Confused about the need for passive sign conventions in sources

[etotheipi]

For passive electrical components, I can understand the need for the passive sign convention - i.e. taking the voltage to be the potential on the side where the current enters (higher potential, for a passive component) minus the potential on the other side. For a resistor, this means the change in potential along the direction of current is then $-iR$. If instead we had an inductor, according to PSC the voltage would be $L\frac{di}{dt}$ and the change in potential moving along the direction of current $-L\frac{di}{dt}$. Ohm's law holds with no negative signs.

In the above scenarios, the PSC is actually fairly helpful - especially when dealing with capacitors and other components where sign errors are more likely. However, using PSC for voltage sources like cells seems needlessly confusing.

I thought it seemed simpler to stick to PSC for passive components and ASC for active components, so that all voltages come out as positive (it seems sort of weird to label a cell as having negative voltage...). However, the relevant Wikipedia article says this with regards to my suggestion:

This convention may seem preferable to the PSC, since the power P and resistance R always have positive values. However it cannot be used in electronics, because it is not possible to classify some electronic components unambiguously as "sources" or "loads".

Which components are they talking about here? Granted, I haven't come across that many electrical components yet however this - to me at the moment, at least - doesn't seem like that convincing an argument.

Thank you!

 

[vanhees71]

It's much easier to remember all these signs by thinking, where Kirchhoff's Laws come from, namely from Maxwell's equations as anything that has to do with the electromagnetic field.

For the "mesh rule", you integrate Faraday's Law,
$$\vec{\nabla} \times \vec{E}=-\partial_t \vec{B}$$
over an area with the closed loop in the circuit under consideration as the boundary curve. The sign of the currents and the loop is arbitrary. Only for the sign of the magnetic flux on the right-hand side of the equation you must keep the right-hand rule in mind (and the additional - sign).

As an example take the most simple circuit consisting of an ideal coil and a resistor, and an AC source in series. Assume you point the direction of the current in the same direction as the loop in the integral. Then the integral of the left-hand side gives $R i - U_0 \cos(\omega t)$ and the right-hand side $-L \dot{i}$, i.e., the equation is
$$R i + L \dot{i}=U_0 \cos(\omega t).$$
The other Kirchhoff law is simply charge conservation, i.e., adding all currents at each node (counting currents flowing into the node positive and those flowing out negative) yields 0.

 

[Dale]

The sign conventions don’t matter a bit. Just be consistent between your circuit diagram and your equations and it will all work out correctly.

 

[etotheipi]

The sign conventions don’t matter a bit. Just be consistent between your circuit diagram and your equations and it will all work out correctly.

Though how we choose the reference directions does affect the defining equations. If we have the reference directions of current and voltage in opposite directions (PSC), Ohm's law becomes $V=IR$. If they are in the same direction (ASC), then we have $V=-IR$. Same goes for power, if the reference directions are opposite then $P=IV$, if they are the same then $P=-IV$.

Before I used to think it was more like $|V| = |I|R$ - and that we'd account for the sign of the change in potential etc. using the circuit diagram - however this isn't what Ohm's law actually says! $V$ is definitely a signed change in potential, but it is the conventions that tell us which side we subtract from the other!

On the plus side, I think this issue only really applies to passive components, since for sources the direction of the voltage is independent of the current and there is no ambiguity when applying KVL.

 

[Dale]

If we have the reference directions of current and voltage in opposite directions (PSC), Ohm's law becomes ...

I never think in terms of modifying Ohm’s law, just KCL. At each node I just make all outward currents positive. So if a current points into a node then it is negative for that node’s KCL equation. That is it.

 

[etotheipi]

I never think in terms of modifying Ohm’s law, just KCL. At each node I just make all outward currents positive. So if a current points into a node then it is negative for that node’s KCL equation. That is it.

I agree with your thinking for KCL, but am struggling to understand why Ohm's law need not be modified.

If a signed current $i$ flows from node $A$ to node $B$ through a resistor $R$, then the voltage $v$ we obtain via $v=iR$ equals $V_{a} - V_{b}$, that is, the reference direction for the voltage is necessarily opposite to that of the current.

So essentially if we write $v = iR$, $v=\frac{Q}{C}$ or $v = L\frac{di}{dt}$, then it is implied the reference direction for $v$ is opposite to the direction of the current.

