Question

A transformer of efficiency \(90 \%\) draws an input power of \(4 \mathrm{~kW}\). An electrical appliance connected across the secondary draws a current of \(6 \mathrm{~A}\). The impedance of device is ....... (a) \(60 \Omega\) (b) \(50 \Omega\) (c) \(80 \Omega\) (d) \(100 \Omega\)

Short answer

The impedance of the device is \(100 \Omega\), which corresponds to option (d).

Step-by-step solution

Calculate the output power of the transformer

We are given the efficiency of the transformer as 90% and the input power as 4 kW. The efficiency of the transformer is defined as the ratio of the output power to the input power. Let's denote the output power as P_out (in kW). Therefore, Efficiency = \(\frac{P_{out}}{P_{in}}\) We are given Efficiency = 0.9 and P_in = 4 kW. We will solve for P_out: 0.9 = \(\frac{P_{out}}{4}\) P_out = 0.9 * 4 = 3.6 kW

Convert the output power to Watts

Since we are given the current drawn by the electrical appliance in Amperes, it is convenient to have the output power in Watts. To convert the output power from kilowatts to watts, we simply multiply by 1000: P_out (W) = 3.6 * 1000 = 3600 W

Calculate the impedance of the device using Ohm's law

Ohm's law relates the voltage V, current I, and impedance (resistance) R as: V = I * R In this case, we want to find the impedance (R) of the device. We also know that the power P_out equals the product of current I and voltage V across the device: P_out = V * I Since we have already found the output power and we are given the current I, we can now find the voltage V across the device: V = \(\frac{P_{out}}{I}\) V = \(\frac{3600}{6}\) = 600 V Now, using Ohm's law, we can find the impedance R: R = \(\frac{V}{I}\) R = \(\frac{600}{6}\) = 100 Ω Therefore, the impedance of the device is 100 Ω, which corresponds to option (d).

Key concepts

Ohm's Law

In electrical circuits, Ohm's Law is a fundamental principle that defines the relationship among voltage (V), current (I), and resistance (R). This law is particularly useful for analyzing simple circuits and understanding how changes in one quantity affect the others.

The basic formula of Ohm's Law is expressed as:

• \( V = I \times R \)

This equation tells us that the voltage across a component is the product of its current and its resistance. If you know two of these values, you can easily solve for the third.

For example, if a device draws a current of 6 Amperes, and we want to know the impedance, or electrical resistance, we can rearrange the formula to solve for R: \( R = \frac{V}{I} \). This rearrangement makes it straightforward to find the impedance once you have the values for voltage and current. In the exercise, using Ohm's Law allowed us to calculate the impedance of the device as 100 Ω.

Electrical Impedance

Electrical impedance extends the concept of resistance to AC (alternating current) circuits. Unlike pure resistance, impedance also includes the effects of capacitance and inductance.

Impedance is often represented as a complex number, especially in AC circuits, to account for the phase differences between voltage and current. However, when dealing with simple devices, complex numbers might not be necessary. For practical purposes, you can often think of impedance as a measure of how much an object resists the flow of AC current, similar to resistance in DC circuits.

In the context of the given problem, impedance (\( Z \)) is calculated using the same formula as Ohm's Law: \( Z = \frac{V}{I} \). This simple approach works well when there's no additional reactive components involved, making it simpler to understand and apply.

AC Circuits

AC Circuits refer to electrical circuits powered by alternating current, where the current periodically reverses direction. This is different from DC circuits, where current flows in a single direction. The standard power systems in most countries are based on AC due to the efficient transmission over long distances.

In AC circuits, both voltage and current take the form of sine waves, characterized by specific frequencies. This periodic fluctuation creates what is called reactance, which affects the overall impedance in the circuit. Components like capacitors and inductors store and release energy, which can lead to phase shifts between voltage and current.

• For example, capacitors can store electrical energy temporarily, affecting how AC current flows through the circuit.
• Inductors, on the other hand, store magnetic energy, which can delay the current flow.

Understanding these components' interactions is crucial for analyzing the behavior of AC circuits but requires more advanced concepts beyond basic resistance and simple impedance calculations. In our exercise, the assumption is that components acting purely as resistors make the impedance analysis using Ohm’s law sufficient.