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Difficulty: Very hardInductors and Transformers

An ideal transformer has a primary coil with 600600 turns and a secondary coil with 150150 turns. The primary coil is connected to a 120 V120\text{ V} AC source, and the secondary coil is connected to a load consisting of two 15 Ω15\ \Omega resistors wired in parallel. What is the total current flowing through the primary coil?

  1. 1 A1\text{ A}Answer
  2. B
    16 A16\text{ A}
  3. C
    4 A4\text{ A}
  4. D
    0.25 A0.25\text{ A}

Answer

The total current flowing through the primary coil is 1 A1\text{ A}.
The correct answer synthesizes multiple principles. First, the transformer step-down ratio is 600:150600:150, or 4:14:1. This drops the primary 120 V120\text{ V} to a secondary 30 V30\text{ V}. Next, the two 15 Ω15\ \Omega resistors in parallel create an equivalent resistance of 7.5 Ω7.5\ \Omega. Using Ohm's Law (I=V/RI = V/R), the secondary current is 30 V/7.5 Ω=4 A30\text{ V} / 7.5\ \Omega = 4\text{ A}. Finally, because a transformer steps current inversely to voltage, a 4:14:1 voltage step-down means a 1:41:4 current step-up from primary to secondary. Therefore, the primary current is 4 A/4=1 A4\text{ A} / 4 = 1\text{ A}.

Step-by-Step Solution

1
Calculate the turns ratio to determine the secondary voltage.
Np/Ns=600/150=4N_p/N_s = 600/150 = 4. Therefore, Vs=120 V/4=30 VV_s = 120\text{ V} / 4 = 30\text{ V}.
In a transformer, the voltage ratio is directly proportional to the turns ratio.
2
Determine the equivalent resistance of the secondary load.
For two equal resistors in parallel, Req=15 Ω/2=7.5 ΩR_{eq} = 15\ \Omega / 2 = 7.5\ \Omega.
Parallel branches provide multiple paths for current, reducing the total resistance of the circuit.
3
Apply Ohm's law to find the total secondary current.
Is=Vs/Req=30 V/7.5 Ω=4 AI_s = V_s / R_{eq} = 30\text{ V} / 7.5\ \Omega = 4\text{ A}.
The total current drawn by the load depends on the secondary voltage and the equivalent resistance of the load.
4
Calculate the primary current using the inverse turns ratio.
Ip=Is/(Np/Ns)=4 A/4=1 AI_p = I_s / (N_p / N_s) = 4\text{ A} / 4 = 1\text{ A}.
In an ideal transformer, power is conserved (Pp=PsP_p = P_s). If voltage steps down by a factor of 4, the current must step up by a factor of 4 from primary to secondary, making the primary current 1/4 of the secondary current.

Key Concept

Transformer turns ratio scaling for voltage and current, combined with parallel circuit analysis.
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Topics

Inductors and Transformers

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