91Ó°ÊÓ

In electrical circuits, impedance is a fundamental concept that combines resistance and reactance into a single complex number, denoted typically by \( Z \). Impedance provides a more comprehensive way to analyze circuits, especially those with alternating current (AC), where phases and magnitudes of currents and voltages vary over time.
Impedance is crucial because it allows engineers to simplify complex circuit analysis using the powerful tools of complex numbers and phasors. Phasors are like arrows representing the magnitude and phase relationship of voltage and current. By using impedance, we can easily compute how voltages and currents behave, and how much energy the circuits store and dissipate."
To calculate impedance \( Z \), if you have voltage \( V \) and current \( I \) phasors, you use the formula:

• \( Z = \frac{V}{I} \)

In the given problem, we were provided \( V = 220 \text{ cis } 34^{\circ} \) and \( I = 5 \text{ cis } 90^{\circ} \), which means that by dividing voltage by current, we obtain \( Z \). This gives us a complex number, providing a clear picture of how the circuit manages energy flow.

Resistance is a measure of how much an object opposes the flow of electric current. It is the real part of the complex impedance \( Z \), representing what is called 'pure resistance.'
In the context of AC circuits, resistance works to dissipate energy in the form of heat, just like in DC circuits. However, it's important to know that in AC circuits, other forces, such as reactance, also play a role.
Let's break it down: resistance (

• is measured in ohms (\( \Omega \))
• stays constant over frequency changes
• gives direct energy conversion (like heat in a resistor)

In our exercise, using the impedance calculated as \( Z = 24.6048 - 36.476i \), the resistance part is 24.6048 ohms, which is the absolute value of the real part of \( Z \). This tells us the amount of energy dissipated as heat in the circuit due to this opposition.

While resistance handles how a circuit dissipates energy, reactance shows how a circuit stores or releases energy. Reactance is the imaginary part of impedance \( Z \). It causes current and voltage to be out of phase. This phase difference is a hallmark of AC circuits where storage and release of energy, typically due to capacitors and inductors, happen frequently.
The interesting thing about reactance:

• It is measured in ohms (\( \Omega \)), just like resistance.
• It can be positive or negative, indicating inductive or capacitive reactance, respectively.
• It depends on the frequency of the source; reactance changes as the frequency changes.

Given the problem's impedance \( Z = 24.6048 - 36.476i \), the reactance is \( |-36.476| = 36.476 \) ohms. This tells us about the energy storage dynamics within the circuit, specifically the shift in current phase relative to voltage. Understanding this shift is vital in engineering applications where timing and phase differences impact performance.