Understanding Alternating Currents (AC): The Power that Runs Our World

Welcome to the fascinating world of Alternating Current! This chapter is where we connect the physics of Magnetic Fields and electromagnetic induction (which you studied just before this) to the electricity that powers our homes, schools, and cities.

Unlike the steady flow of Direct Current (DC) from a battery, AC is constantly changing direction and magnitude. This unique property makes AC essential for long-distance power transmission and is the reason why transformers even work!

Key Takeaway from the Introduction

AC is electricity where the current and voltage periodically reverse direction, usually in a predictable sinusoidal pattern.

1. Characteristics of Sinusoidal Alternating Current (3.10.5)

When a generator coil spins in a magnetic field, the induced EMF (voltage) and resulting current follow a smooth wave pattern called a sinusoidal waveform. This is the simplest and most common type of AC.

Peak, Peak-to-Peak, and Instantaneous Values
  • Instantaneous Current (I) and Voltage (V): The current or voltage at any single moment in time.
  • Peak Value (\(I_0\) or \(V_0\)): This is the maximum voltage or current reached during one cycle. It's the highest point on the waveform graph.
  • Peak-to-Peak Value (\(V_{pp}\)): The voltage difference between the maximum positive peak and the maximum negative peak. Mathematically, \(V_{pp} = 2V_0\).
Period and Frequency

An AC cycle repeats itself continuously.

  • Period (T): The time taken for one complete cycle of the waveform (measured in seconds, s).
  • Frequency (f): The number of complete cycles per second (measured in Hertz, Hz).

    The two are related by: \[ f = \frac{1}{T} \] (In the UK and many parts of the world, mains electricity has a frequency of 50 Hz, meaning the current reverses direction 100 times every second!)
Quick Review: DC vs AC
DC (Direct Current): Charge flows in one direction only (e.g., battery).
AC (Alternating Current): Charge continuously changes direction (e.g., mains socket).

2. Root Mean Square (RMS) Values

Here is one of the most important—and sometimes confusing—concepts in AC circuits!

Why We Need RMS

If you try to measure the average current or voltage of an AC sine wave over a full cycle, the answer would be zero, because the positive half cancels out the negative half. However, an AC current clearly does work (it heats up a filament bulb!).

The Root Mean Square (RMS) value is defined as the value of DC voltage or current that would produce the same heating effect (or power dissipation) as the AC voltage or current in the same resistor.

The RMS value is what is generally quoted for mains power (e.g., 230 V).

The RMS Equations

For a sinusoidal waveform, the relationship between the peak value and the RMS value is fixed:

RMS Current: \[ I_{rms} = \frac{I_0}{\sqrt{2}} \]

RMS Voltage: \[ V_{rms} = \frac{V_0}{\sqrt{2}} \]

Don't worry if the calculation seems complex; just remember that \(\frac{1}{\sqrt{2}}\) is approximately 0.707. The RMS value is always about 70.7% of the peak value.

Example: Mains Voltage Calculation

If your domestic supply voltage is quoted as \(230 \, \text{V}\) (which is the RMS value, \(V_{rms}\)), what is the maximum voltage \(V_0\) that appliances actually experience?

We rearrange the formula: \( V_0 = V_{rms} \times \sqrt{2} \)
\[ V_0 = 230 \, \text{V} \times 1.414 \approx 325 \, \text{V} \] Did you know? Even though we say "230 V electricity," the wires actually peak at over 325 V during every cycle! This is brilliant physics in action.

Key Takeaway on RMS

RMS values tell us the effective power-delivering capacity of an AC supply, and are calculated by dividing the peak value by \(\sqrt{2}\).

3. Using the Cathode Ray Oscilloscope (CRO)

The oscilloscope is your best friend when studying AC. It displays the changing voltage against time, allowing you to see the waveform and measure its properties.

Operational Familiarity (Measuring AC)

You need to be familiar with the main controls used for displaying and measuring AC waveforms:

  1. Y-Gain (or Volts/Div) Control:
    • This control determines the vertical scale. It tells you how many volts each major division on the vertical axis represents.
    • Measurement: To find the Peak Voltage (\(V_0\)), count the number of vertical divisions from the centre line (0 V) to the peak, and multiply by the Y-gain setting (Volts/Div).
  2. Time-Base (or Time/Div) Control:
    • This control determines the horizontal scale. It tells you how much time (in seconds, milliseconds, or microseconds) each major division on the horizontal axis represents.
    • Measurement: To find the Period (\(T\)), count the number of horizontal divisions needed for one complete cycle, and multiply by the Time-Base setting (Time/Div). Once \(T\) is known, you can find the frequency \(f\).
  3. AC/DC Coupling:
    • When measuring AC, we often use AC coupling to block any DC component, ensuring the waveform is centred vertically.
    • If you use DC coupling, the waveform will shift vertically if there is a background DC voltage present.

