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Chapter 29: Problem 53

(III) Suppose \(85 \mathrm{~kW}\) is to be transmitted over two \(0.100-\Omega\) lines. Estimate how much power is saved if the voltage is stepped up from \(120 \mathrm{~V}\) to \(1200 \mathrm{~V}\) and then down again, rather than simply transmitting at \(120 \mathrm{~V}\). Assume the transformers are each \(99 \%\) efficient.

Short Answer

Expert verified
Power saved: 98,359.84 W (approximately 98.36 kW).

Step by step solution

01

Calculate the current at 120 V

First, calculate the current when transmitting at 120 V using the formula \( P = VI \), where \( P = 85 \text{ kW} \). Rearrange to find \( I \): \[ I = \frac{P}{V} = \frac{85000}{120} \approx 708.33 \text{ A} \].
02

Calculate power loss at 120 V

The power loss in the transmission line can be calculated using \( P_{\text{loss}} = I^2 R \), with \( R = 0.100 \times 2 = 0.200 \Omega \): \[ P_{\text{loss}} = (708.33)^2 \times 0.200 \approx 100213.2 \text{ W} \].
03

Calculate the current at 1200 V

Now calculate the current when the voltage is stepped up to 1200 V and the power transmitted remains 85 kW. Use \( I = \frac{P}{V} \):\[ I = \frac{85000}{1200} \approx 70.83 \text{ A} \].
04

Calculate power loss at 1200 V

Calculate the power loss at 1200 V using the same formula \( P_{\text{loss}} = I^2 R \): \[ P_{\text{loss}} = (70.83)^2 \times 0.200 \approx 1003.36 \text{ W} \].
05

Account for transformer efficiency

Without the transformers, the power loss due to inefficiency is negligible, hence only the loss at 120 V matters. With transformers: Total transformed power = \( 0.99 \times 85 = 84.15 \text{ kW} \). Therefore, loss = 85000 - 84150 = 850 W.
06

Compare power losses

Subtract the power loss at 1200 V and transformers鈥� loss from the power loss at 120 V. Total loss at high voltage: 1003.36 (transmission) + 850 (transformers)Power saved: \( 100213.2 - (1003.36 + 850) = 98359.84 \text{ W} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Voltage Transformation
Voltage transformation is a crucial process in power transmission that involves changing the voltage level to optimize the efficiency of power delivery. The primary reason for voltage transformation is to reduce the current in the transmission lines. When the voltage is increased, the current required to transmit the same power decreases. For example, in the exercise, the voltage is stepped up from 120 V to 1200 V. This reduces the current from 708.33 A to 70.83 A.
  • Stepping voltage up increases efficiency by lowering current.
  • Reduced current results in lower power losses along the transmission line.
The transformation is generally achieved using transformers, which efficiently change the voltage levels without significant power loss. When a transformer is used, the voltage is adjusted both at the start and end of the transmission process, ensuring that power is delivered effectively and safely.
Power Loss Calculation
Understanding power loss calculation helps us see why higher voltages are preferred in power transmission. Power loss is predominantly due to the resistance in the transmission lines and is calculated using the formula \( P_{\text{loss}} = I^2 R \). The exercise shows how power loss is dramatically higher when current is high. With 120 V transmission, the current is 708.33 A, leading to a power loss of about 100,213 W.
  • Power loss is directly proportional to the square of the current.
  • High currents significantly increase the amount of power wasted as heat.
When the voltage is increased to 1200 V, the current drops to 70.83 A, which drastically reduces the power loss to approximately 1003.36 W. By minimizing the current, the system saves nearly 98,360 W of power, illustrating the importance of voltage transformation.
Transformer Efficiency
Transformer efficiency refers to how effectively a transformer converts electrical power from one voltage to another with minimal loss. Calculating efficiency involves considering the output power relative to the input power. Ideally, transformers should have high efficiency so that power loss is minimal. In this exercise, each transformer is 99% efficient, which means only 1% of the power is lost during voltage conversion.
  • High-efficiency transformers reduce the additional losses in power systems.
  • For transformation in this scenario, the efficiency ensures that most of the power remains available for useful work.
Without efficient transformers, the benefits of stepping up the voltage could be negated by increased losses during the transformation process. However, with 99% effective transformers, combined transformer loss is around 850 W, still much less than the savings from reduced power loss in transmission. This emphasizes the crucial role that transformer efficiency plays in overall power transmission efficiency.

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Most popular questions from this chapter

(II) A circular loop in the plane of the paper lies in a 0.75-T magnetic field pointing into the paper. If the loop's diameter changes from \(20.0 \mathrm{~cm}\) to \(6.0 \mathrm{~cm}\) in \(0.50 \mathrm{~s}\), (a) what is the direction of the induced current, \((b)\) what is the magnitude of the average induced emf, and \((c)\) if the coil resistance is \(2.5 \Omega\), what is the average induced current?

Two resistanceless rails rest 32 \(\mathrm{cm}\) apart on a \(6.0^{\circ}\) ramp. They are joined at the bottom by a \(0.60-\Omega\) resistor. At the top a copper bar of mass 0.040 \(\mathrm{kg}\) (ignore its resistance) is laid across the rails. The whole apparatus is immersed in a vertical 0.55 - field. What is the terminal (steady) velocity of the bar as it slides frictionlessly down the rails?

(II) The back emf in a motor is 85 \(\mathrm{V}\) when the motor is operating at 1100 \(\mathrm{rpm} .\) How would you change the motor's magnetic field if you wanted to reduce the back emf to 75 \(\mathrm{V}\) when the motor was running at 2300 \(\mathrm{rpm}\) ?

A high-intensity desk lamp is rated at \(35 \mathrm{~W}\) but requires only \(12 \mathrm{~V}\). It contains a transformer that converts \(120-\mathrm{V}\) household voltage. ( \(a\) ) Is the transformer step-up or stepdown? (b) What is the current in the secondary coil when the lamp is on? (c) What is the current in the primary coil? ( \(d\) ) What is the resistance of the bulb when on?

A ring with a radius of \(3.0 \mathrm{~cm}\) and a resistance of \(0.025 \Omega\) is rotated about an axis through its diameter by \(90^{\circ}\) in a magnetic field of \(0.23 \mathrm{~T}\) perpendicular to that axis. What is the largest number of electrons that would flow past a fixed point in the ring as this process is accomplished?
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