Power Transmission and Transformers

Introduction

Efficient power transmission is crucial for delivering electricity from power plants to consumers. Significant power losses can occur during transmission due to the resistance of transmission lines. Transformers play a vital role in minimizing these losses by enabling the transmission of power at high voltages and low currents.

Power Loss in Transmission Lines

Resistance of Transmission Lines

Transmission lines have inherent resistance ($R$) due to the material they are made of (typically aluminum or copper) and their length. This resistance causes a voltage drop and power loss as current flows through them.

Power Loss Formula

The power loss ($P_{loss}$) in a transmission line is given by:
$$P_{loss} = I^2R$$
where:
* $I$ is the current flowing through the line
* $R$ is the resistance of the line

KEY TAKEAWAY: Power loss is proportional to the square of the current. Therefore, reducing the current significantly reduces power loss.

Factors Affecting Power Loss

• Current: Higher current leads to significantly higher power loss.
• Resistance: Longer and thinner transmission lines have higher resistance, leading to greater power loss.

Role of Transformers in Reducing Power Loss

Transformers: Step-Up and Step-Down

Transformers are used to change the voltage of alternating current (AC) electricity. They can either increase the voltage (step-up transformer) or decrease the voltage (step-down transformer).

Ideal Transformer Equation

For an ideal transformer (100% efficiency), the relationship between voltage, current, and the number of turns in the primary and secondary coils is:

$$\frac{V_1}{V_2} = \frac{N_1}{N_2} = \frac{I_2}{I_1}$$

where:
* $V_1$ is the primary voltage
* $V_2$ is the secondary voltage
* $N_1$ is the number of turns in the primary coil
* $N_2$ is the number of turns in the secondary coil
* $I_1$ is the primary current
* $I_2$ is the secondary current

Power Transmission Process

1. Step-Up Transformer at Power Plant: A step-up transformer increases the voltage from the generator to a high voltage (e.g., 132 kV, 220 kV, or even higher) for transmission. This reduces the current.
2. High-Voltage Transmission: The electricity is transmitted over long distances at high voltage. Because the current is lower, the power loss ($I^2R$) is significantly reduced.
3. Step-Down Transformers at Substations: Step-down transformers at substations reduce the voltage to lower levels (e.g., 11 kV) for distribution to local areas.
4. Distribution Transformers: Smaller distribution transformers further reduce the voltage (e.g., 240 V) for use in homes and businesses.

Why High Voltage?

Transmitting power at high voltage reduces current. Since power loss is proportional to the square of the current ($P_{loss} = I^2R$), reducing the current significantly minimizes power loss during transmission.

EXAM TIP: Always explain why high voltage reduces power loss. Simply stating it is reduced is not sufficient. Mention the relationship $P_{loss} = I^2R$.

Transformer Efficiency

Real-world transformers are not perfectly efficient. Some power is lost due to:

• Hysteresis Loss: Energy lost due to the magnetization and demagnetization of the core.
• Eddy Current Loss: Currents induced in the core due to changing magnetic fields, leading to resistive heating.
• Copper Loss: Resistive losses in the windings ($I^2R$ loss).
• Flux Leakage: Not all the magnetic flux produced by the primary coil links with the secondary coil.

COMMON MISTAKE: Forgetting that real-world transformers are not 100% efficient. Calculations often assume ideal transformers, but be aware of the limitations.

Comparing Power Losses with and without Transformers

Consider a scenario:

• Without Transformers: Transmitting power at a low voltage results in high current and significant power loss.
• With Transformers: Stepping up the voltage reduces the current, leading to a much lower power loss.

The percentage reduction in power loss can be significant, often exceeding 90%.

STUDY HINT: Work through numerical examples, both with and without transformers, to quantify the dramatic reduction in power loss achieved by using transformers.

Calculations Involving Power Transmission

Calculating Voltage Drop

The voltage drop ($V_{drop}$) across a transmission line is given by:
$$V_{drop} = IR$$
where:
* $I$ is the current in the line
* $R$ is the resistance of the line

Calculating Power Delivered to the Load

The power delivered to the load ($P_{load}$) is the power supplied by the generator ($P_{supply}$) minus the power loss in the transmission line:
$$P_{load} = P_{supply} - P_{loss}$$

Example Problem

A power plant generates 10 MW of power at 20 kV. This power needs to be transmitted over a transmission line with a resistance of 5 ohms.
a) What is the current in the transmission line if the voltage is not stepped up?
b) What is the power loss in the transmission line if the voltage is not stepped up?
c) If a transformer steps up the voltage to 200 kV, what is the new current in the transmission line?
d) What is the power loss in the transmission line with the stepped-up voltage?

Solution:

a) $P = VI \implies I = \frac{P}{V} = \frac{10 \times 10^6}{20 \times 10^3} = 500 A$

b) $P_{loss} = I^2 R = (500)^2 \times 5 = 1.25 \times 10^6 W = 1.25 MW$

c) $I = \frac{P}{V} = \frac{10 \times 10^6}{200 \times 10^3} = 50 A$

d) $P_{loss} = I^2 R = (50)^2 \times 5 = 12500 W = 0.0125 MW$

The power loss is significantly reduced when the voltage is stepped up.

REMEMBER: Follow the steps: Calculate the current, then calculate the power loss. Always include units.

Summary

Transformers are essential components of power transmission systems. By stepping up the voltage for transmission and stepping it down for distribution, they minimize power loss and enable efficient delivery of electricity over long distances. Understanding the relationship between voltage, current, resistance, and power loss is crucial for analyzing and optimizing power transmission systems.

APPLICATION: Consider the environmental impact of efficient power transmission. Reducing power loss means less energy needs to be generated, leading to lower greenhouse gas emissions.

VCAA FOCUS: Pay close attention to calculations involving power loss, voltage drop, and transformer ratios. VCAA often includes these types of questions on exams. Make sure you can explain the physics behind the formulas, not just plug in numbers.