# Why do we have exactly three parameters to describe relationship between current and voltage in linear elements?

I was simply pondering over everything I learned about circuits and electricity and there was this one question I couldn't give a simple answer to. Why exactly do we need only these three parameters of capacitance, inductance, and resistance to describe every voltage-current relationship in linear circuits?

One may bring up impedance and say the quantity of impedance unifies these three things and there is only one quantity that controls the relation between voltage and current but even then the question remains, why the impedance is said to be made of these three unique parts.

Thanks!

p.s: One may say there are other factors that control the relationship such as the geometry of the object, but that is really captured as a whole in each of these three things already.

• They aren't enough to capture all possible behaviour. If you add voltage sources or equivalently current sources, including dependent sources, you can then describe all possible linear behaviour, but nonlinear elements exist and are particularly important. Jun 17 at 14:02
• why do we ... We don't. Unthink that thought. Jun 17 at 14:17
• Simply because they're not. The same as asking "why isn't all mathematics encompassed by addition and subtraction?" Jun 17 at 14:30
• @Buraian without going all math on you, the numbers and the opeartions on them you know are far from all that math is about, and they're not sufficient to describe a lot of the math underlying everyday things like Wifi. Jun 17 at 14:55
• @Buraian I've done this sort of thing a lot, it's a very common pattern. I start to learn something new, at the 'explain this like I'm five' level, and it's really simple. There's a few fundamental building blocks that explain everything I've been told (so far). I then fixate on these few simple things. Then I get upset at the complications that the subject really has. Jun 17 at 14:55

It seems like you're wondering why there are only three fundamental linear passive elements, so I'll explain that:

You have two physical quantities at play at the circuit level: voltage and current.

Resistance defines a linear relationship between voltage and current.

Inductance defines a linear relationship between voltage and the rate of change of current.

Capacitance defines a linear relationship between current and the rate of change of voltage.

A linear relationship between the rate of change of voltage and the rate of change of current would be equivalent to one between voltage and current, so resistance covers that too.

Now, you may wonder if there's any linear relationship between current and rate of change of current, or between voltage and rate of change of voltage. As it turns out, you can construct these by combining resistors with capacitors or inductors: a resistor in parallel with an inductor gives you an I-I' relationship, and a resistor in series with a capacitor gives you a V-V' relationship.

So we've covered every possible linear relationship up to first order with just the three elements. Combinations that include more inductors and capacitors in more complex combinations can give you second order relationships, and from there you can extend to third order and fourth, and onward arbitrarily far. As I only have a finite amount of time to write this answer, I don't plan on going through the infinite combinations one by one.

I've skipped over one important point here, though. In reality, we can't achieve any possible relationship with just passive elements, because without active elements, you can't have negative resistance, capacitance, or inductance. These would be required to span the full space of linear source-free circuits, yet they don't exist. They can be made by using active components, but if we're restricted to passive components only, you can't have negative impedances.

That's not a limitation of the mathematics, which will quite happily handle negative impedances, but a limitation of physical reality, where they simply can't exist--as passive elements, they would violate conservation of energy.

You will also find that if you try to make, for instance, an ideal voltage integrator using only passive elements, you'll find zeroes and infinities coming out of your equations for what the capacitance and resistance should be. That also, of course, is not physically realizable, and in fact in this case the math starts to break down as well--so you can't get any linear relationship without adding active elements too. But you can get a limited subset of them. I don't have any rigorous definition for what exactly that subset is--that may be an interesting derivation to run through, but I suspect far more complex than it seems.

• "linear relationship up to first order " Why did you mention 'first order' here? Jun 17 at 15:10
• Because I hadn't gone through any second-order relationships, such as V-I''. Just first derivatives. Jun 17 at 15:11
• Following the comments and discussion , I've posted this question Jun 17 at 15:15
• @Buraian You misunderstood my answer--you can get second-, third-, fourth-, fifth-order, as far up as you want, linear circuits with just resistors, capacitors, and inductors. You need active elements if you want anything nonlinear, but you can get whatever order you want. Jun 17 at 15:17
• Wait so... the active elements are causes linearity? And even in those circuits is the same division of quantites? Jun 17 at 16:09

In an electric circuit, we are tracking charge. At any point in a circuit we are keeping track of three things: how much charge is present, how much charge is moving across that point, and how much pressure are the moving charges under.

We call charge Q.

We call amount of charge that is moving, current (I).

We call the pressure behind the moving charge voltage (V). If the circuit is closed and charge is allowed to flow then the pressure determines the speed that charge flows. But when the circuit is open the flow stops and reaches a steady pressure (voltage) at each point.

Just picture a circuit as a pipeline for charge.