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EEVblog 1661 - AC Basics Tutorial Part 5 Time Domain vs Frequency Domain

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EEVblog:
The difference between Time Domain and Frequency domain in AC circuit analysis.

AC Theory Playlist: https://www.youtube.com/playlist?list=PLvOlSehNtuHvD6M_7WeN071OVsZFE0_q-

Nominal Animal:
:-+

Switching or transforming problems between different domains is a key way to solve hard problems in math, physics, computing, and engineering.

In the olden days, before calculators, multiplications and divisions were made easier by using slide rules, transforming \$p = x \cdot y\$ via \$\log(p) = \log(x \cdot y) = \bigl( \log(x) + \log(y)\bigr)\$ to \$p = \exp\bigl(\log(x) + \log(y)\bigr)\$, and similarly \$d = x / y\$ to \$d = \exp\bigl(\log(x) - \log(y)\bigr)\$; essentially transforming numbers into their logarithms, turning multiplication into an addition, and division to a subtraction.  These kind of transforms are ubiquitous everywhere, we only need to know them and when they are useful.  Just because our calculators are fast does not mean we can or should brute-force our way without such tools; it'd be a huge waste of effort.

For this particular video, the transform from (real) time domain to (complex) frequency domain can be done using Fourier transform (for signals in general), or Laplace transform (for electronics).  The difference, really, is whether you use \$s = i \omega = 2 \pi i f\$ or not.  Well, literally that, actually, plus some notational differences and scaling factor conventions between forward and inverse transform, because other than those, the two are the exact same thing.

I can tell from personal experience that knowing something about their practical use, actually having a need or an use for the transforms before you learn the math behind them, will make it immensely easier to understand and learn.  Back in my physics studies, I had been learning and investigating the then-new JPEG compression and decompression –– based on discrete Cosine transform, very closely related to 2D Fourier series and 2D Fourier transform, on 8×8 blocks of samples in YCbCr color space –– when I needed to learn the mathy details on Fourier and Laplace transforms.  It was very easy, because I already had a need for exactly that kind of a tool; so learning the math involved was like listening to an experienced old hand explaining what works, what doesn't, and why and how. 

Note that I am definitely not a mathematician: I only use math as a tool, extensively, and am not at all interested in the theory of math, or how one detail can be derived from another detail, or how to prove these well-known methods are valid.  I cannot memorize even important formulae off-hand; I only remember the ideas and principles, and how to quickly and efficiently find the formulae and other stuff I might need to solve some problem.


Apologies for the slight derail in the following:

For those learning this stuff, I believe mixing in basic 2D vector algebra as a corollary to complex numbers and phasor notation would help.  After all,
$$\vec{v} = \left [ \begin{matrix} v_x \\ v_y \end{matrix} \right] \quad \leftrightarrow \quad \mathbf{v} = v_x + j \, v_y$$
(noting that in electronics \$j\$ is often used as the imaginary unit; elsewhere \$I\$ is more commonly used), and
$$v = \lVert \vec{v} \rVert = \sqrt{v_x^2 + v_y^2}, \quad
v_\varphi = \arg(\vec{v}) = \operatorname{atan2}(v_y, v_x)$$and$$\quad \left\lbrace \begin{aligned}
v_x &= v \, \cos \varphi \\
v_y &= v \, \sin \varphi \\
\end{aligned} \right ., \quad \mathbf{v} = v \, e^{j \, \varphi}$$
phasor notation using \$v\$ for the magnitude and \$\varphi\$ for the angle (in degrees; whereas in math, we typically use radians, \$2 \pi\$ corresponding to 360° and \$\pi\$ to 180°).

Essentially, complex numbers are a way to describe and operate on quantities composed of two components like 2D vectors using Cartesian \$(v_x, v_y)\$ or polar (\$v, \varphi\$) coordinates, the two being transformable (once again!) using the above, and do math on these with analogous rules to when operating on real scalars (one-component values).  "Complex numbers" are just those rules.  (In comparison, 2D vector algebra does not have analogous rules to vector-vector multiplication (there are at least three related but very different operations!) or any at all for vector-vector division.)

For graphics, you then only need to add dot and wedge products (the 2D analog of cross product) and a little bit of linear algebra for rotations (2×2 matrices, matrix-matrix and matrix-vector multiplication, transpose, inverse; and same for 3×3 matrices if perspective projection is used).  Extending to 3D adds one more dimension, but proper coordinate system rotation involves the concept of a four-component orientation description, a bivector (related to wedge product) or an unit-length quaternion (versor, with one scalar and three imaginary components, that when squared will always sum to 1).
Anyone using Euler or Tait-Bryan angles for 3D graphics or 3D rotations should be shot. Versor/unit quaternion/bivector \$(q, i, j, k)\$ representation is superior: unambiguous, more robust, easier, more efficient, and has no gimbal lock or coordinate axis order or preference.

Similar to what Dave does with this series of videos, I've always wondered whether it would be useful to show how to learn 2D and 3D graphics (more properly, the math of 2D and 3D descriptive geometry) from plain physics with minimal initial math skills needed, starting from the physical optical situation like I did in this StackExchange post.

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