Question about Nodal Analysis of non-linear component

[genxium]

When I'm reading a tutorial about Harmonic Balance algorithm (this is a link to the tutorial website), I'm quite confused by the concept it mentions about the nodal analysis of non-linear components, especially for:


The non-linear circuit is modeled by its current function $i(t) = i(v_1, ..., v_P)$ and by the charge of its capacitances $q(t) = q(v_1, ..., v_Q)$ . These functions must be Fourier-transformed to give the frequency-domain vectors Q and I , respectively.

Say for a simple BJT with 3 nodes b,c,e, as conventional notations, I can understand how $i_c=f(v_b,v_c,v_e)$ coming from Early Effect, and $i_b=\frac{i_c}{β \cdot (1+\frac{V_{CB}}{V_A})}$ or anything similar, to indicate "non-linearity", but what's the parameters involved with charge of capacitances?

I really have no idea how Charge of Capacitances is playing a role in nodal analysis,especially why it's regarded as "non-linear" part, any suggestion is appreciated ^_^

 

[Kholdstare]

Hmm... trying to understand Quite Universal Circuit Simulator ? Good thing

Yeah, charge can be thought as non-linear quantity here. Think of a capacitor. The total charge gathered is the time integral of current. The integrated parameter is non-linear as current will vary with time.

These circuit simulators usually model the internal I-V relationship of a device with current equation and the parasitic capacitors are modeled by considering the charge they hold. One can easily find the current contributed by these parasitics by simply differentiating the charge.

 

[Kholdstare]

Oh by the way! Have you heard of modified nodal analysis? It'll be little bit simpler than doing all those Fourier and stuff. However it is complicated than KCL or KVL alone. But that's just for robustness of the method. ^_^

 

[genxium]

Oh by the way! Have you heard of modified nodal analysis

Thanks for the reply and yes, I've learned MNA but it doesn't solved my problems, I think what I'm confused about is that:

I think capacitors, are often regarded as linear components , but in that tutorial, http://qucs.sourceforge.net/tech/node31.html , it's said that non-linear circuits are measured by (1) current function and (2) charges of capacitance, does the word "capacitance" here refer to only the parasitic capacitance of the non-linear components , but not all capacitors?

I also posted my question here and got a similar answer, but this hasn't answered why charges of capacitances are counted "non-linear" here, at least not quite clear and persuading yet.

 

[genxium]

These circuit simulators usually model the internal I-V relationship of a device with current equation and the parasitic capacitors are modeled by considering the charge they hold. One can easily find the current contributed by these parasitics by simply differentiating the charge.

Oh for god sake please forgive me man~ I just didn't see your word "parasitic" and it's what I've been looking for ^_^

So you mean that "charges of capacitances" are actually referring to those parasitic capacitances but not all the capacitors?

 

[Kholdstare]

Well you can't call capacitors and inductors linear component as far as their dc characteristics are concerned. ^_^

Its a standard way to model (all) capacitances by their charge.

 

[genxium]

Well you can't call capacitors and inductors linear component as far as their dc characteristics are concerned. ^_^

OMG...? I'm afraid I'm confused again >_<

I regarded capacitors and inductors "linear" because they have I-V relationships like:

$I=Y \cdot V$

instead of

$I= a \; polynomial \; of \; V$

compared to a diode's or transistor's characteristics, were I wrong all the time before ? >_<

 

[Kholdstare]

You can only write V = ZI only if you express Z as the reactance of capacitor or inductor (1/jwC or jwL), which is valid only for sinusoidal V and I. Thus capacitor and inductor being linear component is only a special case of more general picture (where integration and differentiation is involved >_<). ^_^

 

[AlephZero]

OMG...? I'm afraid I'm confused again >_<

I regarded capacitors and inductors "linear" because they have I-V relationships like:

$I=Y \cdot V$

instead of

$I= a \; polynomial \; of \; V$

compared to a diode's or transistor's characteristics, were I wrong all the time before ? >_<

Kholdstare's definition of "nonlinear" is different from the usual meaning. Stick with your defintion. if "C" and "L" are constant, then your capacitors and inductors are linear components. Of course their impedance depends on the frequency, but that doesn't count as nonlinear behaviour.

"Linear" here means the same as in "linear differential equations" for eaxmple. If you have two input signals $V_1$ and $V_2$ which produce outputs $I_1$ and $I_2$, then "linear" means that any linear combination $aV_1 + bV_2$ will produce output $aI_1 + bI_2$.

 

[genxium]

You can only write V = ZI only if you express Z as the reactance of capacitor or inductor (1/jwC or jwL), which is valid only for sinusoidal V and I. Thus capacitor and inductor being linear component is only a special case of more general picture (where integration and differentiation is involved >_<). ^_^

>_< Thank you so much! I don't know why I always missed simple but important issues, only after I posted my last reply my friend told me the same issue as your reply, and that's really the point.

Thanks again Kholdstare!

 

[genxium]

"Linear" here means the same as in "linear differential equations" for eaxmple. If you have two input signals $V_1$ and $V_2$ which produce outputs $I_1$ and $I_2$, then "linear" means that any linear combination $aV_1 + bV_2$ will produce output $aI_1 + bI_2$.

I think Khold's explanation is OK for me, however , you make me think of another question, the "linearity" you stated : $aV_1+bV_2 \Rightarrow aI_1+bI_2$ would work for capacitors and inductors only when the signals are sinusoidal, but according to Fourier, any signal can be transformed into a superposition of sinusoidal waves, then what kind of capacitors should be regarded "non-linear" according to your idea?

PS: At least some capacitances are regarded non-linear in harmonic balance algorithm, see here

 

[AlephZero]

Real components are always nonlnear to some degree. In capacitors there is the phenomenon of "charge hiding" in the dielectric, which makes the effective capacitance depend on the applied voltage, the frequency, etc. There are some measurements here: http://www.cliftonlaboratories.com/capacitor_voltage_change.htm

These effects can be are large - e.g. a cap marketed as "0.1uF 50V" reduces to 20% of its nominal value with a DC offset of 50V across it!

Some capacitors are intended to be nonlinear - e.g. varactors.

I'm not sure what Kholdstare means by

Well you can't call capacitors and inductors linear component as far as their dc characteristics are concerned. ^_^

Possibly that the DC leakage of a capacitor is a nonlinear function of voltage? But for many design purposes that isn't very interesting, even though it's true. It doesn't (or shouldn't!) mean that the transient behaviour of any capacitor in a DC circuit at switch-on is intrinsically nonlinear, simply because it is a transient.

 

[genxium]

Thanks AlephZero, I'll spend some days on learning more about capacitors ^_^

 

[Kholdstare]

I think you should be careful about the notion of linearity here.
Capacitors and inductors are indeed linear components as they will obey superposition theorem and their I-V characteristics have linear differential equation.
What I meant was the linearity like V and I related by a constant slope as an equation of straight line. It only works for magnitudes in case of sinusoidal signals. Thus this concept is not useful in circuit simulators.
However you can think about capacitors as V = (1/C)*Q and its much easy to incorporate this linear voltage vs charge relationship in circuit simulators.