Switched Capacitor Circuitry

In this circuit, a capacitor is alternately connected in series between a voltage source and a load, then shorted, by means of two MOSFET transistors that are never conducting at the same time:

Note: the φ₁ and φ₂ pulse signals are collectively referred to as a non-overlapping, two-phase clock.

Consider the average amount of current through the load resistor, as a function of clock frequency. Assume that the “on” resistance of each MOSFET is negligible, so that the time required for the capacitor to charge is also negligible. As the clock frequency is increased, does the load resistor receive more or less average current over a span of several clock cycles? Here is another way to think about it: as the clock frequency increases, does the load resistor dissipate more or less power?

Now suppose we have a simple two-resistor circuit, where a potentiometer (connected as a variable resistor) throttles electrical current to a load:

It should be obvious in this circuit that the load current decreases as variable resistance R increases. What might not be so obvious is that the aforementioned switched capacitor circuit emulates the variable resistor R in the second circuit, so that there is a mathematical equivalence between f and C in the first circuit, and R in the second circuit, so far as average current is concerned. To put this in simpler terms, the switched capacitor network behaves sort of like a variable resistor.

Calculus is required to prove this mathematical equivalence, but only a qualitative understanding of the two circuits is necessary to choose the correct equivalency from the following equations. Which one properly describes the equivalence of the switched capacitor network in the first circuit to the variable resistor in the second circuit?

Be sure to explain the reasoning behind your choice of equations.

Identify the polarity of voltage across the load resistor in the following switched capacitor circuit (called a transresistor circuit). Note: φ₁ and φ₂ are two-phase, non-overlapping clock signals, and the switches are just generic representations of transistors.

Identify the polarity of voltage across the load resistor in the following switched capacitor circuit. Note: the only difference between this circuit and the last is the switching sequence.

What difference would it make to the output signal of this operational amplifier circuit if the switching sequence of the switched capacitor network were changed? What difference would it make if the switching frequency were changed?

Research the resistance equivalence equations for each of these switched-capacitor networks (using N-channel MOSFETs as switches), describing the emulated resistance (R) as a function of switching frequency (f) and capacitance (C):

Note: φ₁ and φ₂ are two-phase, non-overlapping clock signals.