The difference between impedance and resistance

Hi Bob,

Re the use of radians vs. degrees, see the section on this page headed "Advantages of measuring in radians," which begins by stating:

In calculus and most other branches of mathematics beyond practical geometry, angles are universally measured in radians. This is because radians have a mathematical "naturalness" that leads to a more elegant formulation of a number of important results.

Most notably, results in analysis involving trigonometric functions are simple and elegant when the functions' arguments are expressed in radians.

I'm not certain, but I believe that the results of differentiating and integrating a sine or cosine function would be much less "clean" and easy to work with if degrees or cycles were used instead of radians.

I understand that sine and cosine are out of phase by pi/2, but why do you say that one leads or lags the other?

What's the point of reference?

Just take any voltage level on one of the waveforms, such as a zero crossing or a peak level, and find the corresponding point on the other waveform. The two points will be pi/2 radians or 90 degrees apart, with one occurring first. Of course, for continuous waveforms a 90 degree lead is the same as a 270 degree lag, but characterizing it that way would not seem sensible physically. That would say that for a capacitor a voltage change precedes the corresponding current change, and for an inductor a current change precedes the corresponding voltage change. Given that capacitors resist instantaneous changes in voltage, and inductors resist instantaneous changes in current, viewing a 90 degree lead as a 270 degree lag would not work for a transient or non-continuous waveform.

Best regards,
-- Al

Hey we go again.. kudos to Al. As methodical and detail as ever. Way to go Prof Al.

The reason radians are used is because the derivative of sin(x) is directly cos(x) when using radians. This is why the formulations using radians come out cleaner.

I would like to add too that you very rarely get 90 degrees in a circuit. The parasitic effects will cause the angle to be all over the map, positive and negative, the extent of which is dependent on the frequency of the signal.

Simply-q: Break-in effects are largely due to parasitic capacitance effects through the air, insulation, cable sheaths, PC boards, etc., surrounding the signal-carrying wire(s). The capacitance is formed between the wire and "ground," which can be just about anything at a lower potential, and everything in between ground and the signal is the dielectric. These effects are part of what I was talking about. The fact the ground is indeterminate is what leads to the mystery of break-in. As time goes on, the more-dominant capacitances charge to a "neutral" level and finally break-in "ends." This is in addition to any physical material changes due to heat or an applied field.

There has been extensive research done by the French government (and the Germans to a certain extent) on this effect as it relates to high-power transmission lines. It is easier to witness when you're dealing with MV instead of mV. (When you stand under a power line, you are quite literally standing inside a capacitor.) They have shown that just the heavy ions in air can have a big impact on the capacitive resonance effects that affect a signal. But they are far from nailing down all the variables in all instances, of course.

Arthur

One way of dealing with any signal is via Fourier analysis. That analysis, along with a host of other signal analysis techniques, benefit greatly from Euler's identity, without which the analysis would be unnecessarily cumbersome. There is simply no good reason to express signals in any other way when engaged in any such analysis.