Transformers (HSC SSCE Physics): Revision Notes

Transformers

Introduction to transformers

Transformers are electromagnetic devices that use induction to convert alternating current (AC) voltages from one level to another. They work by using a changing magnetic field in one coil (the primary coil) to induce a voltage and current in a second coil (the secondary coil). The two coils are not electrically connected—they are linked only through the magnetic field.

Transformers are found in many applications, from small plug packs that charge mobile phones to large industrial units at power stations and electricity substations. They can be used to increase voltage (step-up transformers) or decrease voltage (step-down transformers).

Alternating current (AC)

What is alternating current?

Alternating current is an electric current that periodically reverses its direction of flow. Unlike direct current (DC), which flows steadily in one direction, AC oscillates back and forth in a sinusoidal pattern.

Describing AC signals

AC currents and voltages vary sinusoidally with time, creating a wave pattern that can be described using several key parameters:

Amplitude measurements:

• Peak current ($I_{\text{peak}}$): The maximum current value reached during the cycle
• Peak-to-peak current ($I_{\text{peak-peak}}$): The total variation from maximum positive to maximum negative current; equal to $2 \times I_{\text{peak}}$
• Peak voltage ($V_{\text{peak}}$): The maximum voltage (emf) value
• Peak-to-peak voltage ($V_{\text{peak-peak}}$): The total voltage variation; equal to $2 \times V_{\text{peak}}$

Time variation:

• Period ($T$): The time taken for one complete cycle (measured in seconds)
• Frequency ($f$): The number of cycles per second (measured in hertz, Hz)

The relationship between period and frequency is:

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

Root mean square (RMS) values

Although AC voltage and current oscillate between positive and negative values, they still deliver power to circuits. To quantify this, we use RMS (root mean square) values. The RMS value represents the equivalent DC voltage or current that would deliver the same power.

Mathematically, this gives:

$V_{\text{rms}} = \frac{V_{\text{peak}}}{\sqrt{2}}$

$I_{\text{rms}} = \frac{I_{\text{peak}}}{\sqrt{2}}$

Australian mains power supply

The standard household electricity supply in Australia delivers 230V RMS between the active and neutral terminals. This is an AC supply that oscillates sinusoidally at a frequency of 50 Hz.

Australian power sockets have three connections:

• Active (Live): Brown or red wire; carries the alternating voltage
• Neutral: Blue or black wire; completes the circuit
• Earth (Ground): Green or green-yellow wire; safety connection at 0V

Structure and operation of transformers

Flux linkage

Transformers work through electromagnetic induction via flux linkage—the magnetic flux from one coil passes through the other coil. There is no direct electrical connection between the primary and secondary coils.

Two methods achieve flux linkage:

1. Coaxial coils: One coil placed inside the other (used in some cordless appliances like kettles)
2. Ferromagnetic core: The more common method, using an iron core to link the coils

How ferromagnetic cores work

In a core-linked transformer, the following process occurs:

1. The primary coil wraps around one part of a ferromagnetic (usually iron) core
2. The secondary coil wraps around another part of the same core
3. AC current in the primary coil creates a time-varying magnetic field
4. This magnetises the entire iron core, not just the part inside the primary coil
5. The magnetised core creates a time-varying magnetic field in the secondary coil
6. The changing field in the secondary coil induces an emf (by Faraday's Law)
7. This emf drives a current in the secondary coil

The result is that an AC current in the primary coil induces an AC current in the secondary coil with the same frequency but potentially different amplitude.

Transformer equations

Deriving the relationships

For an ideal transformer with perfect flux linkage, the rate of flux change ($\frac{\Delta\Phi}{\Delta t}$) is the same through both coils. Using Faraday's Law:

For the primary coil: $V_P = -N_P \frac{\Delta\Phi}{\Delta t}$

For the secondary coil: $V_S = -N_S \frac{\Delta\Phi}{\Delta t}$

Where $N_P$ and $N_S$ are the number of turns in the primary and secondary coils respectively.

