Problem 71 - University Physics with Modern Physics

Question

A transformer consists of 275 primary windings and 834 secondary windings. If the potential difference across the primary coil is \(25.0 \mathrm{V},\) (a) what is the voltage across the secondary coil, and (b) what is the effective load resistance of the secondary coil if it is connected across a \(125-\Omega\) resistance of the secondary coil if it is

Step-by-Step Solution

Answer

(a) The voltage across the secondary coil is approximately 75.8 V. (b) The effective load resistance seen by the primary is approximately 1371.2 Ω.

Step 1: Understand the Transformer Turns Ratio

A transformer works based on the turns ratio, which is the ratio of secondary windings (\(N_s\)) to primary windings (\(N_p\)). This ratio determines how voltage is transformed between the primary and secondary coils. For our problem, \(N_p = 275\) and \(N_s = 834\).

Step 2: Apply the Voltage Transformation Formula

The voltage across the secondary coil can be found using the formula: \(\frac{V_s}{V_p} = \frac{N_s}{N_p}\), where \(V_p = 25.0 \text{ V}\) is the primary voltage and \(V_s\) is the secondary voltage. We rearrange to solve for \(V_s\):\[V_s = V_p \times \frac{N_s}{N_p}\]

Step 3: Substitute and Calculate for Secondary Voltage

Substitute the given values into the formula: \[V_s = 25.0 \times \frac{834}{275}\] Calculate \(V_s\) to get the voltage across the secondary coil.

Step 4: Use Transformer Load Resistance Relation

To find the effective load resistance, we use the principle that the resistance of the load \(R'\) seen by the primary is given by:\[R' = (\frac{N_s}{N_p})^2 \times R\], where \(R = 125 \ \Omega\) is the load resistance across the secondary coil.

Step 5: Substitute and Calculate for Effective Load Resistance

Substitute the values into the formula:\[R' = (\frac{834}{275})^2 \times 125\] Calculate \(R'\) to find the effective load resistance.

Key Concepts

Turns Ratio

Transformers operate on a critical concept known as the "turns ratio." This is essentially a comparison between the number of windings, or coils, on the primary side versus the secondary side.

• For the given transformer, the primary windings (N_p) are 275.
• Secondary windings (N_s) are 834.

The turns ratio is crucial because it directly influences how the voltage is changed from the primary to the secondary. By dividing the number of secondary windings by the primary windings, we can determine this ratio as:\[\text{Turns Ratio} = \frac{N_s}{N_p} = \frac{834}{275}\]This ratio serves as the backbone for understanding how the transformer manages the transformation of voltage between its coils, known as voltage transformation.

Voltage Transformation

The process of voltage transformation is what allows transformers to step up or step down voltages based on the turns ratio. This is expressed in the equation:\[\frac{V_s}{V_p} = \frac{N_s}{N_p}\]Where:

• V_s is the secondary voltage, which we want to find.
• V_p is the primary voltage, given as 25.0 V.

Using the previously calculated turns ratio, we can solve for V_s by rearranging and substituting, giving:\[V_s = V_p \times \frac{N_s}{N_p}\]Plugging in the known values:\[V_s = 25.0 \times \frac{834}{275}\]This calculation allows us to find the voltage across the secondary coil, demonstrating the transformer's ability to adjust voltage levels based on its design.

Load Resistance

The load resistance in a transformer is an essential aspect to consider, especially when connecting a known resistance across the secondary coil. In this scenario, a load resistance of 125 \Omega is connected.The effective load resistance (R') viewed from the primary coil is altered by the square of the turns ratio:\[R' = \left(\frac{N_s}{N_p}\right)^2 \times R\]Where R is the load resistance on the secondary side. By substituting the turns ratio and the given resistance:\[R' = \left(\frac{834}{275}\right)^2 \times 125\]This equation helps pinpoint how the load affects the primary side of the transformer, ensuring a complete understanding of how resistance is transformed along with the voltage.

Primary and Secondary Windings

Primary and secondary windings lie at the core of how a transformer functions. They consist of coils of wire wrapped around a core, typically made of iron, which aids in efficiently channeling the magnetic field created by the winding currents.

• Primary winding: connected to the source voltage.
• Secondary winding: delivers the transformed voltage to the load.

These windings are separate but magnetically linked, allowing energy transfer between them without direct electrical contact. The proportion of these windings, the turns ratio, influences both voltage and resistance transformation. Understanding windings is essential not only for voltage transformation but also for impedance matching, which ensures maximum energy transfer and efficiency.