Listen to Professor Yang talk about "water integrator" and learn the most comprehensive switched capacitor filter technology!

As shown below: Figure 1 is a "water integrator" model using "water" as an analogy . The height of the ball valve is like a water resistance R, which together with the water level of the reservoir (analogous to the input potential) determines the water flow rate per unit time (analogous to the current). This water flow is injected into the water container 1 (analogous to the capacitor), causing the water level of the water container 1 (analogous to the potential) to rise. This is an integrator. The water level of the water container 1 is the output of the water integrator, and the water level of the reservoir is the input of the water integrator.

Figure 1: Water model of the integrator. The water resistance determines the speed at which the water level in the water tank 1 rises.

When the input water level remains unchanged and the size of water volume 1 remains unchanged, the rate of the integrator output (water level of water volume 1) can be changed by adjusting the height of the ball valve. This is similar to an integrator with a potentiometer adjusting resistance. Objectively, it can change the time constant of the integrator.

In the water integrator, there is another method to change the integrator time constant - "switch water volume method" , as shown in Figure 2. It no longer uses a continuously adjusted ball valve, but two switches SW1 and SW2 (realized by pulling out and plugging the ball valve), and a water volume 2 is added between the input and output. In the Φ1 stage, SW1 is turned on, SW2 is blocked, and water volume 2 is immediately filled with water to the same level as the reservoir water level - note that since the waterway is fully open when SW1 is turned on, we assume that its water resistance is 0, so this water injection process will be very short and there is no need to consider the water injection process. In the Φ2 stage, SW2 is turned on, SW1 is blocked, and the water in water volume 2 immediately flows into water volume 1. In this way, the water level of water volume 1 is also rising.

Figure 2: Switching water-capacitor model of the integrator. The faster the reciprocating frequency, the faster the water level in water container 1 rises, which is equivalent to a smaller water resistance.

At this time, changing the time constant of the water integrator can be achieved by changing the reciprocating frequency fCLK of Φ1 and Φ2. This looks like using fCLK and water capacitor 2 to jointly simulate a water resistance. The larger the fCLK, the smaller the water resistance, just like the frequency of the porter moving water back and forth increases; the larger the water capacitor 2, the smaller the water resistance, just like the porter moves water in a larger bucket each time.

Completely similarly, the integrator in the circuit, as shown in the right figure of Figure 3, its resistance RSC can also be changed by program control using the above method, that is, the standard integrator on the right is replaced by the circuit on the left.

Figure 3: Switched capacitor module replaces resistor for integrator

On the left side of the figure is an integrator circuit in which a switch capacitor module replaces a variable resistor. The switch capacitor module is the circuit in the green dashed box, which consists of two switches SW1 and SW2 and a capacitor C1. Under the action of the external clock fCLK, two switch control signals are formed - a high level corresponds to a closed switch, and a low level corresponds to an open switch. In this reciprocating process, the switch capacitor module can be regarded as a resistor RSC, whose resistance is related to the external clock frequency fCLK and the capacitor C2:

The above formula can be easily proved:

In the Φ1 stage, there is a process where uI charges capacitor C2, and the charge obtained by C2 is:

In the Φ2 stage, capacitor C2 is connected to the negative input terminal of the integrator op amp through the closure of SW2. The charge in capacitor C2 will be quickly and completely transferred to capacitor C1, making the voltage of C2 0 - the negative input potential of the op amp becomes 0V, so that a virtual short circuit occurs. Of course, the charge of C2 also becomes 0.

Thus, in a complete cycle, the total amount of charge transferred from uI by capacitor C2 is U1C2. If the frequency is fCLK, the total amount of charge transferred from uI by capacitor C2 in 1 second is:

A standard integrator is shown on the right side of the figure. The current flowing through the resistor RSC is:

In 1 second, the total amount of charge transferred to the subsequent circuit is:

For the switched capacitor block to simulate a standard integrator, the two charges should be the same:

Right now:

The switched capacitor integrator is used in the filter to form a switched capacitor filter. So far, we can use a switched capacitor module to form an integrator with a variable time constant, which can be called a switched capacitor integrator. Its time constant can be controlled by the externally provided fCLK . We use it to replace the integrator in the traditional filter, and we can use fCLK to control the key parameters of the filter.

This is the core principle of the switched capacitor filter. As long as there is an integrator in the traditional filter and the integration time constant affects the key parameters of the filter, then replacing it with a switched capacitor integrator will definitely make a programmable filter that "uses external fCLK to control the cutoff frequency", that is, a switched capacitor filter.

For example, the state variable filter shown in Figure 4 contains two integrators, A2 and A3. It can be seen from the transfer function that the time constant of the integrator has a direct impact on the characteristic frequency, and the low-pass output is:

Figure 4: State-variable filter

If only the integrator composed of C1 and R4 is replaced by a switched capacitor integrator, then when the external input clock fCLK changes, its characteristic frequency will change accordingly.

The Biquad filter also has an integrator inside, as shown in Figure 5. It has low-pass and band-pass outputs, and through appropriate addition operations, it can achieve a more colorful filtering effect.

Figure 5: Biquad filter

Most switched capacitor filters use Biquad filters, which replace the integrator in the figure with a switched capacitor integrator. In fact, the resistor formed by the switched capacitor replaces R4 in the figure. The low-pass gain is:

It can be seen from its low-pass expression that changing the resistor R4 can indeed change the filter characteristic frequency.