Cursed circuits: capacitance multiplier - lcamtuf’s thing
lcamtuf’s thing
SubscribeSign in
Cursed circuits #5: capacitance multiplier<br>Capacitor vendors don't want you to know this! Save money on capacitors by spending more on other components<br>Jul 05, 2026
Share
Electronic circuit theory is a frequent theme on this blog. As a part of this sorta-curriculum, I published a number of articles about operational amplifiers. I keep coming back to this topic for two reasons. First, I think these components are usually explained poorly, making them a major stumbling block for folks trying to learn the craft. Second, op-amps have gotten really good, inexpensive, and small, so I think they should be used more.<br>If the component is still a mystery to you, this article is probably not the best place to start; we’ll cover the very basics, but I recommend carving out some time to go over two other write-ups from 2023:
The basics of signal amplification<br>lcamtuf<br>February 7, 2023
Read full story
What's inside an op-amp?<br>lcamtuf<br>February 11, 2023
Read full story
I also cover op-amp theory (and a lot more!) in The Secret Life of Circuits, a lovingly-crafted book that’s available in early access and will be hitting the shelves in about two months.<br>Today, I’d like to talk about a circuit that didn’t make the cut for the book. It’s not nearly as useful as a transimpedance amplifier, an integrator, or a Sallen-Key filter, all of which get in-depth treatments from first principles. At the same time, it’s just too cool not to share.<br>But first, a word from our sponsor
I know that most readers don’t click links, so before we dive in, let’s recap what an op-amp does. If you have it pegged as some sort of a variable-gain amplifier that “reads” the value of a pair of external resistors to configure internal gain, it’s bet to forget all that and start afresh.<br>An ideal op-amp does one thing and one thing only: it calculates the difference between the voltages on its two input pins (Vin- and Vin+), multiplies it by a humongous constant factor (AOL, typically 1,000,000 or more), and then outputs the resulting voltage in relation to the midpoint of the supply (Vmid). We can write this as:<br>\(V_{out} = V_{mid} + (V_{in+} - V_{in-}) \cdot A_{OL}\)
In practical terms, it means that if Vin- is noticably less than Vin+, the output voltage swings toward the positive supply of the chip; conversely, if Vin- exceeds Vin+, the output swings dives toward the negative rail. Intermediate output voltages are possible only in a very narrow, microvolt-range linear region of Vin- ≈ Vin+.<br>The simplest op-amp circuit — and the only one we need today — is the voltage follower:
A simple voltage follower.<br>The circuit loops the output voltage onto one of its differential input pins. If the input signal on Vin+ creeps higher in relation to Vin-, this makes the amplified differential signal more positive, forcing the output voltage to rise until the Vin- ≈ Vin+ equilibrium is restored. In the same vein, if Vin- drops, the difference becomes more negative, sending the amplified signal toward a lower equilibrium point. In effect, the output voltage tracks the input signal with sub-millivolt accuracy.<br>To make sense of the next section, we need just two other tidbits of electronic theory. First, Ohm’s law: the current flowing through a resistor is proportional to the electromotive force (voltage) applied to its terminals, divided by the component’s value (resistance): I = V/R. Second, we need to know that a capacitor subjected a voltage across its terminals will admit (nearly-arbitrary) charging current until the pushback force created by the accumulated charge matches the external voltage. Higher component value (higher capacitance) means that proportionately more electrons can be shuffled around in response to the same electromotive force; in other words, higher C means more current over time.<br>If the above paragraph sounds confusing, you should review the article on core concepts in electronic circuits before venturing forth.<br>A capacitor, with a twist
This brings us to our guest of honor: the capacitance multiplier. It’s the kind of a circuit that usually doesn’t make sense at first glance because we can’t pattern-match it to anything we know:
A basic capacitance multiplier.<br>To unravel the mystery, it suffices to break it down into two mostly-separate building blocks. Section A is just an op-amp configured as a voltage follower. No matter what else is going in the circuit, it takes some voltage from section B and then mirrors that signal on its output leg.<br>Section B is a capacitor that’s charging through a resistor; although the voltage across the capacitor’s terminals (Vcap) will change over time, let’s model this in a freeze-frame view. In this setting, the op-amp is mirroring the capacitor’s current charge state; the voltage on the right terminal of R2 is almost exactly the same as Vcap.<br>Moving onto R1, we can conclude from Ohm’s law that the current...