Of course, we can just negate these if we traverse a component in the opposite direction to the reference voltage direction in order to sum all of the changes in potential as usual when applying $KVL$. I think this is what you are getting at (or I might be totally off!); we don't really need to have a negated version of Ohm's law, so long as we keep in mind which direction the $v$ in $v=iR$ refers to?

 

[vanhees71]

It's because $\vec{E}=-\vec{\nabla} V$ and $\vec{j}=\sigma \vec{E}$. As I wrote earlier in this thread, it's much simpler to think in terms of the integral form of Faraday's Law (for all parts of the circuit at rest):
$$\int_{\partial A} \mathrm{d} \vec{r} \cdot \vec{E}=-\frac{\mathrm{d}}{\mathrm{d} t} \int_{A} \mathrm{d}^2 \vec{f} \cdot \vec{B}.$$
The only important sign convention then is that the surface-normal vectors $\mathrm{d}^2 \vec{f}$ are oriented relative to the boundary-trangent vectors $\mathrm{d} \vec{r}$ according to the right-hand rule (as in Stokes's theorem which is here used to derive the integral form of Faraday's Law from the corresponding local Maxwell equation).

 

[Dale]

am struggling to understand why Ohm's law need not be modified.

If a signed current i flows from node A to node B through a resistor R, then the voltage v we obtain via v=iR equals Va−Vb, that is, the reference direction for the voltage is necessarily opposite to that of the current.

Yes. And you can always write it that way ($V_a-V_b$) when applying KCL in node voltage analysis. I guess in that sense I always follow the passive convention in KCL regardless of how my currents are drawn in the circuit diagram.

Of course, we can just negate these if we traverse a component in the opposite direction to the reference voltage direction in order to sum all of the changes in potential as usual when applying KVL. I think this is what you are getting at (or I might be totally off!);

I think so, although I always definitively recommend node voltage analysis over mesh current analysis, so KCL is my primary tool. My KVL is now quite rusty.

 

[vanhees71]

I don't understand what you mean here. For a AC-network analysis you need both Kirchhoff rules, the one applying at the nodes, saying that the sum of all currents is 0 (with the appropriate signs, i.e., with all currents running into the node with + and all the currents running out of the node with -; you have the opposite convention to mine, but that doesn't matter of course) and the mesh rule, which is nothing else than Faraday's Law in integral form with an integration along the mesh.

Of course it involves the electromotive force (not voltages), but that's another story correcting some unfortunate slang in the EE as well as physics literature.

 

[Dale]

For a AC-network analysis you need both Kirchhoff rules,

Really? I have done plenty of AC circuits with just KCL. I wonder if there is a specific category of circuits that requires both. If so then my school assignments carefully avoided that category. I have never needed more than KCL and node voltage analysis.

 

[vanhees71]

What precisely do you label with KCL? Perhaps I understand what you mean when you just take a simple case of two impedances in parallel and tell us how you analyze it with only one of the Kirchhoff rules.

 

[etotheipi]

Right, I think I've cleared up what was causing the confusion.

All that is meant by the PSC is that the reference directions for current and voltage are defined in opposite directions. With this convention, $V_{psc} = IR$ and $P = IV_{psc} = I^{2}R$ for a load element, and sources have negative power.

ASC applies when the reference directions are defined in the same direction. This results in $V_{asc} = -IR$ and $P = IV_{asc} = -I^{2}R$ for a load element, and sources have positive power. This convention is apparently hardly ever used in electrical engineering.

Effectively, if we use PSC then there is no negative sign in any of the defining v-i equations for passive elements. If we use ASC, then we need to negate the defining v-i equations for passive elements. Note, however, that the power is $P=IV$ regardless. It's just that for a specific element, PSC will give a power of $k$ and ASC will give a power of $-k$.

To conclude, it seems wise to just stick to PSC and define the reference directions oppositely, so that the power comes out as per the convention and the v-i relations are stated as usual. Though if the reference directions are (somehow) the same, ASC is being used; so the power convention is flipped (sources vs loads), and the v-i relations are also negated.