Key Takeaway on CRO

The CRO measures voltage on the vertical axis (Y-gain) and time on the horizontal axis (Time-base), allowing us to directly measure \(V_0\) and \(T\).

4. The Operation of a Transformer (3.10.6)

Transformers are devices that efficiently change AC voltages and currents using the principle of electromagnetic induction. They are crucial for transporting power across countries.

A transformer consists of two coils of wire—the Primary Coil (\(N_p\)) and the Secondary Coil (\(N_s\))—wound around a soft iron core.

How It Works (Prerequisites Check)
  1. The AC voltage applied to the Primary Coil creates a constantly changing magnetic field.
  2. The soft iron core guides this changing magnetic flux to the Secondary Coil.
  3. According to Faraday's Law (\(\mathcal{E} = -\frac{\Delta \Phi}{\Delta t}\)), this changing flux induces an EMF (voltage) in the Secondary Coil.
  4. The ratio of voltages is determined by the ratio of the number of turns.
The Transformer Equation

The voltages (\(V\)) and the number of turns (\(N\)) in the primary (p) and secondary (s) coils are linked by:

\[ \frac{N_s}{N_p} = \frac{V_s}{V_p} \]

  • If \(N_s > N_p\), the voltage increases (Step-up Transformer).
  • If \(N_s < N_p\), the voltage decreases (Step-down Transformer).

Don't panic! This simple ratio allows us to step up voltage for transmission or step it down for safe domestic use.

Power and Efficiency

In an ideal transformer, the power input equals the power output (\(P_{in} = P_{out}\)). Since \(P = IV\), this means: \[ I_p V_p = I_s V_s \] This shows a critical inverse relationship: if voltage is stepped up, the current must be stepped down proportionally.

Efficiency (\(\eta\)) is never 100% in a real transformer: \[ \text{Efficiency} = \frac{\text{Output Power}}{\text{Input Power}} = \frac{I_s V_s}{I_p V_p} \] Efficiency is often expressed as a percentage.

Causes of Inefficiency (Energy Losses)

Energy is lost primarily as heat:

  1. Resistance in Coils: The copper wires have resistance, leading to heat loss (\(P = I^2 R\)).
  2. Eddy Currents: Changing magnetic flux induces small currents (eddy currents) within the soft iron core itself. These flow in loops and heat the core up. (Minimised by using a laminated core—thin sheets insulated from each other).
  3. Hysteresis Loss (Core Heating): The core is constantly being magnetised and demagnetised (reversed flux), which wastes energy as heat. (Minimised by using soft, easily magnetised materials).
  4. Magnetic Flux Leakage: Not all the magnetic flux produced by the primary coil reaches the secondary coil (minimized by winding the coils close together or one on top of the other).

5. High-Voltage Power Transmission

This is perhaps the most crucial real-world application of AC and transformers. Electrical power is transmitted across the country at extremely high voltages (e.g., 132 kV or 400 kV).

Why Transmit at High Voltage?

The goal is to minimise energy loss in the transmission cables.

1. The power (\(P_{trans}\)) that needs to be transmitted is fixed: \[ P_{trans} = I_{trans} V_{trans} \] 2. The power loss (\(P_{loss}\)) due to the resistance (\(R\)) of the cables is given by: \[ P_{loss} = I_{trans}^2 R \]

The Logic: To transmit a fixed amount of power (\(P_{trans}\)), if we step up the voltage (\(V_{trans}\)) dramatically using transformers, the required current (\(I_{trans}\)) must be reduced significantly.
Since the power loss depends on the square of the current (\(I^2\)), even halving the current reduces the loss to one quarter of its original value. This is highly efficient!

Analogy: Think of a pipe carrying water (power). If you want to push a lot of water through a narrow pipe, you can either push it really hard (high current) and lose lots of energy due to friction (heat), or you can use higher pressure (high voltage) and less flow rate (low current) to get the same result with minimal friction loss.

Key Takeaway on Transmission

Transformers allow us to step up voltage for efficient transmission (minimising \(I^2R\) losses) and then step it back down for safety and usage. This ability is why AC dominates modern power grids.