A seasoned electronics engineer explains the surprisingly simple solution for impossible-to-find capacitors.
Some folks find “impossible burgers” to be a tasty plant-based alternative to classic hamburgers. But nobody likes “impossible” capacitors—high value, high precision capacitors that are extremely hard to find. I’ve seen many engineers search far and wide for these rare components with increasing frustration.
But there’s an easy way to make impossible capacitors possible, and it’s a trick I discovered early on in my electronics engineering career.
The Professor’s Answer to Impossible Capacitors
One day a young engineer named Tom approached my desk at Sundstrand Corporation, where I worked part time as an aerospace engineer. I spent the rest of my time at Rock Valley College as an electronics engineering professor and department head. My dual career awarded me the nickname that Tom was about to call me.
“Professor, where can I find large-value precision capacitors?” he asked.
Grinning, I countered: “Do I look like a distributor—Digikey, Mouser, or Newark?”
Capacitance is dependent on a capacitor’s physical dimensions. Large-value capacitors tend to be physically large with larger tolerances, while small-value capacitors are generally physically small with tighter tolerances. Large-value yet low-tolerance capacitors (say, ±1%) are nearly impossible to find. At best, you’ll wind up with an expensive and bulky capacitor with a very long lead time.
I asked Tom to sketch his application circuit, a buffered, low-pass filter circuit with a corner frequency (pole) of 479 mHz. It looked like this:
Tom explained that resistor R1 is the largest permissible value to minimize the required capacitor value. If the resistor value is made much larger, op amp DC bias current and parasitic capacitances can become significant. Tom declared that precision is critical in his application. The resistor has a tolerance of 0.1 percent, which is available readily at a reasonable price. The required capacitor must have a tolerance of 1 percent or better. Given the precision of the resistor, the variation in the pole frequency tolerance is dominated by the capacitor. Tom continued, “With a capacitor tolerance of ±1 percent, the overall pole frequency tolerance is about ±1.2 percent, which is acceptable.”
Since a 1-µF, 1 percent capacitor is nearly impossible to find, I suggested that a better approach would be to specify a small-value precision capacitor and use an impedance multiplier to inflate its value.
With apologies, I explained that I needed to put on my “professor hat” (technically, that would be my mortar board). I have seen many projects go wrong because an engineer jumped at a solution without understanding it completely. So, I started with the essence of the circuit I had in mind.
Faking the Impossible Capacitor
The circuit incorporates positive feedback and two voltage followers. In this case, the goal is to make a particular resistor, Rx, look like an open circuit. The resulting equivalent resistance is presented to the signal source Vin. With reference to the diagram, the top of resistor Rx has a potential equal to the input voltage Vin. The positive feedback drives the bottom of the resistor to the same potential. This means the voltage drop across resistor Rx is reduced to zero. A resistor with zero volts across it will also have a current of zero through it. Because an open circuit passes no current, the input voltage source “sees” an open circuit.
We can control the multiplied value of Rx by making the circuit less good. A voltage divider accomplishes this nicely. The voltage divider works such that only a controlled fraction of the input voltage is delivered to the bottom terminal of resistor Rx. There will be a voltage drop across Rx, which means current will be drawn from the input voltage source. It will “see” a finite value of input resistance.
Applying a little circuit analysis and just enough algebra to render a slight headache, we achieve the result:
The 100kΩ resistance has been multiplied by 1.67. While resistances are indicated, the circuit performs in a similar fashion when we include impedances (resistance, inductive reactance, capacitive reactance and combinations).
Tom’s eyes lit up, “I got it! We replace Rx with a capacitor and we are all set to go!”
I responded, “Whoa—lightspeed, slow down! Let’s think things through. If we replace Rx with a capacitor, the capacitor’s impedance will be multiplied by a factor k. Because capacitive reactance is 1/(2πfC), what happens to the capacitance when we multiply the reactance by k?”
Tom puzzled for just a brief moment and concluded, “We will be dividing C by k, making C smaller.”
I confessed to Tom that I had been pondering the circuit for a while. I moved the input signal to the right side of the circuit. Tom saw the attraction of the modified circuit arrangement. The input signal is applied to a voltage follower. The output of the op amp AR2 drives a low-pass filter consisting of resistor R3 and capacitor C1.
Tom indicated eagerly that we must be close to the solution. I agreed that we were nearly done, but we needed to perform a bit of circuit analysis to expose the nuances embodied in the circuit. I explained to Tom that one analysis approach is to treat Vout and Vin as independent voltage sources and use the superposition theorem. I suggested to Tom that he run through the derivation. He promised he would do so with the same sincerity of a 16-year-old son borrowing $25 from his father with an assurance to pay it back.
For reference, we look at the simple low-pass filter that follows:
Comparing the results allows us to determine the capacitor’s multiplication.
Using the component values indicated in our example, we can determine the equivalent capacitance.
The required capacitor value! After a quick search, I shared, “The 8200pF capacitor has a tolerance of 1 percent, which means the 1.00µF multiplied value also has a tolerance of 1 percent.” I pointed out to Tom the exact capacitor he needed, which is readily available from Mouser.
And the high-end corner frequency was what it needed to be:
Tom was thrilled. He had a new design tool: use an impedance multiplier and a small, precision capacitor to produce a larger value precision capacitor. He had an impedance multiplier solution that would provide the required corner frequency and employ a relatively low-cost, small off-the-shelf capacitor.
As he left clutching the paper holding his solution, I called to him, “Don’t forget to do that derivation!” I did not know until that very moment that the rolling of eyes makes a distinctive sound (I figured it must be due to the friction between the eyeballs and dry eyelids).