Since $\frac{\Delta\Phi}{\Delta t}$ is the same for both:

$\frac{V_S}{V_P} = \frac{N_S}{N_P}$

For an ideal transformer (100% efficient), power in equals power out:

$P_{\text{in}} = P_{\text{out}}$

$V_P I_P = V_S I_S$

This gives:

$\frac{I_P}{I_S} = \frac{N_S}{N_P}$

The transformer equations

Combining these relationships gives the transformer equations:

$\frac{V_S}{V_P} = \frac{I_P}{I_S} = \frac{N_S}{N_P}$

These can be rearranged to find specific values:

• Secondary voltage: $V_S = V_P \frac{N_S}{N_P}$
• Secondary current: $I_S = I_P \frac{N_P}{N_S}$

Step-up and step-down transformers

Step-up transformers

A step-up transformer increases the voltage from primary to secondary. This occurs when the secondary coil has more turns than the primary coil ($N_S > N_P$).

From the transformer equations: $\frac{V_S}{V_P} = \frac{N_S}{N_P}$

If $N_S > N_P$, then $V_S > V_P$ (output voltage is greater than input voltage).

However, energy is conserved. The increased voltage comes at the cost of decreased current: $\frac{I_S}{I_P} = \frac{N_P}{N_S}$

If $N_S > N_P$, then $I_S < I_P$ (output current is less than input current).

Step-down transformers

A step-down transformer decreases the voltage from primary to secondary. This occurs when the secondary coil has fewer turns than the primary coil ($N_S < N_P$).

If $N_S < N_P$, then $V_S < V_P$ (output voltage is less than input voltage).

And: $I_S > I_P$ (output current is greater than input current).

Electricity distribution in Australia

Transformers play a crucial role in electricity distribution networks. The journey of electricity from power station to home involves both step-up and step-down transformers:

1. Power station: Generators produce electricity at relatively low voltage
2. Step-up transformer: Increases voltage to around 100kV or more for transmission
3. Transmission lines: High voltage reduces current, minimising energy loss in long-distance lines
4. Substation: Step-down transformers reduce voltage for distribution
5. Distribution lines: May carry high voltage (around 8-50kV) for longer distances or lower voltage (230V) for local supply
6. Local transformers: In regional areas, pole-mounted transformers reduce high voltage to 230V for individual houses
7. Household supply: 230V RMS delivered to homes and businesses

Real transformers and energy losses

The transformer equations assume ideal conditions: perfect flux linkage and 100% efficiency. Real transformers deviate from these assumptions, losing energy in several ways:

Imperfect flux linkage

Not all magnetic flux from the primary coil passes through the secondary coil. Some "stray field" extends beyond the core and can:

• Induce eddy currents in nearby materials (transformer case, supports)
• Cause heating in these materials
• Create vibrations, producing the characteristic buzzing sound near large transformers

Resistive heating in coils

The wire in transformer coils has resistance. When current flows, energy is dissipated as heat according to Ohm's Law. This is why phone chargers and other transformers become warm during use.

Eddy currents in the core

The changing magnetic field induces eddy currents within the ferromagnetic core itself. By Lenz's Law, these currents oppose the flux changes, wasting energy as heat.

Magnetisation losses

Each time the AC current reverses direction, the core must be magnetised in the opposite direction. This repeated magnetisation and demagnetisation process consumes energy, appearing as heat in the core.

Modern transformers are typically 95-99% efficient, with the lost energy appearing primarily as heat. Large transformers may require cooling systems to dissipate this heat.

Investigation: Transformers

Aims

This investigation explores how changing magnetic flux in a transformer's secondary coil is produced by varying the voltage across the primary coil. It also verifies the relationship between voltage ratios and turns ratios for both step-up and step-down transformers.

Safety considerations

Risk	Management Strategy
Power supplies can cause electric shocks	Limit voltages to maximum 12V. Ensure all equipment is checked by teacher before use

Apparatus

• 2 air-core solenoids with known number of turns
• Iron rod (ferromagnetic core)
• Variable AC/DC power supply or separate AC and DC supplies
• Connecting leads
• Cathode ray oscilloscope (CRO)

Experimental setup

Method overview

Part A: Investigates the difference between DC and AC in electromagnetic induction

• Connect solenoids end-to-end with iron rod through centres
• Apply DC voltage to primary solenoid
• Observe oscilloscope trace when switching DC on/off
• Compare with constant DC

Part B: Investigates the transformer voltage relationship

• Replace DC with AC supply
• Measure output voltage for different input voltages
• Swap primary and secondary coils
• Compare experimental results with theoretical predictions

Analysis

The investigation involves:

1. Converting peak voltage measurements to RMS values
2. Calculating expected RMS output using transformer equations
3. Comparing experimental and theoretical results
4. Explaining discrepancies based on non-ideal behaviour