# A Graphical Notation for Circuitry

Common practice in electronics lies in understanding, bootstrapping, and further manipulating symbols to represent signal filtration systems. First, someone understands the basic premises of a circuit. Then, that person chains together circuits to make new circuits. Then, hopefully, the person achieves the output effect sought after.

Sounds simple, right?

I mean, how inconspicuous could this possibly be? It's just a resistor. The resistance observed by the battery is just 4700 Ohms.

A bit more complex, but the algebra shouldn't be too bad. It's just a parallel resistance of 4700 Ohms and L. All in all, it's:

\[ \frac{4700*sL}{4700+sL} \]

Okay, not bad. Just a bit of math.

Here's where it starts to get bad. There's a **lot** more manipulation necessary, and it's easy to get lost. The equation \(Z_{out}=\frac{1}{\sqrt{\frac{1}{R}^2+(\frac{1}{s*L}-\frac{1}{\frac{1}{s*C}})^2}}\) isn't forgiving if you forget a symbol.

And a bit past here, a lot of engineers I've met decide to use auto-solvers (e.g. LTSpice).

Don't get me wrong, those tools are handy, and this approach makes sense. But I get the sense, from talking to practicing engineers, that algebra's been the primary way of navigating this complexity. It makes sense: it's quick to throw around, and it's quick to navigate with. But when things get complex, the path of least resistance becomes avoiding wasting time "verifying the computer" and instead just letting the computer create the equation.

Here's my attempt at a prototype of a graphical circuit analysis toolkit. Instead of relying on systems of equations to understand a circuit reaction, I'll derive geometric representations of reactions. From there, I'll show you how to quickly analyze these circuits.

Let's start with our three basic building blocks: an inductor, a resistor, and a capacitor.

###### Resistor

This part's easy. Resistors don't have any changes based on frequency, so it's just a flat line set at the value of the resistor.

For an equation of \(R=100\):

This horizontal axis is the frequency, and the vertical axis is how the element changes its reactance to that frequency.

###### Inductor

Inductors gain impedance as their frequency increases. That makes sense; let's chart this as a line graph that matches the equation for an inductor.

For an equation of \(Z_L=s*100uH\):

Okay, so this shouldn't surprise anyone. The equation is a line graph, so the resistance increases as the frequency increases. That makes sense.

Two caveats: at 0 frequency (a DC signal) in steady state, no inductor carries resistance. This makes it synonymous to a wire storing energy in this state. At infinite frequency, the inductor acts as an open.

###### Capacitor

Capacitors lose impedance as their frequency increases. That also makes sense: the less electrons shifting between the plates of a capacitor, the less it even looks like the capacitor is more than a wire. Let's model this.

For a capacitor of \(C=100uF\), the equation \(Z_C=\frac{1}{s*100uF}\):

This is my poor rendition of an asymptotic graph. The capacitor impedance equation acts like an asymptotic plot; this roughly resembles its reaction.

###### Upgrading from our pinpoints

Okay, so the basic graph building blocks exist. Where does the road lead from here?

First, the reactances should be straight lines: they're quick to draw and lead to easy prototyping and inference of circuit behavior. Then, the bottom portion of this graph should scale by decades: frequencies by the single digit increment don't matter as much as frequencies by the 10x increase or decrease.

These upgrades are reflected below:

You might have observed that we've gone from my (incredibly) basic graphs to this diagramming system. Let me explain: we've taken our equations, piped them through a \(20log_{10}(f(x))\) transformation, and indexed along the bottom axis. The above graph displays a capacitor reactance displayed along this.

Let's try quickly interpreting those above circuits again.

Okay, so we have an inductor and a resistor in parallel. The natural intuition would let us guess that the inductor is easier to pass through than the resistor until a certain frequency. Afterwards, the resistor's easier to pass through. We just draw two lines, and already there's a ton of insight: at around 10kHZ the resistor takes over, and the inductor needs to be a smaller value if you want to change that crossover frequency. That fat green line is our final path.

Okay, so this is a bit more confusing. Let's decipher it.

The inductor gives us less resistance at low frequencies. That makes sense. The capacitor gives us an open wire at high frequencies. The complex part is factoring in how the resistor affects the LC combination. Because a resonance frequency is hit around 1kHZ, the resistance between the inductor and capacitor are even. But then, the resistor supersedes either path. This makes that Q point jump to the value of the resistor. That fat green, curved line is our final path.

From that, we immediately have a few parameters to play with. First, to drop the Q point, the resistor value can be dropped. A resistor value below 700Ohms or so is dead weight. The resonance will be at around 1kHZ, and we have two levers to play with to shift this.

Consider the speed and efficiency of this approach vs this equation:

\[ Z_{out}=\frac{1}{\sqrt{\frac{1}{R}^2+(\frac{1}{s*L}-\frac{1}{\frac{1}{s*C}})^2}} \]

Can you tell off-cuff where the crossover frequency is? Or what's the optimal value for a resistor, in this case, to not modify a Q point?

Okay, so this gets a bit trickier: we need to modify out conceptual understanding to daisy-chain together circuits. Consider breaking this up into two distinct portions: the capacitor and the combination of the inductor and resistor. I've done this below and merged the two in a graph. The capacitor is the darker line, and the inductor-resistor combo is the lighter line.

Okay, so we have our two inductances. But they're in series, not parallel. Therefore, **the element with the most resistance determines the dominating effect of the impedance**. This is reflected below:

That final mark might have come in a bit too high, but you get the point: at low frequencies, the capacitor acts as the primary resistance. Because the inductor is involved, the first part of the circuit is basically an open wire. Then, as frequency increases, eventually the capacitor acts like an open wire while the resistor dominates the circuit.

Here's the equation:

\[ Z_{out}=\frac{RsL}{R+sL}+\frac{1}{sC} \]

I don't know how good you are at math, but I certainly can't tell all of that information from the above.

###### Conclusion

Multiplication in the Roman era was thought of only to be reserved for geniuses. The syntax was clunky, people didn't see the purpose, and it was time consuming. But when Arabic numerals were introduced, people's eyes were opened to how easy and expansive multiplying and dividing numbers can actually be. Now, this is done by the youngest in our generation.

On that same note, vomiting equations for circuits does little to actually *tell* me what a circuit does, or how it's supposed to react. Instead, I'll go to a computer and need to trust it in order to figure out how to play with the circuit. But with this new notation, it takes me about thirty seconds to see how a circuit functions (in terms of frequency to resistance), what key levers exist in the circuit to modify functionality, and what I need to say to a designer to ask whether I can modify a circuit.

That's pretty special.