Question

The impedance of a component is \(Z=12-12 \mathrm{j}\). State (a) the resistance, (b) the reactance, (c) the phase of the voltage relative to the current.

Short answer

Answer: The resistance is 12 Ohms, the reactance is -12 Ohms (capacitive), and the phase difference between the voltage and current is -45 degrees (the voltage lags the current by 45 degrees).

Step-by-step solution

Calculate the Resistance

The resistance (R) can be taken as the real part of the complex impedance. Given that the impedance is Z = 12 - 12j, the resistance R = 12 Ohms.

Calculate the Reactance

The reactance (X) is the imaginary part of the complex impedance. Given that the impedance is Z = 12 - 12j, the reactance X = -12 Ohms. Note that the negative sign indicates that it is a capacitive reactance.

Determine Phase of Voltage Relative to Current

To find the phase (Φ) of the voltage relative to the current, we can calculate the angle between the impedance vector and the real axis of the complex plane. This can be done using the following formula: Φ = arctan(X/R), where X is the reactance and R is the resistance. Plugging in the values, we get Φ = arctan(-12/12) = arctan(-1) = -45 degrees. Thus, the phase difference between the voltage and current is -45 degrees, which implies that the voltage lags the current by 45 degrees.

Key concepts

Resistance Calculation

When dealing with complex impedance, understanding resistance is straightforward because it corresponds to the real part of the impedance. In this exercise, the complex impedance is given as \(Z = 12 - 12j\).

• The real part is the number without any imaginary component.

In this case, we can clearly see that the resistance \( R \) is 12 ohms.
Resistance is a measure of how much a component opposes the flow of electric current.
It doesn't matter if you are working with AC or DC circuits; resistance remains the same.
In AC circuits, it's just one piece of the puzzle of impedance.

Reactance Determination

Reactance is slightly different from resistance as it deals with the imaginary part of impedance. In the formula \(Z = 12 - 12j\), the imaginary unit 'j' acts like the special sauce of the equation.

• Reactance can be either inductive or capacitive. Inductive reactance is positive and capacitive reactance is negative.
• Our reactance value here is \(-12\) ohms.

The negative sign is key.
It tells us that this reactance is capacitive.
In simple terms, capacitive reactance causes the current to lead the voltage in phase.
So, in this setup, the component reduces the overall current in the circuit but in a different way than plain resistance.

Phase Angle Calculation

The phase angle is like the compass of your AC circuit, showing how far off or "out of sync" your voltage and current are. This angle can be determined by looking at how the reactance and resistance relate to one another via the formula: \[\Phi = \arctan\left(\frac{X}{R}\right)\]Plug in the values: \(X = -12\) ohms and \(R = 12\) ohms,

• \(\Phi = \arctan\left(-1\right)\)

This calculates to \(-45\) degrees.
A phase angle of \(-45\) degrees means the voltage lags behind the current by 45 degrees.
This information is crucial for applications such as power factor correction and circuit tuning. Understanding the phase angle helps in effectively managing energy consumption and ensuring the stability of power systems.