Several years post grad, I've been studying from Halliday et al.'s "Physics Volume 2, 5ed." and various articles online including Network Analysis, particularly on Impedance (i.e. not a Homework question). I've come across the notion that AC voltage sources take the form:
$$V(t) := |V(t)|e^{j(\omega t+\phi_V)}.$$
If we assume one dimension of oscillation is along the wire that transmits the current, there is still a 2D disc cross-section that leaves a choice for the other oscillation direction. Am I missing something? How does AC voltage propagate?
If instead, we assumed it oscillated only in the cross-sectional directions, the electrons couldn't make it down the line to energize the other circuit elements? Any thoughts on this are appreciated
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3 Answers
Why is the dimension of AC oscillation 2 and not 3?
An alternating current does not oscillate in 2 or 3 dimensions - it is an oscillation of voltage (or alternatively of current) which can be positive or negative but does not have a direction in 2 or 3 dimensional space. So AC only oscillates in one dimension.
However, it is useful to model an AC voltage mathematically as if it were the projection onto one axis of a point rotating around a circle in two-dimensional space. And a convenient way to represent this mathematically is to introduce a complex valued variable $e^{i \omega t}$ and then take the voltage to be proportional to the real valued component of this.
In the same way, a particle executing simple harmonic motion (such as a simple pendulum) only has one degree of freedom, so it is only oscillating in one dimension. But its displacement can be modelled mathematically as the real valued component of a complex valued variable which has two degrees of freedom. So we can represent the displacement $x(t)$ as the real valued part of $A e^{i \omega t}$ where $A$ is the pendulum's amplitude and $\frac {\omega}{2 \pi}$ is its frequency.
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The " $A e ^ {i\omega t}$" formula is the equation for a helix; it has a real and imaginary extent, as well as propagating along a time axis. Imagine the projection of this kind of curve (picture a Slinky toy and observe its shadow on a wall); that's a sine wave.
The equation for a resistor current versus voltage is not time-dependent, but the equations for capacitors and inductors contain time derivative and time integrals, So, a network of those components requires a lot of math know-how to understand.
The use of the helix formulation is that the derivatives and integrals of AC sine wave components are in proportion to the imaginary parts of that helix (called also a gyrator). Those curves are contained in the helix's OTHER shadows, thus can be represented by simple linear equations. When one applies Kirchoff's rules for setting up equations, the stage is set for treating circuitry as if it were coupled linear equations, which has been well-studied: you make a matrix equation, and solve it (inverting a matrix, typically) in a mechanical fashion. Charles Proteus Steinmetz Theory and Calculation...taught this over a century ago, and half a century ago it was formulated into simulation software: SPICE, notably.
Adding in diodes, transistors, vacuum tubes, spark gaps: that requires baby-step equation solving in a slightly different way, because those components are only linear for small signals over short timescales: your SPICE program may have to readjust every microsecond of time to a new matrix , adjusting the coefficients of the matrix for each new operating point of each curve (because those elements treat AC in a nonlinear way, and are best approximated by different linear equations at different times). That's called transient analysis, and takes lots of SPICE time to do well.
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It does not propagate. AC steady state circuits belongs to the quasi-static regime. The values in voltage and current in the various branch of the circuit do not propagate with a finite time. They change instantaneously in the entire circuit, without transmission or retardation effects.
Transmission lines show propagation. While still modeled by using lumped circuit elements (for each of which the respective quasi-static approximation - which uses incomplete Maxwell's equations - is still valid) they can show propagation in terms of the exchange of energy between adjacent lumped elements cells in a 'chain' that extends in the direction of the line.
A full-fledged EM description of the propagation would require all Maxwell's equations in their full form and it is usually too complex and numerically intensive to solve, so it is not the first choice to tackle this sort of problems. (And yet, if you were to have a light-second long, say one meter wide circuit with source and load facing each other in the middle, you'd better resort to the full EM simulation, because a transmission line model would miss the propagation in the transversal direction).
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