 

[Dale]

What precisely do you label with KCL? Perhaps I understand what you mean when you just take a simple case of two impedances in parallel and tell us how you analyze it with only one of the Kirchhoff rules.

Sure. Since the parallel circuit with a voltage source is too easy I will assume that you intend a current source of complex magnitude and phase $i_0$. The parallel impedances are $Z_1$ and $Z_2$ with currents $i_1$ and $i_2$ and the unknown voltage across the source and impedances is $v$. All of the above are complex phasors at frequency $\omega$.

Using Ohms law we get $i_1=v/Z_1$ and $i_2=v/Z_2$. Since it is a parallel configuration there are only two nodes, one will be grounded and the other is the one at the unknown voltage $v$. At that node we write KCL:
$$-i_0+v/Z_1+v/Z_2=0$$
Which we solve for $v$ to get $v=i_0 (Z_1 Z_2/(Z_1+Z_2))$.

 

[vanhees71]

Well, so you are indeed using both Kirchhoff rules, not only one, which cannot be sufficient from the mathematics of the Maxwell equations (in quasistationary approximations) underlying the entire theory.

 

[Dale]

you are indeed using both Kirchhoff rules

I never once calculated the sum of the voltage drops around a loop. So I did not use KVL.

Of course, the result is consistent with KVL, but I did not use it. I used only Ohm’s law and KCL.

 

[vanhees71]

But you implicitly did right that (though in an elegant final form of AC circuit theory with complex impedances). What you used is thus not Ohm's Law (which holds only for DC and resistances) but Faraday's Law of induction with EMFs.

 

[Dale]

But you implicitly did right that (though in an elegant final form of AC circuit theory with complex impedances). What you used is thus not Ohm's Law (which holds only for DC and resistances) but Faraday's Law of induction with EMFs.

That is not KVL. KVL is that the sum of the voltages around a closed loop is zero. In $i_1=v/Z_1$ there is neither a loop nor a zero sum. There simply is no way to call that KVL.

I have no problem with you saying that $i_1=v/Z_1$ involves Faraday's law* if you like, but that does not make it KVL. Faraday's law and KVL are different, KVL is derived from Faraday’s law it is not identical to Faraday’s law. Without a loop and without a sum there is no KVL.

*I don’t object to you saying it that way, but I would not say it that way at all. It is a generalization of Ohm’s law to complex resistances. This generalization applies equally to resistors, capacitors, and inductors, whereas Faraday’s law would only be important for inductors. So since the form is that of Ohm’s law and the use is a generalization of the use of Ohm’s law I would call it Ohm’s law.

 

[vanhees71]

This is semantics. Of course everything is contained in Maxwell's equations at the most fundamental level.

 

[Dale]

This is semantics. Of course everything is contained in Maxwell's equations at the most fundamental level.

Of course it’s semantic. The phrase “you can analyze a circuit using KCL rather than KVL” is generally understood to mean “you can calculate all of the currents and voltages in a circuit by writing and solving the equations summing the currents leaving each node to zero rather than by writing and solving the equations summing the voltages around each loop to zero”. That is the generally understood meaning, which is of course semantics.

By the way, the non-semantic reason that I always recommend node voltage analysis (using KCL) rather than mesh current analysis (using KVL) is that mesh current analysis requires a planar circuit and cannot analyze non-planar circuits. So I do know of situations that you cannot use KVL (usual semantics) to completely solve a circuit.

I don’t know if any similar restriction for node voltage analysis. So I was surprised to learn (I thought) from you that there were some circuits that could not be analyzed with node voltage analysis with KCL. I have to admit that I was fairly disappointed to learn that you were simply making that claim based on non-standard semantics and that there was not a specific class of circuits that you were aware of that could not be solved with node voltage analysis.

 

[vanhees71]

My claim is you need both Kirchhoff laws not only one, but as I said, it's semantics whether you simply use only one by naming it and the other without naming it ;-)).

 

[Dale]

My claim is you need both Kirchhoff laws not only one, but as I said, it's semantics whether you simply use only one by naming it and the other without naming it ;-)).

The way everyone else in the circuits world uses the term, if you don’t write down the sum of all of the voltages around a loop then you have not used KVL.

The semantic argument is yours. Yes, you can use the term differently from everyone else, but you can expect that will cause unnecessary confusion as we had here.