Problem 22 - Applied Physics

Question

A transformer steps down \(66 \overline{0} 0 \mathrm{~V}\) to \(12 \overline{0} \mathrm{~V} .\) (a) If the secondary current is \(14.0 \mathrm{~A}\), what is the primary current? (b) Find the power in the primary coil.

Step-by-Step Solution

Answer

Primary current is 0.255 A. Power in the primary coil is 1683 W.

Step 1: Understanding the Transformer Terms

A transformer changes an AC voltage level up or down. In this case, we're using a step-down transformer. The primary voltage \( V_p \) is given as 6600 V, and the secondary voltage \( V_s \) is given as 120 V. The current in the secondary coil, \( I_s \), is 14 A.

Step 2: Calculate the Primary Current

Transformers work on the principle that the product of voltage and current remains constant in an ideal transformer (ignoring losses). Given by the formula:\[V_p \times I_p = V_s \times I_s\]Rearrange to solve for primary current \( I_p \):\[I_p = \frac{V_s \times I_s}{V_p} = \frac{120 \times 14}{6600}\]\[I_p = 0.2545\text{ A} \approx 0.255\text{ A}\]

Step 3: Calculate the Power in the Primary Coil

Power in a coil is calculated using the formula:\[P_p = V_p \times I_p\]Substitute the values of \( V_p \) and calculated \( I_p \):\[P_p = 6600 \times 0.255\]\[P_p = 1683\text{ W}\]Thus, the power in the primary coil is 1683 Watts.

Key Concepts

Step-Down Transformer

Transformers are devices used to change the voltage levels of alternating current (AC) in electrical circuits. In the case of a step-down transformer, it reduces, or "steps down," the voltage from a higher level to a lower level. This is particularly useful when a lower voltage is required for various electrical devices.

The transformer consists of two coils known as the primary and secondary coils. The primary coil is connected to the higher voltage source, while the secondary coil delivers the lowered voltage to the desired application.

• Primary Voltage ( V_p ) is the initial voltage supplied to the transformer.
• Secondary Voltage ( V_s ) is the voltage output from the transformer.

This principle allows devices requiring lower voltage inputs to operate efficiently in systems with higher voltage sources.

Primary Current Calculation

In a transformer, the current on the primary side can be determined using the relationship between the voltages and currents on both sides of the transformer. The relationship can be described with the following ideal transformer equation:\[ V_p \times I_p = V_s \times I_s \]Where:

• V_p is the primary voltage.
• I_p is the primary current.
• V_s is the secondary voltage.
• I_s is the secondary current.

By rearranging the equation, the primary current (I_p ) can be calculated as:\[ I_p = \frac{V_s \times I_s}{V_p} \]Inserting the given voltages and current as found in the exercise, I_p will give you the current flowing through the primary coil. This calculation ensures that energy conservation principles are respected as it relates to the voltages and currents of both coils.

Power Calculation in Transformers

Calculating power in a transformer involves understanding the rate at which energy flows through the coils. Power in the primary coil (P_p ) can be determined by the formula:\[ P_p = V_p \times I_p \]Where:

• P_p is the primary power.
• V_p is the primary voltage.
• I_p is the primary current.

Once you have determined the primary current (I_p ) using the previous calculations, you can substitute it in to find the primary power. This calculation helps to indicate how much electrical energy the primary coil transfers over time, often measured in watts (W). Power calculations ensure that the transformer operates within safe limits and aids in the design of efficient electrical systems.

Transformer Equations

The functioning of transformers revolves around a few key equations that stem from the conservation of energy and electromagnetic induction. These equations allow us to determine the behavior and performance of transformers:

• Voltage Transformation Ratio: \( \frac{V_p}{V_s} = \frac{N_p}{N_s} \) where N_p and N_s are the numbers of turns in the primary and secondary coils respectively.
• Current Ratio: \( \frac{I_p}{I_s} = \frac{N_s}{N_p} \)

These equations demonstrate the inverse relationship between the voltage and current ratios concerning the number of turns. This relationship is fundamental in designing transformers to ensure that they meet specific voltage and current requirements. Understanding these relationships aids in troubleshooting issues related to transformer operations and optimizing their use in electrical systems.