In the last decade or so many active filters with resistors and capacitors have been replaced with a special kind of filter called a switched capacitor filter. The switched capacitor filter allows for very sophisticated, accurate, and tuneable analog circuits to be manufactured without using resistors. This is useful for several reasons. Chief among these is that resistors are hard to build on integrated circuits (they take up a lot of room), and the circuits can be made to depend on ratios of capacitor values (which can be set accurately), and not absolute values (which vary between manufacturing runs).
The Switched Capacitor Resistor
To understand how switched capacitor circuits work, consider the circuit shown with a capacitor connected to two switches and two different voltages.
If S_{2} closes with S_{1} open, then S_{1} closes with switch S_{2} open, a charge (q is transferred from v_{2} to v_{1} with
.
If this switching process is repeated N times in a time (t, the amount of charge transferred per unit time is given by
.
Recognizing that the left hand side represents charge per unit time, or current, and the the number of cycles per unit time is the switching frequency (or clock frequency, f_{CLK}) we can rewrite the equation as
.
Rearranging we get
,
which states that the switched capacitor is equivalent to a resistor. The value of this resistor decreases with increasing switching frequency or increasing capacitance, as either will increase the amount of charge transfered from v_{2} to v_{1} in a given time.
The Switched Capacitor Integrator
Now consider the integrator circuit. You have shown (in a previous lab) that the input-output relationship for this circuit is given by (neglecting initial conditions):
We can also write this with the "s" notation (assuming a sinusoidal input, Ae^{st}, s=jw)
If you replaced the input resistor with a switched capacitor resistor, you would get
Thus, you can change the equivalent w' of the circuit by changing the clock frequency. The value of w' can be set very precisely because it depends only on the ratio of C_{1} and C_{2}, and not their absolute value.
The LMF100 Switched Capacitor Filter
In this lab you will be using the MF100, or LMF100 (web page, datasheet, application note). This integrated circuit is a versatile circuit with four switched capacitor integrators, that can be connected as two second order filters or one fourth order filter. With this chip you can choose w' to either be 1/50 or 1/100 of the clock frequency (this is given by the ratio C_{1}/C_{2} in the discussion above),. By changing internal and external connections to the circuit you can obtain different filter types (lowpass, highpass, bandpass, notch (bandreject) or allpass).
Filter Type Transfer Function Low Pass High Pass Band Pass Notch (Band Reject)
The pinout for the LMF100 is shown below (from the data sheet):
You can see that the chip, for the most part, is split into two halves, left and right. A block diagram of the left half ((and a few pins from the right half) is shown below.
The pins are described on page 8 of the datasheet. I will describe a few of them here:
- 50/100 - determines if the value of w' is w_{CLK}/100, or w_{CLK}/50.
- CLK_{A} - is w_{CLK}.
- INV_{A} - the inverting input to the op-amp
- N/AP/HP_{A} - an intermediate output, and the non-inverting input to the summer. Used for Notch, All Pass or High Pass output.
- BP_{A} - another intermediate output, the output of the first integrator. Used for Band Pass output.
- LP_{A} - the output of the second integrator. Used for Low Pass output.
- S1_{A} - an inverting input to the summer.
- S_{AB} - determins if the switch is to the left or to the right. That is, this pin determines if the second inverting input to the summer is ground (AGND), or the low pass output.
The two integrators are switched capacitor integrators. Their transfer functions are given by,
where w' is w_{CLK}/100, or w_{CLK}/50, depending on the state of the 50/100 pin. Note that the integrator is non-inverting.
A Typical Circuit.
The diagram below shows one of the modes (mode 1) of operations (pages 13 through 20 of the datasheet).
Let's analyze this circuit and try to derive the filter specifications as given in the datasheet, and given below.
The low pass (LP_{A}) output is easily given in terms of the band pass output (BP_{A}), as well as the band pass output as a function of the summer (SUM, not labeled on diagram).
The summer output (SUM) is simply the output of the op amp (N_{A}) minus the lowpass output (LP_{A}). However we can see that the op amp is set up as the inverting summing circuit. So
Replace SUM on the left hand side using equation (2) from above, and LP_{A} using equation (1).
Rearranging brings
Equating this with the transfer function for a bandpass circuit
yields,
which is what we were trying to derive.
Similarly, the relationship between low pass and band pass, equation (1), can be used to find the low pass transfer function. The notch filter transfer function is derived in the same way.
The datasheet gives several other ways to connect the chip to realize other sorts of filters.
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