(II) For the electric power transmission system shown in Fig. 29–26, what is the ratio N_s/N_p for (a) the step-up transformer, (b) the step-down transformer next to the home?

Verified step by step guidance

1. Understand the problem: The question involves determining the ratio of turns (Nₛ/Nₚ) for two transformers in an electric power transmission system. Transformers work based on the principle of electromagnetic induction, where the ratio of the secondary voltage (Vₛ) to the primary voltage (Vₚ) is equal to the ratio of the number of turns in the secondary coil (Nₛ) to the number of turns in the primary coil (Nₚ). This can be expressed as: $\frac{N_s}{N_p}=\frac{V_s}{V_p}.$
2. For part (a), the step-up transformer increases the voltage from the power plant to the high-voltage transmission lines. Identify the primary voltage (Vₚ) and the secondary voltage (Vₛ) from the problem or figure. Use the formula: $\frac{N_s}{N_p}=\frac{V_s}{V_p}.$ Substitute the given values for Vₛ and Vₚ to find the ratio Nₛ/Nₚ for the step-up transformer.
3. For part (b), the step-down transformer reduces the voltage from the transmission lines to a lower voltage suitable for home use. Again, identify the primary voltage (Vₚ) and the secondary voltage (Vₛ) from the problem or figure. Use the same formula: $\frac{N_s}{N_p}=\frac{V_s}{V_p}.$ Substitute the given values for Vₛ and Vₚ to find the ratio Nₛ/Nₚ for the step-down transformer.
4. Verify the units and ensure that the voltages are consistent (e.g., both in volts). If necessary, convert the units to match before substituting into the formula.
5. Summarize the results: The calculated ratios Nₛ/Nₚ for both the step-up and step-down transformers provide the required information about the number of turns in the secondary and primary coils for each transformer. These ratios are crucial for understanding how the transformers adjust voltage levels in the power transmission system.

Key Concepts

Transformers

Transformers are electrical devices that transfer electrical energy between two or more circuits through electromagnetic induction. They consist of primary and secondary coils, where the voltage is increased (step-up transformer) or decreased (step-down transformer) based on the turns ratio of the coils. This principle is crucial for efficient power transmission over long distances.

Turns Ratio

The turns ratio of a transformer is the ratio of the number of turns in the primary coil (Nₚ) to the number of turns in the secondary coil (Nₛ). This ratio determines how the voltage changes between the primary and secondary sides. For a step-up transformer, Nₛ/Nₚ is greater than 1, while for a step-down transformer, it is less than 1.

Voltage and Power Relationships

In transformers, the relationship between voltage and power is governed by the equation P₁ = P₂, where P represents power. This means that the power input to the primary coil equals the power output from the secondary coil, assuming no losses. Consequently, the voltage and current are inversely related, allowing for efficient energy transfer while maintaining power conservation.