General Complex Impedance (Z)

Impedance (Z) in AC circuits is the measure of how much opposition a circuit offers to the flow of alternating current (AC). Unlike direct current (DC), where resistance is the only factor, AC circuits also involve reactance due to inductors and capacitors. Reactance is the opposition to changing currents and can either be inductive or capacitive. Therefore, impedance combines both resistance and reactance into a complex number, which can be expressed as Z = R + jX, where R is the resistance and jX is the reactance. This helps us analyze AC circuits in the same way we analyze DC circuits by simplifying calculations using phasors.

Ohm's Law applies to AC circuits, but we need to consider phasors because the current and voltage change continuously over time in an AC signal. Instead of using simple voltage (V) and current (I), we use phasors, which are representations of these quantities that include both magnitude and phase angle. In this context, we can express Ohm's Law as V = IZ, where V is the phasor representation of voltage, I is the phasor representation of current, and Z is the complex impedance. This relationship indicates that voltage is the product of current and impedance, making it easier to analyze AC circuits in terms of their sinusoidal functions.

In AC circuits, resistors are straightforward components. Their impedance is purely real and given by ZR = R ∠0°, meaning the impedance does not introduce any phase shift between voltage and current. This indicates that if you plot voltage and current against time, they will reach their maximum and zero points simultaneously (in phase). As such, resistors consume electrical energy and convert it primarily into heat, with no energy storage.

Inductors oppose changes in current due to the magnetic fields they create when current flows through them. This opposition is quantified as inductive reactance, XL = ωL, where ω is the angular frequency, and L is the inductance. When analyzing circuits, the impedance for inductors is expressed as ZL = jXL, indicating that it introduces a +90° phase shift. Thus, the voltage across an inductor reaches its peak before the current does, creating a lagging condition in the phase relationship.

In circuits with resistors (R), inductors (L), and capacitors (C) together, the total impedance combines their individual effects. The formula Z = R + j(XL - XC) reflects that we add the real part (resistance) and the imaginary part (the difference between inductive and capacitive reactances). This distinguishes whether the overall circuit exhibits inductive behavior (positive net reactance) or capacitive behavior (negative net reactance). The flexibility of this representation allows for more complex analysis and understanding of AC circuits with mixed components.

The magnitude of the impedance gives a numerical measure of how much the circuit resists current flow, which can be calculated through the formula that involves squaring the resistance and the difference in reactances, then taking the square root. The impedance angle (θ) reveals the phase difference between voltage and current, calculated using the arctangent of the ratio of net reactance to resistance. Knowing both the magnitude and angle allows further insights into the AC circuit's behavior, particularly in terms of how voltage and current variations relate over time.