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Alternating Current question

2013 · Shift 2 · Q53
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Alternating Current question

2013 · Shift 2 · Q53

JEE AdvancedPhysicsAlternating CurrentMCQ+3 / −1
A thermal power plant produces electric power of 600 kW at 4000 V, which is to be transported to a place 20 km away from the power plant for consumers' usage. It can be transported either directly with a cable of large current carrying capacity or by using a combination of step-up and step-down transformers at the two ends. The drawback of the direct transmission is the large energy dissipation. In the method using transformers, the dissipation is much smaller. In this method, a step-up transformer is used at the plant side so that the current is reduced to a smaller value. At the consumers' end, a step-down transformer is used to supply power to the consumers at the specified lower voltage. It is reasonable to assume that the power cable is purely resistive and the transformers are ideal with power factor unity. All the currents and voltages mentioned are rms values.In the method using the transformers, assume that the ratio of the number of turns in the primary to that in the secondary in the step-up transformer is 1 : 10. If the power of the consumers has to be supplied at 200 V, the ratio of the number of turns in the primary to that in the secondary in the step-down transformer is
  1. A
    200 : 1
  2. B
    150 : 1
  3. C
    100 : 1
  4. D
    50 : 1
View written solutionFree

Correct answer: A

Step-by-step Solution:

  1. Identify the given information:

    • Power produced by the plant, P=600 kW=600×103 WP = 600 \text{ kW} = 600 \times 10^3 \text{ W}P=600 kW=600×103 W.
    • Initial voltage at the plant, Vplant=4000 VV_{plant} = 4000 \text{ V}Vplant​=4000 V.
    • Turns ratio of the step-up transformer (primary to secondary), Np,up:Ns,up=1:10N_{p,up} : N_{s,up} = 1 : 10Np,up​:Ns,up​=1:10.
    • Required voltage for consumers, Vconsumer=200 VV_{consumer} = 200 \text{ V}Vconsumer​=200 V.
    • The transformers are assumed to be ideal.

  2. Calculate the voltage after the step-up transformer: The step-up transformer is located at the power plant. Its primary coil is connected to the plant's output.

    • Input voltage to the step-up transformer, Vp,up=Vplant=4000 VV_{p,up} = V_{plant} = 4000 \text{ V}Vp,up​=Vplant​=4000 V.
    • For an ideal transformer, the ratio of voltages is equal to the ratio of the number of turns: Vs,upVp,up=Ns,upNp,up\frac{V_{s,up}}{V_{p,up}} = \frac{N_{s,up}}{N_{p,up}}Vp,up​Vs,up​​=Np,up​Ns,up​​
    • From the given ratio, Ns,up/Np,up=10/1=10N_{s,up} / N_{p,up} = 10 / 1 = 10Ns,up​/Np,up​=10/1=10.
    • The voltage at the secondary of the step-up transformer, which is the transmission voltage VtransmissionV_{transmission}Vtransmission​, is: Vs,up=Vp,up×Ns,upNp,up=4000 V×10=40000 VV_{s,up} = V_{p,up} \times \frac{N_{s,up}}{N_{p,up}} = 4000 \text{ V} \times 10 = 40000 \text{ V}Vs,up​=Vp,up​×Np,up​Ns,up​​=4000 V×10=40000 V
    • So, the power is transmitted at Vtransmission=40000 VV_{transmission} = 40000 \text{ V}Vtransmission​=40000 V.

  3. Determine the input voltage for the step-down transformer: The high voltage VtransmissionV_{transmission}Vtransmission​ is carried by the power cable to the consumer's end. At the consumer's end, a step-down transformer is used.

    • The primary coil of the step-down transformer is connected to the transmission line.
    • Although the problem mentions a resistive power cable which would cause a voltage drop, no information about the cable's resistance is given. In such problems, it's standard to assume the voltage drop is negligible, or that the given transmission voltage is the voltage available at the primary of the step-down transformer.
    • Therefore, the input voltage to the step-down transformer is Vp,down=Vtransmission=40000 VV_{p,down} = V_{transmission} = 40000 \text{ V}Vp,down​=Vtransmission​=40000 V.

  4. Calculate the turns ratio for the step-down transformer: The step-down transformer must reduce the voltage from 40000 V to the required consumer voltage of 200 V.

    • Input voltage (primary), Vp,down=40000 VV_{p,down} = 40000 \text{ V}Vp,down​=40000 V.
    • Output voltage (secondary), Vs,down=Vconsumer=200 VV_{s,down} = V_{consumer} = 200 \text{ V}Vs,down​=Vconsumer​=200 V.
    • For an ideal transformer, the relationship between voltages and turns is: Np,downNs,down=Vp,downVs,down\frac{N_{p,down}}{N_{s,down}} = \frac{V_{p,down}}{V_{s,down}}Ns,down​Np,down​​=Vs,down​Vp,down​​
    • Substituting the values: Np,downNs,down=40000 V200 V=4002=200\frac{N_{p,down}}{N_{s,down}} = \frac{40000 \text{ V}}{200 \text{ V}} = \frac{400}{2} = 200Ns,down​Np,down​​=200 V40000 V​=2400​=200
    • The ratio of the number of turns in the primary to that in the secondary is 200 : 1.

  5. Conclusion: The required ratio for the step-down transformer is 200 : 1. This corresponds to option A.

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