# Capacitance Multiplier?

Some friends (thanks Mark and Matt) pointed me at a handy little power supply filter called a Capacitance Multiplier. It’s useful for filtering audio frequency noise (ripple) on power supply rails, for example in sensitive audio or RF circuits like VCOs.

The Wikipedia Capacitance Multiplier article suggests it effectively multiplies the capacitance by the transistor current gain Beta. I’ve done some analysis and have another interpretation, backed by LTSpice simulations and calculations.

Here are two circuits of a power supply (12VDC, 1 ohm impedance) powering a 100 ohm load. The power supply has 100mVpeak of 100Hz ripple. The upper circuit is a standard power supply filter formed by a parallel 1uF capacitor across the load. The lower circuit uses a capacitance multiplier to enhance the filtering effect of the 1uF capacitor.

The 1uF/1M combination on the right hand side of each circuit is a DC blocking circuit that I use to measure AC voltages with LTSpice (I’m sure there’s a better way to do this).

Analysis

The standard power supply filter is a voltage divider, the output noise voltage Vo across the load is:

```Vo = Vi*Z/(Z+Rs)
```

where Z is impedance of the parallel C and Rl combination, Rs is the power supply noise impedance, Vi is the power supply noise voltage, and Rl is the load resistance. Rl must be much larger than Rs, otherwise the load current would drag the DC voltage down and it would be a crappy power supply. So Rl doesn’t have much impact.

So it’s basically a low pass filter with a 3 dB cut off at f=1/(2*pi*Rs*C). As Rs is low, you need a lot of C to push the cut off frequency down and filter out that low frequency noise. If we plug in the values above the 3dB cut off frequency is 1/(2*pi*1*1E-6) = 159 Hz.

Lets look at the Capacitance Multiplier circuit. I’ve drawn it to make the emitter follower topology clear. We have a load resistor Rl, and a RC circuit driving the base. The base resistor Rb provides DC bias, and also feeds the power supply noise voltage to the base of the transistor.

An emitter follower has a voltage gain of roughly 1. So the AC output voltage on the emitter Ve, is the same as the voltage on the base Vb. Now the emitter is across the load Rl, so the output noise voltage across the load is Vo=Ve=Vb.

In the emitter follower configuration the base of the transistor “sees” a (rather high) impedance of Beta*Rl. So we use voltage division again to find Vo:

```Vo = Vb = Vi*Z/(Z+Rb)
```

where Z is the parallel combination of the capacitors reactance and Beta*Rl. Rs is really small compared to Rb so we can ignore it.

In practice Beta*Rl is big (e.g. 100*100=10k) so we can ignore it. So the Capacitance Multiplier is a power supply noise filter with a 3dB cut off f=1/(2*pi*Rb*C). This can do a much better job than the standard power supply filter circuit above as Rb can be much larger than Rs. For our test values f = 1/(2*pi*5000*1E-6) = 31Hz, 5 times lower than the standard filter.

Note Beta is nowhere to be found in the expression for power supply noise filtering. This analysis suggests we do not have a capacitance multiplier effect at all. It’s an active filter. The product Rb*C defines the filter, not Beta*C.

Results

Here are the LTSpice simulated and calculated output noise values (mVrms) for 100mVp (71mVrms) input noise. Click on the links for the LTSpice and GNU Octave source code.

Circuit SPICE Calculated
Standard 68 71
Cap Mult 19 21

The calculated and LTSpice values are quite close.

Small Rb and Saturation

With small Rb there may be an issue operating the transistor at or near saturation. The assumption of an emitter follower assumes it is operating in a linear small signal mode. With a small Rb, Vce will be close to 0.7V. This may still be enough to operate linearly and filter the rather small noise voltages on the input, however the expressions above suggest the noise filtering will be poor with a small Rb. So I suggest biasing the transistor with a Rb of a few k-ohm and a Vce of a few volts. A Vce of a few volts is also suggested in the Wikipedia article.

Here are some plots of Vout for C=47uF, with Rb=50 (green) and Rb=5000 (blue):

And the table of simulated and calculated values of Vo (mVrms):

Cap Mult C=47uF SPICE Calculated
Rb=50 36 40
Rb=5000 0.48 0.5

Note the shift in the DC output voltage, but also the improvement in filtering. The LTSpice simulation is pretty close to the small signal calculations, suggesting saturation is not a problem even when Rb=50. It would be interesting to follow this up with some measured results on real parts.

With C=47uF, Rb=5k, we have a filter attenuation at 100Hz of 20*log10(71/0.5) = 43dB. Not bad!

[1] A few years ago I played with a similar circuit, called a Gyrator.

[2] Found this analysis of the capacitance multiplier linked from the SolderSmoke web site. In contrast to my analysis this does relate the noise attenuation to Beta, and approximates C as a short circuit to ground in their AC model. I haven’t read through it and reconciled it with my approach above. They make the same point about operating out of saturation. Very useful to have more than one look at the same circuit.

## 2 thoughts on “Capacitance Multiplier?”

1. I never realised anyone had given an actual name to that technique.

2. John says:

Hi David,

We are missing the obligatory microprocessor :).

Happy Holidays to all.

John