Complex form for Capacitors and Inductors - Page 1
Forum section: electronics / beginners

author: sahko123
author_rank: Frequent Contributor
timestamp: April 26, 2022, 01:51:43 am
content: So Dave released his video at a perfect time for me. I'm looking into making an LCR meter and it just so happens I want to use the mathematical V-I plot method rather than balance bridge.So we know that for Capacitors current leads voltage and for Inductors voltage leads current. So the phase angle between ideal inductors and capacitors is always 90 degrees in whichever direction. But how does this work for complex impedances then? If you want to find both the ESR and Reactance of a capacitor you end up with no dissipation factor with a phase angle of 90 degrees. So does this mean that say for a real life capacitor the phase angle actually ends up being <90 degrees to account for series inductance which then results in a X an ESR value as opposed to only an X value?Gotta love engineering maths.I'm basically trying derive all the parameters possible from simply voltage, current, frequency and phase. Wrapping the maths around my head I've managed to understand deriving the actual capacitance or inductance but dont understand how to get a complex impedance when you just get a single Ohms value in an ideal situation.To re-state my questions. Is the phase angle of a real life capacitor/inductor the same as that of an ideal model? If not to derive the complex impedance from Z is it simply a case of converting the polar form (Z,angle(<90)) to rectangular with resulting values being the resistance and reactance? Do I understand this correctly?

author: TimFox
author_rank: Super Contributor
timestamp: April 26, 2022, 02:52:29 am
content: The V-I method of measuring impedances basically measures the current through the device and divides it by the voltage across it.However, it is not merely the (single value) ratio of the magnitudes (which would be the magnitude of the admittance, which is the reciprocal of the impedance).By the use of phase-sensitive detectors (two synchronous demodulators, one driven in phase with the drive voltage and the other driven in phase quadrature), the in-phase ("real") and quadrature-phase ("imaginary") components of the current are measured separately.A real capacitor or inductor can be modeled, at a single frequency, as a resistance in series with the reactance, or as a resistance in parallel with the reactance (actually, the conductance in parallel with the susceptance).The equivalent values (e.g., series capacitance and equivalent series resistance) can be computed from these two components of the current waveform.If you have a capacitance in series with an inductance, the reactance will be negative (capacitive) at low frequencies and positive (inductive) at higher frequencies.

author: T3sl4co1l
author_rank: Super Contributor
timestamp: April 26, 2022, 08:39:52 am
content: Yes, the phase of a real, dissipative 1-port (i.e., it has two terminals you can put anywhere, or one with respect to GND) is within +/-180°, indeed strictly within (exclusive exactly +/-180) if nonzero loss.This is true no matter what internal structure the element might have: at any given frequency, all you can measure is phase and magnitude, or real and complex, which are equivalent complete descriptions of the condition.(What internal structure describes, then, is how it can vary with respect to frequency. Consider a fixed R in series with C: this is constant ESR, so the Q factor varies inversely with frequency: in series, Q = X/R = 1/(ωRC). Whereas a real capacitor might have Q closer to a fixed value, meaning ESR is dependent on ω. So we can get different Z behavior as F varies; but Z will always be within this range [right half plane].)A greater phase (< -180 or >180) implies the element is generating power, which implies negative resistance, or you've connected to a transmitter that happens to be synchronous to your impedance meter's excitation. (That is, the same sort of thing when you hook a DMM on resistance range, across a resistor that has some bias already in it: the meter can't know if the voltage drop is due to its supplied bias current, or the circuit's. It assumes its own exclusively. The AC version of that is, the frequencies must match -- otherwise, the phase will rotate over time, of course, which also means the extra source can be averaged out over time, but this is not the case when they match exactly. Accordingly: when the frequency is exact, you can't know if it's due to the component's apparent negative resistance, or an external source -- not just from a single measurement, anyway.)As for working with complex impedance, it is what it says: an impedance Z = a + jb ohms is resistance a in series with inductive (if positive) reactance b. So the series equivalent circuit is trivial. There is also a parallel equivalent circuit, which should prove a good exercise: convert c || jd into a series equivalent. That is, simply evaluate it until real and complex terms are separable. (The only trick you will need is the complex conjugate: note that (a + jb)(a - jb) = a^2 + b^2.)Tim

