# How exactly does switched capacitor system work, what exactly is the detail of the primitive that switches

I have been seeing these switched capacitor circuits for some time now and have even seen Z-transform being used to model how these circuits behave. While I can see that there is some capacitor that switches, what exactly is the mystery behind this 3 terminal device one terminal of which comes from some sort of clock. How is this clock generated?

I am wondering how do these things work from within a circuit but don't know how to get a good thorough answer.

• I have seen MOSFETs being used at one place, are there any other methods by by which "switching" is performed? Where does the clock come from? I wonder why people only developed this like around 1980s as far as I know if it is really a simple concept. Commented May 16, 2013 at 16:26
• Example of this "switched capacitor circuit"? Clocks are traditionally generated from the resonant frequency of crystals (for accuracy) or R/C oscillators (if accuracy is not important). Commented May 16, 2013 at 16:47
• Often the clock is user-provided. Very convenient to change a corner frequency in a filter by changing the clock frequency. Commented May 16, 2013 at 21:00
• its just that while I did read about them at University I never came across a real circuit to see how it works. Commented May 16, 2013 at 22:11

Switched cap circuits are typically used in chip design because it's far far easier to get small, high value caps that match each other than it is to get resistors that meet all those same criteria. Until there was a need the circuit technology and theory wasn't developed, once circuits got smaller the need arose and bingo! it appeared.

The relationship between resistance and capacitance in a switched cap circuit is:

$R_{equ}=\dfrac{1}{Cf_{clk}}$

If I want to emulate a $10M\Omega$ resistor I can do it with a $0.5pF$ cap and two switches running at $200 kHz$. the switches can be a CMOS transmission gate of 1 PMOS and 1 NMOS. This might take up $0.001 mm^2$ in a $0.5 \mu m$ process. In comparison a $10 M\Omega$ diffused resistor in the same process might take $1.0 mm^2$. This is a factor of ~$1000$.

• +1, interesting insight into feature dimension as a decision factor for something so simple. Commented May 17, 2013 at 8:09

If you have a black box load across a voltage source and you looked at the average current into that black-box, you would not be able to say whether the black-box contained a resistor or a switched capacitor to ground.

It's all based on R = V/I and Q=CV. I = dQ/dt and dQ = CdV/dt etc...

The switching frequency of the capacitor has to be beyond the highest frequency that your voltage source can produce i.e. it has to obey the nyquist criterion but if you are dealing with audio (for instance) this is not hard to implement. I remember MF10 ICs coming out and today (like @rawbrawb says) the capcitors are inside the chip and things are easier.

I mentioned nyquist and you mentioned Z-transforms - you have to treat this type of filter with some care - you are now sampling the input at a clock rate and this requires some analysis in the Z-plane but mainly, chips like MF10s and their successors make it easy for you to design pretty good high order filters.

Picture S2 closing w/ S1 open, and then S1 closing and S2 opening.

The charge transfer is $$\ \Delta q=C_1(V_2-V_1) \$$ .

If you repeat this N times in some time, t, the amount transferred is $$\ \frac{\Delta q}{\Delta t}=C_1(V_2-V_1) \frac{N}{\Delta t}\$$

The term on the left is simple current (charge per time), and $$\ \frac{N}{\Delta t}\$$ is simply the clock frequency of the switching, $$\ f_{clk}\$$

So, rearranging a bit, we get $$\ \frac{(V_2 - V_1)}{i} = \frac{1}{C_1 f_{clk}} = R\$$

As this is a sampling system, one must be wary of aliasing. Further, clock noise can bleed through your whole board if you're not careful. Switched Cap filter IC's are often designed such that the corner frequencies are much lower than the clock freq, so you can filter out the clock with a simple RC filter.