Abstract – Analysis of AC power systems operating under steady-state conditions are drastically simplified by transforming time domain sinusoids into their frequency domain counterparts. While these transformations are a fairly common occurrence in modern-day power system analysis, the underlying mechanism or building blocks that facilitate these transformations are not readily understood by the practicing engineers who interface with them – namely complex numbers. In an effort for the reader to understand and appreciate the role that complex numbers play in the realm of AC power systems, this paper advances through the following topics:

• Explains how the change in magnetic flux is the catalyst for AC power systems
• Defines, in a clear and simple way, what a complex number is
• Extends the definition of complex numbers to organically incorporate the phasor concept
• Lastly, illustrates the geometry modeled by complex numbers as ideal for capturing the AC sine wave

Introduction – The defining characteristic of AC power systems is the sinusoidal footprint that drives these systems and permits them to perform useful, efficient, and predictable work. During steady-state conditions, voltage and current waveforms mimic periodic sine waves. The natural question arises, what is the fundamental relationship between AC power systems and the mathematical sine function? Furthermore, how do these voltage and current sine waves spawn? In answering these questions, we will forgo the calculus of electromagnetic induction and instead focus on the geometry of magnetic flux, which the voltage waveform, and by extension, the entire power system is predicated on.

Flux – At the heart of electric power generation is a magnetization device, such as a permanent magnet, which makes up the rotor of the power source and a tightly wound coil of wire, which makes up the stator – the stator winding encloses the magnetized rotor. According to Faraday’s Law, the electromotive force or voltage induced across the terminals of the stator winding is negatively proportional to the time derivate of flux. A subtle but important distinction needs to be made between the magnetic field generated by the rotor and the magnetic flux outlined in Faraday’s Law – magnetic flux is the component of the rotor’s magnetic field that penetrates the open surface enclosed by the stator winding and has an orientation or direction that is normal (perpendicular) to the aforementioned surface. In short, magnetic flux is the projection of the magnetic field that contributes to inductance – the rest of the rotor’s magnetic field is wasted and has no contribution to Faraday’s Law. This distinction is critical in grasping where the voltage sine wave initiates. If we draw the rotor’s magnetic field and the associated magnetic flux component on the cartesian plane, we’ll note they comprise the hypotenuse and base of a right triangle, respectively. From trigonometry, the right triangle invokes the cosine function – as the rotor spins with respect to the stator, the strength of the magnetic flux component various sinusoidally with time as given by

𝜑=|𝑩| |𝑨| cos (𝜔𝑡)      [1]

where 𝜑 is the magnetic flux, |𝑩| is the magnitude of the rotor’s magnetic field vector, and |𝑨| is the magnitude of the vector normal representing the cross-sectional area of the surface enclosed by the stator winding. In determining the vector 𝑨, we are assuming the stator windings are tightly wound and uniform such that each turn encloses a flat, two-dimensional circle. Inspection of Eq. 1 reveals that magnetic flux is defined by the cosine function – as time elapses, the magnetic flux waveform oscillates sinusoidally between positive |B||A| and negative |B||A|. The sinusoidal behavior of the magnetic flux waveform is the catalyst for the sinusoidal behavior of the entire AC power system. Revisiting Faraday’s Law, the voltage induced across the terminals of the stator winding is not negatively proportional to the magnetic flux but rather to the time change in flux. Therefore, executing the time derivate of Eq. 1, and recalling the negative sign associated with Len’s Law, as well as accounting for the stator winding being composed of 𝑁 identical turns, generates the AC voltage waveform that drives the power system.

𝑣=𝜔 𝑁 |𝑩| |𝑨| 𝑠𝑖𝑛 (𝜔𝑡)      [2]

Note, the AC voltage waveform induced across the terminals of the stator winding in Eq. 2 is a scaled and shifted replica of the sinusoidal flux waveform in Eq. 1. Furthermore, if the power system is made up of passive circuit elements, therefore satisfying the homogeneity property of linearity, the system’s current response will mimic the characteristics of the voltage waveform driving the system. It should now be evident, AC power systems derive their sinusoidal attributes from the geometry of the right triangle comprised of the rotor’s magnetic field vector and the associated magnetic flux component. We will revisit this important reality to substantiate the role that complex numbers play in capturing this right triangle, and by causality, the AC sine wave.