author: Doctorandus_P
author_rank: Super Contributor
timestamp: April 26, 2022, 10:55:21 am
content: First:This sentence does not make any sense to me. I'm guessing you do understand, but wrote it down a bit too quickly.Quote from: sahko123 on April 26, 2022, 01:51:43 amIf you have any confusion about capacitors and inductors, then remember: A capacitor stores an electrical charge, and you have to put in some current before there is a voltage over the capacitor. Inductors can store energy in a magnetic field. If you put a voltage over an inductor, then the current will gradually increase. From these basic facts you can always deduce the direction of the phase shift.Quote from: sahko123 on April 26, 2022, 01:51:43 amFor the overall behavior of an electric circuit, you have to add the surrounding components into the equations.Take for example a capacitor parallel to a resistor.At very low frequencies (DC) the capacitor has a very high impedance and does nothing, and the impedance is therefore just he same as the resistor value. At very high frequencies the capacitor is a short (with a phase shift) and the contribution of the resistor is negligible. So what happens in between you may ask. Well, it's also pretty simple. The impedance of a capacitor is frequency dependent. So at some frequency the impedance of the capacitor will be equal to the resistance of the resistor. At that frequency both the resistor and the capacitor have the same current, and the phase shift between voltage and current will be 45 degrees. It's the corner frequency in bode plots of filters.Quote from: sahko123 on April 26, 2022, 01:51:43 amCapacitors do have parasitic ESR and inductance. A piece of straight wire also has resistance inductance, and capacitance to everything that surrounds it.It's nearly impossible to calculate everything to exact numbers, and the trick is to figure out what is the dominant part of the circuit.Take for example the same parallel combination of a capacitor and resistor. If you're looking at frequencies that are roughly an order of magnitude below the corner frequency, then you may as wel remove the capacitor because it's contribution is negligible.For frequencies an order of magnitude above the corner frequency, the contribution of the resistor is negligible.If you draw an impedance plot of a capacitor, then capacitance, ESR and series inductance are the main factors (but there are others such as dielectric absorption)These three parameters can each be modeled by a straight line in an impedance (vertical scale) to frequency (horizontal scale) plot.Capacitance will be a downward slope, then a horizontal line for the series ESR, and at high frequencies the upward slope of the parasitic inductance dominates.(ESR is mainly a thing for electrolytic capacitors, for most other capacitor types it's so low that it does not have a clear dominant frequency range in the graph and only the downward and upward slopes remain).Each of these is best measured in a frequency range where it dominates. You simply can not measure the series inductance of a capacitor at low frequencies, your result will not be accurate.If not to derive the complex impedance from Z is it simply a case of converting the polar form (Z,angle(<90)) to rectangular with resulting values being the resistance and reactance? Do I understand this correctly?

author: mawyatt
author_rank: Super Contributor
timestamp: April 26, 2022, 04:50:22 pm
content: Just a couple of general comments. Anything that has physical length beyond a point has some inductance, even if you can't directly measure it. The same can be said for capacitance, anything that has physical size beyond a point has capacitance. We've used these often to help model devices where we couldn't measure all the detailed characteristics (example chip features). This is why some SMDs intended for higher frequency use have the terminations on the long sides rather than the ends, the current flow has a shorter distance to travel across the SMD rather than thru end to end, and thus a lower intrinsic inductance.Also, wavelength can cause issues with impedances, even repeating. Take for example a very large film capacitor with long leads, this can have a very complex impedance at frequencies where the physical length becomes a significant portion of the measurement wavelength, and may have repeated results beyond a wavelength.Certainly no expert on impedance measurements nor equipment, but the wavelength issue may be why some of the higher level LCR meters include a cable length parameter.Best,

author: Doctorandus_P
author_rank: Super Contributor
timestamp: April 26, 2022, 05:12:07 pm
content: A generic umbrella term for non ideal properties of wires and coping with them falls into the category "transmissin line".https://html.duckduckgo.com/html?q=transmission+linehttps://en.wikipedia.org/wiki/Transmission_line

author: mawyatt
author_rank: Super Contributor
timestamp: April 26, 2022, 05:36:05 pm
content: A transmission "line" doesn't give a good analogy to help understand a chip bond pad, nor a large film capacitor, as a "line" for these is hard to visualize. These are best described by other means more in tune with circuit types, rather than lines or fields.Agree that the "transmission effect" includes all sources of electrical energy transfer, including radiated emissions & suspectance, no matter how small or large.Best,