What Is Complex Power? Active, Reactive, and Apparent Power Explained - MATLAB & Simulink
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    What Is Complex Power? Active, Reactive, and Apparent Power Explained

    In AC electrical systems, phase-shifting between voltage and current introduces the concept of complex power, which has active power, reactive power, and apparent power as its components. Understanding complex power is fundamentally important for AC electrical system analysis, operation, and control.

    You will learn:

    • The relationship between voltage and current in resistors, inductors, and capacitors
    • What instantaneous power looks like in resistors, inductors, and capacitors
    • How to use vector representations of voltage and current to calculate complex power
    • How complex power is separated into active power, reactive power, and apparent power
    • How power factor gives insight into system efficiency

    Published: 13 Apr 2022

    Hello, everyone. My name is Graham Dudgeon. And today, we are going to discuss what complex power is in AC electrical systems. Let's begin by looking at the relationship between AC voltage and current in the three fundamental passive components in electrical systems-- resistors, inductors, and capacitors.

    What we see here are two views of voltage and current. On the left, we see the instantaneous waveforms as time progresses. On the right, we see a vector representation of the waveforms, with the vectors rotating at system frequency.

    Voltage is colored blue and current is colored orange. For our resistor, voltage and current are in phase. And the relationship between voltage and current is described by V equals I times R, where V is voltage, I is current, and R is resistance.

    Being in phase means that the zero crossing of the waveforms coincide and the vector angles are overlaid. For an inductor, current lags voltage by 90 degrees. And the relationship between voltage and current is described by V equals I times j omega L, where j is the complex operator, omega is the frequency of the system, and L is the inductance. The 90-degrees phase lag is clearly seen in the vector representation, where the current vector is 90 degrees behind the voltage vector relative to the direction of rotation.

    For a capacitor, current leads voltage by 90 degrees. And the relationship between voltage and current is described by V equals I times 1 over j omega C, where C is the capacitance. The 90-degrees phase lead is clearly seen in the vector representation, where the current vector is 90 degrees ahead of the voltage vector relative to the direction of rotation.

    Next step is to look at instantaneous power, where power is voltage times current. Looking at instantaneous power will give us a level of insight into how power is consumed and supplied in AC electrical systems.

    To generate the system responses we need in order to explore power further, I simulated some basic circuits where I isolate the behavior of the fundamental components. With resistance, voltage and current are in phase. Instantaneous power, which is shown here in green, is strictly non-negative.

    You can see that it varies from 0 to 0.5 peak, in this case. And mean power, shown in magenta, is half the peak power. With inductance, current lags voltage by 90 degrees.

    For each period, instantaneous power is equally positive and negative for each half-cycle. Power is consumed as the magnetic field of the inductor is established. And power is resupplied as the magnetic thick field collapses. Mean power is 0.

    For capacitance, current leads voltage by 90 degrees. For each period, instantaneous power is equally positive and negative for each half-cycle. Power is consumed as the electric field of the capacitor is established. And power is resupplied as the electric field collapses. Mean power is 0.

    Instantaneous power does not typically yield sufficient insights to inform effective system analysis, operation, and control. Mean power is of no value for power measurement on capacitors and inductors, as it measures 0. We need a better way to measure power in AC systems.

    This is where vector representations prove invaluable, leading to the concept of complex power. Complex power, which has the notation, bold S, is determined by multiplying the voltage and current vectors. When multiplying vectors, you need to pay attention to conventions.

    For complex power, we multiply the root mean square, RMS, of the voltage vector by the RMS of the conjugate of the current vector. The RMS value is 1 over square root 2 of the peak value.

    I'm showing voltage and current and complex form here, where we have the real and imaginary components. V RMS is 1 over root 2 times V real plus j times V imaginary. And I RMS star, the conjugate of I RMS, is 1 over root 2 times I real minus j times I imaginary.

    Note that complex power has a 0.5 multiplying factor due to the RMS values. You'll also see complex power equations written in trigonometric form in the literature.

    Let's start with looking at complex power of a resistor. For complex power calculation, we reference the real axis to the voltage angle. The component of complex power along the real axis is known as active power, notation P. Resistive active power measures positive along the real axis. Positive measurement means power is consumed.

    Well now look at complex power in an inductor. The component of complex power along the imaginary axis is know as reactive power, notation Q. Inductive reactive power is measured positive along the imaginary axis.

    And so we see that an inductor consumes reactive power. Note that this is convention due to the way that we construct the complex power equation. With a capacitor, capacitive reactive power is measured negative along the imaginary axis. And so we say that a capacitor supplies reactive power. Again, this is convention.

    To complete our story, we need to consider situations where both active power and reactive power exist, which is what normally happens. And for this, we introduce apparent power. Apparent power, notation S, is the magnitude of the complex power.

    Power factor is active power divided by apparent power, P over S. It's a measure of what ratio of electrical power is converted to real work. In a purely resistive network, power factor is 1. In a purely reactive network, power factor is 0. Most systems line somewhere in between.

    In summary, active power leaves our network. In the case of a resistor, active power is converted to heat. Reactive power remains in a network. We saw how power is both consumed and supplied each half-cycle for both inductors and capacitors.

    Reactive power is only imaginary in a mathematical sense. You may hear power engineers say that reactive power is imaginary. They're not claiming it doesn't exist. It's a very real phenomenon. They are referring to the mathematical representation in the complex domain.

    Resisters consume active power, and active power measures positive for consumption. Inductors consume reactive power. This is convention. As inductive reactive power measures positive, we refer to it as consumption.

    Capacitors supply reactive power. Again, this is convention. As capacitive reactive power measures negative, we refer to it as supply.

    Apparent power is the magnitude of complex power. Power factor is active power divided by apparent power. And it's a measure of what ratio of electrical power is converted to real work.

    Active power in a component is always referenced to the voltage angle across that component. Reactive power is in quadrature, 90 degrees leading or lagging, to that voltage angle.

    I hope this information has proven useful. Thank you for listening.

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