# Is current/resistance a parabola? (For lack of a better word)

So it occurred to me today that is resistance is the load that draws current, but also the thing that limits it, wouldn't a graph of current to resistance increase, but then decrease?

• There is a clear and simple formula given by Ohms Law. Do you know it? – Eugene Sh. Jan 3 '17 at 22:32
• Yeah, it's V = I/R, but if current I'd drawn by resistance, and then limited by it, then isn't there only a certain amount of resistance that can be applied to a circuit before the current drawn and current limited are equal? – Emmett P Jan 3 '17 at 22:34
• What you're thinking about isn't current but power. Look up the "maximum power transfer theorem": the load resistance should match the source resistance for maximum power transfer. This is why a switch dissipates no power when it is off or on, but dissipates power if it has a finite resistance. (And it's not a parabola, but does go to zero at the ends.) – Ken Shirriff Jan 3 '17 at 22:35
• Is this what you have in mind? That is to do with power transfer with a source that has some resistance, showing power delivered vs load resistance. – Tom Carpenter Jan 3 '17 at 22:36
• So it's not current that is drawn by resistance, it's power – Emmett P Jan 3 '17 at 22:38

here is a graphical inverse relationship when Voltage is constant (10V) and $R=\frac{V}{I}$ However sometimes current is constant then means V is proportional to R using same formula above or rearranged V=IR or in the EU, U=IR

Here is a another way to show same data

Here are 3 circuits with resistance and reactance with R,L,C & f chose to give the same impedance , current and power but at different phases.

The series 100R current and power for each circuit is shown in same order as schematic. Which one dissipates the most real most total?

The scope only shows power in the 1st resistor.

Nothing actually "draws" current (or power). You might say that current is pushed by voltage, and that current is limited by resistance.

Try playing with the Power equation $P=VI$ in terms of Ohm's law $V=IR$. You may be looking for the following intuitive effect:

If you replace $I$, for example, with the equivalent $\frac{V}{R}$, you get $P=\frac{V^2}{R}$:

As you decrease the resistance in a real system, however, eventually the power source won't be able provide enough current. Then it will fail / shut down / burn up :)

To expand on my comment, you're probably thinking about the behavior of power. For a fixed current, more resistance means more power. But more resistance also means less current. How does this play out? As you suspected, the power goes up with resistance and then down.

To be specific, the Maximum power transfer theorem says that maximum power is transferred when the load resistance matches the source resistance.

Suppose you have the situation below, where $R_1$ is the load resistor you are varying, and $R_2$ is the fixed source resistance. You get the maximum power for $R_1$ when $R_1=R_2$. Specifically, current $I = V/(R_1+R_2)$ and power $W = I^2 R_1 = V^2 R_1 / (R_1 + R_2)^2$. If $R_1$ is "too small", the numerator drops and you lose power. If $R_1$ is "too big", the denominator gets too large and you lose power. The peak is at $R_1=R_2$. At this point you're getting 50% efficiency, since half the power is dissipated by $R_1$ and half by $R_2$. You get higher efficiency as $R_1$ gets larger, but less power transferred overall.

simulate this circuit – Schematic created using CircuitLab

The following image is from Wikipedia: Maximum power transfer theorem. The red line shows how the power transferred goes up and then down, with the maximum when the load resistance and source resistance match. (Although it's not the parabola you suspected.)

I find real-world applications of this interesting. If a switch is off (infinite resistance) or on (zero resistance), there is no power dissipated in the switch. But if the switch has some resistance, then it dissipates power, which is generally bad. This is why to dim lights you normally use triac dimmer switches (which rapidly switch on and off) rather than a variable resistance. But for, say, a toaster, you want its resistance to be in the middle so it heats up: too much resistance or too little resistance and you're in the low part of the curve.

Another application is MOSFETs. You generally want a MOSFET to have very low resistance when it's on and very high resistance when it's off, so you avoid dissipating power. It's the resistances in the middle that cause it to overheat.