Question

A transformer connected to a 120-V (rms) ac line is to supply 12.0 V (rms) to a portable electronic device. The load resistance in the secondary is 5.00 \(\Omega\). (a) What should the ratio of primary to secondary turns of the transformer be? (b) What rms current must the secondary supply? (c) What average power is delivered to the load? (d) What resistance connected directly across the 120-V line would draw the same power as the transformer? Show that this is equal to 5.00 \(\Omega\) times the square of the ratio of primary to secondary turns.

Short answer

The turns ratio is 10. The secondary current is 2.4 A. The power delivered is 28.8 W. The equivalent resistance is 500 Ω, confirming \(5.00 \Omega \times 10^2 = 500 \Omega\).

Step-by-step solution

Identify the Given Values

Start by identifying the given values from the problem.- Primary voltage: \(V_p = 120\, \text{V (rms)}\)- Secondary voltage: \(V_s = 12.0\, \text{V (rms)}\)- Load resistance: \(R = 5.00\, \Omega\)

Calculate the Transformer Turns Ratio

The turns ratio \(n\) for a transformer is given by the formula:\[\frac{N_p}{N_s} = \frac{V_p}{V_s}\]Substitute the given values:\[\frac{N_p}{N_s} = \frac{120}{12} = 10\]

Determine the RMS Current in the Secondary

The RMS current in the secondary \(I_s\) can be calculated using Ohm's Law:\[I_s = \frac{V_s}{R}\]Substitute the given and calculated values:\[I_s = \frac{12.0}{5.00} = 2.4\, \text{A}\]

Calculate the Average Power Delivered to the Load

The average power \(P\) delivered to the load can be calculated using the formula:\[P = V_s \cdot I_s\]Substitute the values for \(V_s\) and \(I_s\):\[P = 12.0 \times 2.4 = 28.8\, \text{W}\]

Determine the Equivalent Resistance Across 120-V Line

The equivalent resistance \(R'\) that draws the same power from the 120-V line can be calculated using:\[R' = \frac{V_p^2}{P}\]Substitute the values:\[R' = \frac{120^2}{28.8} = 500\, \Omega\]Verify that \(R'\) equals \(5.00\, \Omega \cdot \left(\frac{N_p}{N_s}\right)^2\):\[R' = 5.00 \cdot 10^2 = 500\, \Omega\]This shows that the calculation aligns as described in the problem statement.

Key concepts

Transformer Turns Ratio

In the study of electrical transformers, one crucial concept to grasp is the transformer turns ratio. This ratio \(\frac{N_p}{N_s}\), where \(N_p\) is the number of turns in the primary coil and \(N_s\) is the number of turns in the secondary coil, determines how voltages and currents are transformed between the primary and secondary sides.

In this case, the voltage transformation is directly proportional to the turns ratio, given by the formula:
\[ \frac{N_p}{N_s} = \frac{V_p}{V_s} \]
This means if a transformer is supplied with a primary voltage \(V_p\) and needed to output a secondary voltage \(V_s\), the turns ratio will necessary scale the voltages.

• If the turns ratio is greater than 1, the transformer is a step-down transformer.
• If less than 1, it's a step-up transformer.
  

This makes it clear how the 10:1 ratio calculated here is used to step down from 120 V (rms) to 12 V (rms). Understanding this principle helps design and utilize transformers appropriately in various applications.

RMS Current Calculation

Calculating the root mean square (RMS) current is essential for understanding the behavior of AC circuits. In transformers, knowing the RMS current on the secondary side is crucial for ensuring that electrical devices receive the correct current flow. The formula to calculate RMS current (\(I_s\)) in the secondary is given by Ohm's Law as:
\[ I_s = \frac{V_s}{R} \]

This equation shows that the RMS current depends on both the secondary voltage \(V_s\) and the load resistance \(R\).

For a load resistance of 5.00 Ω and a secondary voltage of 12.0 V (rms), \(I_s\) is calculated to be 2.4 A. This value indicates how much current flows to the device from the transformer.
Understanding and calculating RMS current ensures that the electronic device receives a stable and adequate power supply.

Ohm's Law Applications

Ohm's Law is vital for analyzing AC circuits, including those utilizing transformers. This fundamental law, expressed as \(V = IR\), relates voltage, current, and resistance in a circuit. In the context of transformers, it plays a significant role:

• Calculating Current: As seen in the solution, it's used to derive the secondary RMS current by rearranging it as \(I = \frac{V}{R}\).
• Determining Power: Power delivered to the load, using secondary voltage and current, depends on these relationships.
  
• Equating Resistance: It's also applied in establishing equivalent resistance of \(\frac{V_p^2}{P}\).

In this scenario, Ohm's Law facilitates understanding the power dynamics in the circuit, offering insights into essential design and operational standards. Mastery of Ohm's Law allows students to troubleshoot and optimize circuit performance effectively.

Power Calculation in Circuits

Power calculation in circuits, especially in those involving transformers, is crucial for determining energy transfer efficiency and proper functioning of connected devices. The average power \(P\) delivered to an electrical load can be computed with the formula:
\[ P = V_s \cdot I_s \]

In this exercise, using a secondary voltage \(V_s\) of 12.0 V and current \(I_s\) of 2.4 A, the average power is calculated to be 28.8 W. This result indicates the rate at which energy is transferred to the electronic device and ensures it operates within safe parameters.

Additionally, identifying the equivalent resistance across the primary voltage that draws the same power (in this case 500 Ω) underscores its role in ensuring consistent energy distribution and optimizing energy consumption.
Understanding power calculations allows for predicting and managing energy consumption effectively, which is invaluable for creating efficient and sustainable electrical systems.