Question

A transformer has 800 turns in the primary coil and 40 turns in the secondary coil. a) What happens if an AC voltage of \(100 . \mathrm{V}\) is across the primary coil? b) If the initial \(A C\) current is \(5.00 \mathrm{~A}\), what is the output current? c) What happens if a DC current at \(100 .\) V flows into the primary coil? d) If the initial DC current is \(5.00 \mathrm{~A}\), what is the output current?

Short answer

Answer: For the AC input, the output voltage is 5V and the output current is 100A. For the DC input, both the output voltage and current are 0, because transformers do not work with DC input.

Step-by-step solution

a) Find the output voltage for AC input.

The transformer equation is: $$\frac{V_{primary}}{V_{secondary}} = \frac{N_{primary}}{N_{secondary}}$$, where \(V_{primary}\) and \(V_{secondary}\) are the primary and secondary coil voltages, and \(N_{primary}\) and \(N_{secondary}\) are the primary and secondary coil turns. Given \(V_{primary} = 100V\), \(N_{primary} = 800\), and \(N_{secondary} = 40\). To find the output voltage \(V_{secondary}\), we can rearrange the formula as: $$V_{secondary} = \frac{N_{secondary} * V_{primary}}{N_{primary}}$$ Plug in the values and calculate \(V_{secondary}\): $$V_{secondary} = \frac{40 * 100}{800} = 5V$$. So, when an AC voltage of 100V is across the primary coil, the output voltage across the secondary coil is 5V.

b) Find the output current for AC input.

Using the principle of conservation of energy, for an ideal transformer, the power input at the primary coil is equal to the power output at the secondary coil. Power \(P = IV\), where \(I\) is the current and \(V\) is the voltage. From part a, we found the output voltage \(V_{secondary} = 5V\). Now, we are given the input current \(I_{primary} = 5A\) and need to find the output current \(I_{secondary}\). We can set up the following equation: $$V_{primary} * I_{primary} = V_{secondary} * I_{secondary}$$ Rearrange the formula to solve for \(I_{secondary}\): $$I_{secondary} = \frac{V_{primary} * I_{primary}}{V_{secondary}}$$ Plug in the values and calculate \(I_{secondary}\): $$I_{secondary} = \frac{100 * 5}{5} = 100A$$. So, when the initial AC current is 5A, the output current is 100A.

c) Find the output voltage for DC input.

Transformers only work for AC input and not for DC input. This is because transformers depend on the changing magnetic field produced by AC current to induce a voltage in the secondary coil. In the case of DC current, the magnetic field does not change, so there will be no induced voltage in the secondary coil. Therefore, the output voltage for DC input is 0V.

d) Find the output current for DC input.

As mentioned in part c, transformers do not work for DC input. Consequently, there will be no output current for DC input, so the output current for DC input is 0A.

Key concepts

AC Voltage

Alternating Current (AC) voltage is the type of electricity that frequently reverses direction. It is the standard form of electricity delivered to homes and businesses. AC voltage is distributed using high voltages over long distances, as it efficiently reduces power loss.

• AC voltage causes the alternating magnetic field necessary for transformers to operate.
• This alternating magnetic field induces voltage in the transformer's secondary coil, which transforms the input voltage to a new output voltage.

In this exercise, when an AC voltage of 100V is applied to the primary coil, it results in a lower output voltage of 5V in the secondary coil due to the given transformer ratio.

DC Current

Direct Current (DC) current flows in a single direction and is commonly found in batteries and electronic devices. Unlike AC, DC does not cause a changing magnetic field necessary to operate a transformer. Thus, it cannot be effectively transformed to other voltages using traditional transformers.

• Since a transformer relies on the changing magnetic field produced by AC, applying DC voltage results in no output current or voltage.
• In this exercise, a DC current at 100V inputs to the primary coil, but no voltage or current outputs from the transformer.

This limitation of transformers is crucial, as it highlights the importance of AC in power distribution and equipment design.

Transformer Equation

The transformer equation plays a fundamental role in determining how transformers change voltages. It establishes the relationship between the number of turns in the primary and secondary coils and their respective voltages:\[\frac{V_{primary}}{V_{secondary}} = \frac{N_{primary}}{N_{secondary}}\]

• By rearranging this formula, you can determine unknown values such as the output voltage:\[ V_{secondary} = \frac{N_{secondary} \times V_{primary}}{N_{primary}} \]
• The transformer in the example transforms a 100V primary voltage to a 5V secondary voltage because of the turn ratio of 800 to 40.

Understanding how to manipulate this equation allows us to predict how different transformers will behave.

Energy Conservation

The principle of energy conservation ensures that energy in a closed system remains constant. For ideal transformers, power on the input side equals power on the output side. Power is calculated as the product of current and voltage:\[ P = I \times V \]

• For AC, this means: \[ V_{primary} \times I_{primary} = V_{secondary} \times I_{secondary} \]
• In the example, a primary current of 5A with a 100V input leads to a secondary current of 100A at a 5V output, preserving the power across the transformer.

This concept is essential for understanding how transformers manage energy and change current and voltage levels while maintaining overall power balance.