Introduction to Switched Capacitor Circuits

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					          Introduction to Switched Capacitor Circuits
                                 Created by Kat Kim


1    Introduction
Switched capacitor circuits are commonly used configuration to replace a resistor with
switches and capacitors. There are many different configurations, but the main idea is
to pass charge into and out of a capacitor by controlling switches around it. This worksheet
will cover some basics on switching, explain a simple switched capacitor configuration, and
step you through some example circuits.


2    Switching
The basic idea of a switch is a connection that can be opened and closed by something
controlling the circuit. A Metal-Oxide Semiconductor Field Effect Transistor (MOSFET)
is a good example of a switch because it allows current to flow when its gate input is high
and very little current when the gate input is low or 0V . Ideally, the switch would have no
resistance and have an instantaneous response. No switch is truly ideal, but we will assume
they are for ease of calculation.

Let’s say that we have two signals φ1 and φ2 that are producing identical square waves. The
period of the wave is T , and the duty cycle is 50%. We shift φ2 forward by T /2, so that
only one switch is open at any time (we will see why this is important in the next section).
Moving out the ideal switch world for a second, assume that changing voltages take a certain
amount of time. If we want to ensure that only one switch is on at any one time, we need
to decrease the duty cycle on both signals. The resulting waveforms are shown in the figure
below. These are the waveforms that will control the MOSFET switches in the switched
capacitor.




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3    Basic Switched Capacitor
First, let’s think of a single resistor R between two voltages V 1 and V 2 as shown below.
Assuming that current is flowing from V 1 to V 2, we know that the IV characteristic is
(V 1 − V 2) = IR or R = (V 1−V 2) . Keep this in mind as we look at the next circuit.
                              I




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The most basic switched capacitor circuit is shown below. The two MOSFETs are controlled
by the wave forms described in the previous section. The circuit operates in these steps:

  1. Switch 1 closes. The capacitor is charged to V 1.

  2. Switch 1 opens. Charge remains in the capacitor.

  3. Switch 2 closes. The capacitor gives off enough charge to adjust to V 2.

  4. Switch 2 opens. Charge remains in the capacitor. Return to step 1.




After step 2, the charge on the capacitor is Q = C(V 1). After step 4, the charge is
Q = C(V 2). The total charge moving through the system is the difference of the charges:
∆Q = C(V 1 − V 2). Current is charge over time, and we know that this charge flows through
the system in the period of the waveform T , so overall I = C (V 1−V 2) or C = (V 1−V 2) .
                                                                  T
                                                                           T
                                                                                   I




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By comparing the IV characteristics of the resistor and switched capacitor we see that
R = C = f1 , where f is the frequency of the waveform. Because of this important rela-
      T
            C
tionship, the effective resistance of a switched capacitor can be changed by changing the
capacitance or simply changing the frequency of the control waveform.

One thing to keep in mind is that if the input voltage V 1 changes with time, the signal
will have some frequencies. In order for the switched capacitor to function properly, it must
switch at a must faster rate that the highest frequency of the input voltage.




4     Examples
First, we will look at the kind of equivalent resistances we can achieve with a switched ca-
pacitor. Then, we are going to walk through some common circuits that can be reconfigured
to use switched capacitors.


    1. I go to the stock room and pick up a 10 pF capacitor. If I use it with a 100 kHz
       switching frequency, what is the equivalent resistance?



                                              Req =




    2. I have a stockroom full of a wide range of capacitors , and I want to emulate a 100kΩ re-
       sistor, but my frequency generator is for some reason stuck between 50Hz and 20kHz.
       I am not sure what frequencies my input voltage will have, but I want the switched
       capacitor to be as accurate as possible. What frequency and capacitance should I use?
       Why?




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3. An RC inverting integrating circuit is shown below. Remember than an op-amp is in
   negative feedback so the output will do whatever it can to make the negative input
   node match the positive input node.




   (a) Assuming the op-amp is able to work properly in negative feedback, what is the
       voltage on V− ?
   (b) Assuming V 1 is positive, draw current lines through the components.
   (c) Write the KCL equation.
   (d) Use this equation to write V 2 in terms of V 1. (Hint: this is a inverting integrating
       circuit)




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4. The switched capacitor version of an inverting integrator is shown below. We will use
   the same methodology to show that this circuit is essential the same as the previous
   circuit.




   (a) When working with switches, it is useful to redraw the circuit for each phase.
       Draw the circuit when only switch 1 is closed.
         i. Draw the direction of the current and the polarity on the capacitor.
         ii. What is the full charge on the capacitor?
   (b) Draw the circuit when only switch 2 is closed. Include the polarity on the capacitor
       for the previous phase.
         i. Right after the transition, V− is not the value it “wants” to be. What is its
            voltage? What voltage does it “want” to be?
         ii. Draw current lines for each capacitor, and write the KCL equation. Re-
             member that if the current is in the “wrong” direction it will be a negative
             current.
        iii. We know that all the charge in C2 will be discharged because once the op-
             amp adjusts, voltage over the capacitor will be 0V . All that charge will be
             converted to current over the switching period T . What is the current through
             C2? Keep in mind the direction of the current.
        iv. Since we do not know the initial charge on C1, we should think in terms of
            differentials. Write the current equation for C1.
         v. Use the KCL equation and the current across C1 and C2 to solve for V 2 in
            terms of V 1.
   (c) The final equations for the RC and switched capacitor circuits should be very
                                       T
       similar. In fact, using the R = C relation, they should be exactly the same.

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5. Below are the RC and switched capacitor configurations for a circuit that integrates
   the difference between two voltages. Do a similar analysis on these two circuits to
                                                            T
   prove that they are indeed the same, by the relation R = C . Also, notice the difference
   in number and type of components between the two configurations.




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5    Further Reading
For basic information on switched capacitor circuits, Linear Circuits by M.E. Van Valkenburg
and B.K. Kinariwala, Chapter 17, is a good source. It has a number of circuit configurations
that can be worked through with some basic circuit analysis skills.

For a more advanced take on switched capacitor circuits, look at CMOS Analog Circuit
Design by Phillip Allen and Douglas Holberg, 2nd edition Chapter 9. This book comes from
more of a signal analysis perspective and assumes a high level of circuit analysis.




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