# What exactly is the intuition behind physical circuits being linear?

After learning mesh-current method for solving circuits, I started reading the wikipedia article on linearity to understand more on the "why?"

Reference.

In this article, I sort of get why resistors are linear - because they obey both properties of a linear object which is

$$f(x +y ) = f(x) + f(y)$$

$$f(ax) = af(x)$$

as in,

$$i = \frac{v}{R}$$

$$i(v_1 + v_2) = \frac{ v_1 + v_2}{R}$$

$$i(a v_1 ) = a \frac{v_1}{r}$$

But, I don't understand why this would be true for inductors and capacitors. Why do these physical systems obey this mathematical structure? I mean I was shocked that the mesh current method was actually a legitimate technique.

Note: The kind of answer that I'm looking for one is one which uses both physical intuition and mathematics to explain but not entirely explain the thing based on both.

• Do you know the equations for capacitors and inductors? (like how V=IR is the equation for resistors??) Commented Jul 16, 2020 at 9:57
• Yes, for inductor it is V=L di/dt and, for capacitor it is C=Q/V
– Babu
Commented Jul 16, 2020 at 9:59
• so if you multiply the i in an inductor, does it multiply the V by the same amount? Commented Jul 16, 2020 at 10:01
• If the super position is holds for the current, does this imply super position holds it's time derivative as well? I think I get it but I don't get the intuition for why it should
– Babu
Commented Jul 16, 2020 at 10:19
• If you multiply the i in an inductor, do you multiply the di/dt as well? Commented Jul 16, 2020 at 10:31

Some with a sense of humor said

"On a small enough scale, all behaviors are linear."

In circuit design, we do not have exact polynomial descriptions for any component, so we go with "it is linear".

And somewhat later, we learn about Taylor Series approximations to the Exponential Diode behaviors.

And we learn about 2_nd order and 3_rd order intercept points and P1 compression points, input referred and output referred.

And we learn to use spreadsheets to keep track of the distortion, and use worst case or RSS modeling.

All because our components are real structures with imperfect crystalline manufacturing, and incomplete numeric descriptions in the models.

And we get paid to release a design to the FAB or to the manufacturing floor.

So we, over years, become pragmatic, becoming engineers, accepting the wonderful variability of life, and highly reliable systems become available to customers.

It all works.

==============================

Then we get paid to convert all the systems_on_PCB to systems_on_silicon, and the fun learning starts all over again, not forgetting the "ground" is outside the Integrated Circuit, several millimeters away. Thus vast differential circuits become our tools. And wide substrate/well rings, to minimize gradients on silicon surface, become a tool.

Then, with everything located only 100 microns from any other circuit (well, maybe 1,000 micron or 5,000 microns), we get to rethink Heavyside and Maxwell, and realize magnetic fields are just a result of slight delays in the Efields, thus we can SHIELD magnetic fields using the SKIN EFFECT.

Thus circuit design and system design require constant digging down into the fundamentals, to rethink what we have been taught. There is a need to detect "folklore", and understand why the folklore exists, to explore that particular design space and find exceptions to the rules, and exploit them. Folklore about substrate noise control is plentious; challenge them.

Also we might ponder "harmonics". Harmonics are merely an expression of the correlation between a non-linear behavior and various linearly_spaced sinusoids. A step function correlates with ANYTHING, to some degree. We happily indicate the peaks and nulls in the amount of correlation, because of lossless energy storage(infinite Q); this realization lets us ponder FINITE_Q systems, and realize harmonics do not exist; harmonics are just the result of "correlation".

Get thee to the lab, with a Spectrum Analyzer and Pulse Generator. And play with pulse_frequency and pulse_width and pulse_risetime, and learn about "harmonics". Respect the input power limit of the SA.

Most importantly, alter the Resolution Bandwidth and Video Bandwidth of the SA, which lets you in essence alter_the_Q of the system; this may be the crucial takeaway thought from your lab tinkering.

• this post makes me want to become an electrical engineering... Heaviside the best!!
– Babu
Commented Jul 16, 2020 at 20:27

What exactly is the intuition behind physical circuits being linear?

One simple rule that I think of when judging if a circuit is linear is to consider if there might be harmonics produced when driving the circuit with a sine wave. If harmonics are produced with a sinusoidal input, then the circuit is somewhat non-linear. The more harmonic amplitude that is produced (compared to the original sine wave amplitude) then the more non-linear it is.

• what exactly is a harmonic?
– Babu
Commented Jul 16, 2020 at 9:26
• A harmonic is a multiple frequency of the original sine wave. For instance, a square wave contains multiple harmonics added on top of a basic sinewave. The sum of all these harmonics makes a square wave. Look at this Commented Jul 16, 2020 at 9:29
• I downvoted this because it's not intuitive. Harmonics are a result of non-linearity when interpreted in the frequency domain, not vice-versa. Commented Jul 16, 2020 at 10:32
• It may not be intuitive to you dear boy but for many it totally is. Please do leave an answer to this question if you are able @user253751 Commented Jul 16, 2020 at 11:29
• @user253751, harmonics are there no matter which domain you're using. In audio systems you can hear them. Ask any rock guitarist. Commented Jul 16, 2020 at 17:43

I'm not sure what the problem is. If you do the exact same thing you did for the resistor with the inductor and capacitor, you'll find that the inductor and capacitor are linear as well.

$$v=L\frac{di}{dt}$$ If $$\ i=i_1+i_2\$$, then $$\ v=L\frac{d}{dt}(i_1+i_2)=L\frac{di_1}{dt}+L\frac{di_2}{dt}\$$.

If you multiply $$\ i\$$ by some constant $$\ \alpha\$$, then $$\ v=L\frac{d}{dt}(\alpha i)=\alpha L\frac{di}{dt}\$$.

You can do the same thing with the capacitor.

However, I don't know what this has to do with the mesh-current technique. That technique is based on KVL and has nothing to do with the linearity of circuit components. You can still use the mesh-current technique with non-linear components (like a diode), but the resulting set of non-linear equations will be much harder to solve by hand.

• Ok,that seems legit but I can't still get the intuition behind it
– Babu
Commented Jul 16, 2020 at 20:29

For every electronic system, the output voltage or current is limited. So if the response gets very close to those limitations, the system is not linear. A system may be linear if the response is much smaller than those limits.

• What do you mean when you say the word response and limitations?
– Babu
Commented Jul 16, 2020 at 20:26
• Response is the output voltage or output current. If an opamp is supplied with +15 V and - 15 V, the output could not get higher than +15 V, the system behaves unlinear when the output gets close to +15 V, for instance above 14 V.
– Uwe
Commented Jul 16, 2020 at 21:24