Question

In the transmission of a.c. power through transmission lines, when the voltage is stepped up n times, the power loss in transmission, (a) Increase n times (b) Decrease n times (c) Increase n \(^{2}\) times (d) Decrease \(\mathrm{n}^{2}\) times

Short answer

The power loss in the transmission decreases \(\mathrm{n}^{2}\) times when the voltage is stepped up n times (Option d).

Step-by-step solution

Find the power loss before stepping up the voltage

Considering a transmission line, we have the initial power loss equation: \(P_{loss1} = I_1^2R\), where \(I_1\) is the initial current and R is its resistance. Now, let's assume power remains the same after stepping up the voltage n times. Hence, \(P_1 = P_2 = V_1I_1 = V_2I_2\), where \(V_1\) is the initial voltage and \(V_2 = n \times V_1\).

Find the power loss after stepping up the voltage

Rearrange the power equation for current and plug in the values: \(I_2 = \frac{V_1I_1}{n \times V_1} = \frac{I_1}{n}\) We now calculate the power loss after stepping up the voltage: \(P_{loss2} = I_2^2R = (\frac{I_1}{n})^2R\)

Compare the initial power loss equation with the final power loss equation

We now compare the equations we found for power loss before and after stepping up the voltage: \(P_{loss1} = I_1^2R\) \(P_{loss2} = (\frac{I_1}{n})^2R\) Dividing \(P_{loss2}\) by \(P_{loss1}\), we get: \(\frac{P_{loss2}}{P_{loss1}} = \frac{(\frac{I_1}{n})^2R}{I_1^2R} = \frac{1}{n^2}\) Power loss decreases by a factor of \(\frac{1}{n^2}\). From the given options, power loss in the transmission decreases \(\mathrm{n}^{2}\) times (d) when the voltage is stepped up n times.

Key concepts

Power Loss

Power loss in AC transmission lines is a critical factor to consider. This is because it directly impacts the efficiency of electrical energy delivery from power plants to end-users. Power loss occurs when electricity travels through wires and some of the energy is converted into heat. The loss can be expressed using the formula:

• \( P_{loss} = I^2 R \),
  where \( I \) is the current and \( R \) is the resistance of the transmission line.

The higher the current, the more heat generated, leading to increased power loss. That is why reducing current over long distances is a common practice to minimize power loss. In our exercise, the power loss decreases proportionally to the square of the increase in transmission voltage. The reason lies in the relation \( I = \frac{P}{V} \); therefore, if voltage increases \( n \) times, the current decreases \( n \) times, and consequently, power loss decreases \( n^2 \) times. This concept helps us understand why stepping up voltage is crucial in efficient power transmission.

Voltage Stepping

Voltage stepping is an essential concept designed to enhance the efficiency of power transmission over long distances. Voltage stepping involves using transformers to increase (step up) or decrease (step down) voltages based on the requirements of transmission or distribution.The primary advantage of stepping up the voltage is to decrease the current, thus reducing power loss, a significant aspect covered in AC power transmission theory. High voltage allows electricity to travel longer distances without significant loss. By stepping up the voltage, we achieve:

• Reduced power loss \( \propto \frac{1}{n^2} \) due to decreased current.
• Improved efficiency in the use of materials, as smaller cables can carry the same power at higher voltages.
• Extended reach, delivering power to far-off areas with minimal losses.

The ability to control voltage levels easily makes transformers indispensable for power distribution systems across the globe. By increasing the voltage, the efficiency of the entire grid improves, significantly reducing costs and conserving resources.

Transmission Lines

Transmission lines are the backbone of electrical power distribution systems. They are the medium through which electrical power is transmitted from generating stations to substations, and then to end users.A transmission line's efficiency is determined by several factors:

• Resistance \( (R) \), which affects power loss as described by Ohm's Law.
• Voltage level, where higher voltages reduce the current and associated losses.
• Conductor material and design, impacting resistance and loss.

Transmission lines can either be overhead lines using poles and towers or underground lines known for their reliability but at a higher installation cost. To mitigate power loss, maintaining the components of transmission lines is crucial, including regular inspection and adjusting voltage levels appropriately. Moreover, advancements in materials and technology continue to improve the efficiency and reliability of these lines, helping reduce both operational costs and environmental impact.

Ohm's Law

Ohm's Law is fundamental in understanding circuitry and power transmission. It describes the relationship between voltage, current, and resistance in an electric circuit:

• \( V = I \times R \)

It signifies that the voltage across a conductor is directly proportional to the current flowing through it, provided the temperature is constant. Ohm's Law assists in calculating how changes in resistance or current can influence voltage, a crucial aspect when handling power systems.Understanding Ohm's Law helps in recognizing how stepping up or down the voltage impacts current flow and thus power loss. For example, by stepping up the voltage (thereby reducing current), it is possible to decrease the loss of power due to the resistance of the transmission line, as power loss \( (P_{loss} = I^2 R) \) is greatly affected by the magnitude of the current.In electric power distribution, Ohm's Law allows engineers to design systems that optimize power transmission by balancing resistance, voltage, and current efficiently. This understanding is critical for maintaining the reliability and cost-effectiveness of power delivery systems.