Talking about the "imaginary part" in circuit and signal processing

Why do we use imaginary numbers in the impedance expressions of capacitors and inductors?
Why are there imaginary numbers in Fourier transforms?
Why do imaginary numbers appear in the transfer function of filters?

I first encountered the concept of "imaginary numbers" in high school mathematics. I roughly viewed it as an extension of one-dimensional (real numbers) to two-dimensional. In the textbook, it was just two coordinate axes drawn to represent complex numbers. So why is there only one imaginary unit i, and no more advanced complex numbers are defined to describe three-dimensional space? I never thought about it.

After studying advanced algebra in college and supplementing the knowledge of complex number domain and polynomials, I roughly understood why complex numbers were invented to supplement the shortcomings of real numbers. As for the expansion of dimensions, it is not the purpose of complex numbers. Later, after learning Fourier transform, I still did not understand why complex numbers were used to represent signals. There is no problem with this in mathematics. Mathematics is a description of the world and is abstracted. Concepts such as straight lines and space are also abstracted. But the physical quantities in the real world, currents and voltages are real, so where does the imaginary part come from (don’t talk about quantum physics, which is not within the scope of electronic engineering discussion)? Later, when I was taking the "Wavelet Analysis" course, a classmate asked this question during class. The teacher said that it meant "taking two signals". However, after all, complex signals and two-dimensional signals are not the same processing method. After

self-studying some circuit basics, I found that using imaginary numbers in circuits provides great convenience. If imaginary numbers are not used, Ohm's law cannot be used for capacitors and inductors.

By borrowing imaginary numbers, the reactance components in inductance and capacitance (that is, the AC characteristics that are different from pure resistance) are calculated in ohms.


Then, Ohm's law, Thevenin's theorem, etc. can be used as usual, which is very convenient, except that voltage and current must also be converted into complex numbers. What, voltage can have an imaginary part? Can an oscilloscope see this imaginary part? On the other hand, if imaginary numbers are not introduced, how to deal with the inductance and capacitance in the circuit? Then it is necessary to use differential equations and integral equations to express the relationship between voltage and current. It is indeed inconvenient. It takes too much effort to solve the steady-state circuit. With the help of Laplace transform tools, if you look at the relationship between input and output, imaginary numbers will appear again. Alternating current changes with time. If it is expressed by a sine function, u= A sin( ωt+θ ) contains three quantities: amplitude, frequency and phase. Capacitors and inductors will change the phase of alternating current, so when analyzing frequency characteristics, it is not enough to describe it only with amplitude. Because imaginary numbers can be expressed in the form of amplitude and phase angle, they can just describe the relationship between alternating current input and output. In the final analysis, this is still a mathematical tool to explain the world. A sine wave is a sine wave. Does it have to be an imaginary number? Euler formula








It looks very beautiful, but for AC signals, what does the imaginary part mean? Can the electric field be the real part, and the magnetic field be the imaginary part? No. A few years after graduation, I explained this question as follows: There are two most basic forms of motion in the world: one is uniform linear motion, and the other is uniform circular motion (rotation). Rotation has a radius and a period. If uniform linear motion is considered a constant state, then uniform rotation is also constant - it repeats itself over and over again . As long as you know its movement for an infinitely short period of time, you can know its past and future. In other words, a single-frequency signal can be represented by a circular motion with a constant speed. Looking at this circular motion on a plane, its trajectory is a circle.








If we add a time dimension, imagine what it would look like?

A spiral of circles, right? Along the time axis.
Look at it from the side to see it more clearly:

Another angle:

What do you see when you "look" at this circular motion perpendicular to the time axis?

Sine type! You can also see it from another angle like this:

Note that the phase of the sine function has changed.

When we observe a sinusoidal signal (voltage, current, or other physical quantities), what we observe is considered to be its real part . Assume (use your imagination ) that this signal is actually a rotating signal, and it has a corresponding imaginary part that is invisible. After the sine signal passes through a linear system (black box), in addition to the possible change in the rotation radius (amplitude), the rotation angle will also shift, so the waveform we observe also has a phase difference. As we observe from different angles, the initial phase may also be different, but the phase difference between input and output is stable.

In summary: The signal that can be observed is real, but after adding a non-existent imaginary part, the signal changes from a back-and-forth oscillation to a simpler and more basic circular motion. Complex signals can also be decomposed into the superposition of many or even infinite circular motions, and we always observe them from a fixed angle. When describing the relative relationship between two circular motions of the same frequency (such as input and output), using imaginary numbers can more conveniently express the difference in amplitude and phase angle.