{1}{f}$$

Frequency

Number of complete cycles per second.

Unit: hertz (Hz)

Angular Frequency

$$\omega =2\pi f$$

Peak Value

Maximum magnitude of current or voltage:

$$I_0,V_0$$

Peak-to-Peak Value

Difference between positive and negative peaks:

$$2I_0,2V_0$$

Mean Value Over One Full Cycle

For a sinusoidal AC:

$$\text{mean current}=0$$

The positive and negative halves cancel.

RMS Value Overview

Because mean current over a cycle is zero, it does not describe heating effect.

The useful measure is the root mean square value.

It is the value of steady DC current that would produce the same average heating effect in a resistor.

For sinusoidal AC:

$$I_{\text{rms}}=\frac{I_0}{\sqrt{2}}$$ $$V_{\text{rms}}=\frac{V_0}{\sqrt{2}}$$

See RMS and AC Power.

Heating Effect Equivalence to DC

The heating effect of current in a resistor depends on:

$$P=I^{2}R$$

So negative current still gives positive power dissipation. That is why AC with zero mean current can still transfer energy and heat a resistor.

Rms values are defined so that:

• a DC current of $I_{\text{rms}}$ gives the same average heating effect as the AC current
• a DC voltage of $V_{\text{rms}}$ across a resistor gives the same average heating effect as the AC voltage

Power in Resistive Loads

For a resistor:

$$P=IV$$

Since current and voltage vary continuously, instantaneous power also varies.

For a pure resistor:

$$P_{\text{avg}}=I_{\text{rms}}^{2}R$$

Also:

$$P_{\text{avg}}=V_{\text{rms}}{I}_{\text{rms}}$$ $$P_{\text{avg}}=\frac{V_{\text{rms}}^2}{R}$$

For sinusoidal AC in a resistor:

$$P_{\text{avg}}=\frac{1}{2}{P}_{\max }$$

Use of RMS Values in Calculations

In practical AC questions involving resistive loads, use rms values directly in the familiar power formulas:

• $P=V_{\text{rms}}{I}_{\text{rms}}$
• $P=I_{\text{rms}}^{2}R$
• $P=\frac{V_{\text{rms}}^2}{R}$

Do not substitute peak values into these formulas unless the question explicitly asks you to derive power from instantaneous expressions.

Transformer Overview

A transformer changes AC voltage using electromagnetic induction.

See full topic: Transformers

Figure: Ideal transformer showing primary and secondary coils linked by a common iron core.

Ideal Transformer Relations

$$\frac{N_s}{N_p}=\frac{V_s}{V_p}$$ $$V_p{I}_p=V_s{I}_s$$

where:

• $N_p,N_s$ = turns in primary and secondary coils
• $V_p,V_s$ = primary and secondary p.d.
• $I_p,I_s$ = primary and secondary current

Step-Up Transformer

• $N_s>N_p$
• voltage increases
• current decreases

Step-Down Transformer

• $N_s<N_p$
• voltage decreases
• current increases

Power Transmission at High Voltage

For transmission cables, the average heating loss is:

$$P_{\text{loss}}=I_{\text{rms}}^{2}R$$

For fixed transmitted power:

$$P=V_{\text{rms}}{I}_{\text{rms}}$$

Increasing transmission voltage reduces line current, so cable losses decrease greatly.

Hence power is transmitted at high voltage, then stepped down for consumers.

Figure: High-voltage transmission reduces current in the cables and therefore reduces $I_{\text{rms}}^{2}R$ power loss.

Transformer Efficiency Overview

For an ideal transformer:

$$V_p{I}_p=V_s{I}_s$$

So ideal input power equals ideal output power.

In practice, efficiency is less than 100% because of losses such as:

• copper loss in coils
• eddy currents
• hysteresis
• flux leakage

Detailed treatment belongs in Transformers.

Rectification Overview

Many electronic devices require DC supply.

Rectification converts AC to DC using diodes.

See Rectification.

Figure: Comparison of alternating input with typical rectified outputs.

Half-Wave Rectification

• one half-cycle passes
• one half-cycle is blocked

Full-Wave Rectification

• both half-cycles are used
• output always has the same polarity across the load

The output is pulsating DC.

Pulsating DC Output

Rectified output is unidirectional, but it still varies with time.

So it is not steady DC like a battery. It is:

• one direction only
• varying in magnitude
• repeated in pulses

Household Mains Interpretation

When a wall socket is labelled 230 V or 240 V, this refers to:

$$V_{\text{rms}}$$

It does not mean peak voltage.

Peak voltage is:

$$V_0=\sqrt{2}{V}_{\text{rms}}$$

So for 240 V mains:

$$V_0\approx 340\text{ V}$$

Singapore mains frequency is typically:

$$50\text{ Hz}$$

Common Exam Traps Overview

1. Confusing Peak with RMS

Use rms values in power calculations unless stated otherwise.

2. Using Mean Current for Heating

Mean current over a full cycle is zero, but mean power is not zero.

3. Wrong Transformer Current Ratio

Voltage ratio follows turns ratio, but current ratio is inverse.

4. Assuming a Transformer Increases Power

An ideal transformer changes voltage and current combination, not total ideal power.

5. Forgetting Cable Loss Depends on $I^2$

Doubling current quadruples line loss.

See Alternating Current Common Exam Traps.

Exam Relevance

Alternating-current questions are often straightforward if quantities are interpreted correctly. Most lost marks come from using peak values instead of rms values, reversing the transformer current ratio, or describing rectified output as steady DC. The safest approach is to identify whether the question is about waveform description, power calculation, transformer ratios, or transmission losses before choosing formulas.

Summary

Key results:

$$I=I_0\sin (\omega t)$$ $$V=V_0\sin (\omega t)$$ $$T=\frac{1}{f}$$ $$\omega =2\pi f$$ $$I_{\text{rms}}=\frac{I_0}{\sqrt{2}}$$ $$V_{\text{rms}}=\frac{V_0}{\sqrt{2}}$$ $$P=V_{\text{rms}}{I}_{\text{rms}}$$ $$P_{\text{loss}}=I_{\text{rms}}^{2}R$$

Alternating current is central to modern electricity generation, transmission, voltage conversion, and power supply systems.

Links

• Current Electricity Fundamentals
• DC Circuits
• Electromagnetic Induction
• Alternating Current Generators
• Transformers
• Semiconductors and Diodes
• RMS and AC Power
• Rectification
• Alternating Current Common Exam Traps