Complex Numbers – A complex number is broadly defined as a number that contains both a real and imaginary component. It can be expressed in multiple forms but is perhaps easiest to digest in the rectangular form,

𝑎+𝑗𝑏      [3]

where the real and imaginary components are explicitly stated. The variables 𝑎 and 𝑏 are real numbers where 𝑎 represents the real component of the complex number while 𝑏 represents the imaginary component. As you’ll note, there is nothing imaginary about the variable 𝑏 – what makes 𝑏 unique is that it follows the 𝑗 operator. The 𝑗 operator often causes confusion, therefore special attention should be paid to its meaning – in particular, its utility in the analysis of AC power systems. Algebraically, the 𝑗 operator is

𝑗=√−1      [4]

which clearly, no real number can satisfy. More importantly, specifically for power system applications, the real and imaginary axis of the complex plane correspond to the x and y axis of the cartesian plane. If we treat the variables 𝑎 and 𝑏 as vector quantities, instead of scalar values, and plot them on the complex plane, we’ll note that vector 𝒂 falls on the horizontal real axis, while vector 𝒃 follows the trajectory of the vertical imaginary axis. Effectively, the label 𝑗 operates on vector 𝒃 to enforce a 90∘ CCW rotation in angular direction with respect to the reference real axis, bringing it into alignment with the imaginary axis. The 90∘ CCW rotation in angular direction attributed to the 𝑗 operator, translates to a 90∘ phase displacement in the time domain. The application of the 𝑗 operator in electrical engineering is vast, for example, when analyzing the terminal behavior of passive circuit elements such as inductors or capacitors, we can elegantly model the 90∘ phase shift between the voltage and current waveforms in the time domain by the 𝑗 operator in the complex number domain, also referred to as the frequency domain. Now that we’ve defined complex numbers and outlined the importance of the 𝑗 operator, lets extend this definition by introducing the concept of the phasor.

Phasors – In 1740, The Swiss mathematician Leonhard Euler composed one of the most remarkable theorems in mathematics [1] – his theorem has deep rooted and profound application in modern-day AC power systems and the tools application engineers use to analyze them. Regarded as one of the most elegant discoveries in the fields of mathematics, Euler’s formula relates the complex exponential function to the trigonometric functions as given by

𝑒𝑗𝜗=cos𝜗+𝑗𝑠𝑖𝑛 𝜗      [5]

where 𝑒𝑗𝜗 is a complex number expressed in polar form. The right side of the equation is a complex number expressed in rectangular form – take note of the 𝑗 operator that precedes sin𝜗. Euler’s formula in Eq. 5 is important because it gives us a way to express the cosine and sine functions in terms of the complex exponential function, where the trigonometric functions are the real and imaginary components of 𝑒𝑗𝜗. Consequently, when performing AC power system analysis, instead of manipulating sinusoids, we can conveniently represent sine functions as complex numbers in accordance with Euler’s formula – this representation significantly reduces our workload, since complex algebra is easier than trigonometry. We now formally define the phasor according to engineering literature – a phasor is a complex number that carries the amplitude and phase angle information of a sinusoidal function [2]. The amplitude of the sine function is captured in the magnitude of the phasor while the phase angle is carried in the direction of the phasor. Unfortunately, this definition is dubious and frankly, does not distinguish a phasor from a vector – a vector is also uniquely quantified by its magnitude and direction or orientation. The phasor definition, as outlined above, only provides a snapshot of the sinusoid at the moment we begin to measure time. In order to accurately represent a continuous-time sinusoid, the phasor needs to be in a state of rotation, thereby inheriting an infinite number of directions as time elapses. The frequency-domain rotation of the phasor accurately reflects the time-domain argument of the sine function passing through all its possible values from 0−2𝜋. Furthermore, the rotation action of the phasor forces the real and imaginary components of the complex exponential in Eq. 5 to evaluate to every steady-state value the AC power system takes on. Informally, a phasor is a rotating complex number rather than a static complex number.

Summary – In the opening section of this paper, we concluded the catalyst for the sinusoidal behavior of AC power systems is the right triangle, comprised of the rotor’s magnetic field vector and the associated magnetic flux component. In the section that followed, we learned that complex numbers contain both a real and imaginary component, distinguished by the 𝑗 operator. As discussed, the 𝑗 operator enforces a 90∘ CCW rotation onto the imaginary component with respect to the real component. Due to this 90∘ CCW rotation, the components of a complex number form the base and height of a right triangle. Lastly, we learned that a phasor is a rotating complex number, and because the quadrature nature of the complex components, a phasor is also representative of a right triangle. The efficacy for complex numbers to generate right triangles, due to the 𝑗 operator, makes them ubiquitous for application in AC power systems, which attribute their sinusoidal characteristics to the right triangle.

References
[1] John Stillwell (2002). Mathematics and It’s History
[2] James W. Nilsson, Susan Riedel (2014). Electric Circuits – 10th Edition