Question

A transformer with 400 turns in its primary coil and 20 turns in its secondary coil is designed to deliver an average power of \(1200 .\) W with a maximum voltage of \(60.0 \mathrm{~V}\). What is the maximum current in the primary coil?

Short answer

Answer: The maximum current in the primary coil is 1 A.

Step-by-step solution

List given information

- Number of turns in primary coil: \(N_p = 400\) - Number of turns in secondary coil: \(N_s = 20\) - Maximum voltage in secondary coil: \(V_s = 60.0 V\) - Average power output: \(P = 1200 W\)

Determine voltage in primary coil

We will first find the voltage in the primary coil (\(V_p\)). For transformers, the voltage ratio is given by the ratio of the number of turns in the respective coils: $$ \frac{V_p}{V_s} = \frac{N_p}{N_s} $$ Solve for \(V_p\): $$ V_p = \frac{N_p}{N_s} \times V_s $$ Plugging in the given values: $$ V_p = \frac{400}{20} \times 60 = 20 \times 60 = 1200 V $$

Calculate power in the primary coil

For an ideal transformer, the power input to the primary coil is equal to the power output of the secondary coil (\(P_p = P_s\)). The power in the primary coil is given by \(P_p = IV_p\), where \(I\) is the current in the primary coil. Since \(P_p = P_s\), we can write: $$ IV_p = P $$

Solve for current in the primary coil

Now we can solve for the current in the primary coil (\(I\)). Rearrange the equation from Step 3: $$ I = \frac{P}{V_p} $$ Plugging in the values of \(P\) and \(V_p\) from the previous steps: $$ I = \frac{1200 W}{1200 V} = 1 A $$ The maximum current in the primary coil is 1 A.

Key concepts

Transformers

Electrical transformers are fascinating devices that can 'transform' the levels of voltage and current in an alternating current (AC) circuit. The basic principle behind a transformer is electromagnetic induction, which allows for the transfer of electrical energy between two or more coils.

A transformer consists of a primary coil, which receives the input voltage, and a secondary coil, where the transformed voltage is outputted. These coils are wrapped around a core, typically made from a magnetic material, which helps to efficiently transfer energy between the coils without direct electrical connection.

One crucial aspect of transformers is that they operate only with AC power. This is because the changing current produces a changing magnetic field, which is required to induce voltage in the secondary coil. There are various types of transformers, including step-up transformers, which increase voltage from the primary to the secondary coil, and step-down transformers, which do the opposite.

Primary Coil Current

In the context of transformers, the primary coil is where the input voltage is applied. The current that flows through this coil, known as the primary coil current, is a critical component that determines how much energy can be transferred to the secondary coil.

The current in the primary coil is influenced by both the input voltage and the impedance presented by the primary coil. These factors combined determine the amount of power that can be drawn from the power source.

Importance of Primary Coil Current

It's essential to understand how the current interacts with the turns of wire in the coil. The more turns there are, the greater the coil's ability to induce a voltage in the secondary coil, however, this also increases the coil's impedance which can affect the current flow.

An increase in the primary coil's current, under the condition of a constant input voltage, would potentially mean more power is available to be transferred to the secondary coil. But it's not only about quantity; the quality of the current (in terms of stability and purity) is also crucial in maintaining the transformer's efficiency and performance.

Voltage Ratio

The voltage ratio is a fundamental concept in the operation of transformers and is dictated by the number of turns in the primary coil compared to the number of turns in the secondary coil. This ratio determines how much the input voltage is increased or decreased when it's 'transformed' to the output voltage.

To find this ratio, you divide the number of turns in the primary coil, denoted as N_p, by the number of turns in the secondary coil, denoted as N_s. If a transformer has a greater number of turns in the primary coil than the secondary coil, it will step down the voltage, which is common in power distribution to consumers.

On the other hand, if there are more turns in the secondary coil, the transformer will step up the voltage, which is useful when needing to transmit power over long distances. Transformers are designed to have a certain voltage ratio to meet the requirements of specific applications, making them incredibly versatile devices in the field of electrical engineering.