*Switched-Capacitor
Resistor*
CIRCUIT
SWCAP_RES.CIR
Download the
SPICE file
So what's the big headache about fabricating an RC filter on a MOS
integrated circuit? For starters, the biggest capacitor you'll get will be
around 100 pF! Next, that means to create a low-pass filter with fc = 100
Hz, you'll need a resistor of R = 16 MΩ! Now, you've just lost a big chunk
of real estate for one resistor! To make things worse, the resistor is
highly non-linear and the tolerances are horrible. So what's the solution?
If you can't have a real resistor, why not a *simulated* one? All
you need is a couple of switches and a capacitor - both readily available in MOS. And to
boot, the capacitor ratios can be tightly controlled. Today switched-capacitor circuits are
thriving in the field as mult-pole filter circuits and programmable analog ICs.
SIMULATED RESISTOR
First, let's define what our resistor needs to do. Given a potential
difference in volts, get a proportional current to flow in Amps. Of course
Amps is how many Coulombs per second (Q / s) will move through the resistor.
In the switched-capacitor circuit, S1A and S1B are alternately closed. As
a result C1 is charged alternately to V1 and
V2. Now, in order to change C1's voltage from V1 to V2, a charge (Coulombs) must flow from
C1.
Δ*q = *Δ*V ∙
C = *(*V*2* - V*1)* ∙ C*
The same charge flows through S1A and S1B in
sharp pulses each time one of the switches is closed. If these switches are
opened / closed at a regular time interval Δt, you get an *average current*
flowing. You can define this average current by the charged moved Δq divided by the time interval Δt.
Comparing this equation to ohms law
we see a simulated resistance!
The bottom line? Switching the capacitor moves a charge proportional to the
voltage difference. The resistor achieves the same function only in a
continuous manner. Using a water analogy, we can imagine a couple of
scenarios: 1) a steady water flow, or 2) the same water delivered rapidly in
buckets. Both create the same flow of water - on average!
SWITCHED-CAPACITOR CIRCUIT
Let's run a switched capacitor resistor, then compare it to a standard
resistor.
Switches S1A and S1B are turned ON and OFF by non-overlapping pulse generators VSA and VSB.
The "resistor" is strung across voltage sources: V1 = 5V and V2 = 1V.
Suppose you need to simulate a 20 kΩ
resistor. Given C1 = 1000 pF capacitor, how fast do you need to run
the switches?
Δt = R ∙ C
= 20 kΩ
∙ 1000pF
= 20 μs
Given our "resistor", we expect a current to
flow of (5 - 1) / 20 kΩ
= 200 uA.
CIRCUIT INSIGHT
Run a SPICE simulation of SWCAP_RES.CIR. Plot the A clock
phase for S1A by adding trace V(10). To see the current through S1A, add a
plot window and add trace
I(S1A). Notice a current pulse every time V(10) goes HI (S1A ON). This
represents the charge moved to change C1's voltage from V2 to V1. Likewise,
plot the B phase clock V(11) and I(S1B). Aha! - a similar current pulse as
C1 changes from V1 to V2.
Now the question remains - given the current spikes, how much current is
delivered on average? To do this, simply remove traces I(S1A) and I(S1B) and
then plot AVG( I(S1B)) . (Many of SPICE's waveform viewers, like PSPICE, provide a number
of math functions you can apply to the traces such as AVG( ).) The
average current looks to be around around 200 μA.
Now, let's compare this to the continuous current through a standard 20 kΩ resistor
across the similar voltages V3 and V4. Plotting I(R2) we shouldn't be too
surprised to see a current of 200 μA. It
appears the creation of a simulated resistor was a success!
HANDS-ON DESIGN
Need a different value resistor? Just change C or
Δt. As an example, you can create
a resistor that's half the original by doubling C or halving
Δt. To cut Δt in half, change the A
and B phase clocks to look like:
VSA 10 0 PULSE(0V
5V 0US 0.1US 0.1US 2.5US 10US)
VSB 11 0 PULSE(0V 5V 5US 0.1US 0.1US 2.5US 10US)
Don't forget to change R1 to 10 kΩ
for comparison. Did the current go up for both the standard and simulated
resistor?
LOW-PASS FILTER
Okay, sharp current pulses aren't very useful in real circuits. But many applications are
RC filter types where other capacitors smooth or average these current pulses into useful
voltages. Let's try a simple RC low-pass filter. ( Reality Check: Most RC
switched-capacitor filters in practice are op amp based. But this simple RC
example illustrates how simulated resistors can approximate standard
resistors in a filter circuit.)
First, replace V2 and V4 with 10 nF capacitors C2 and C4.
There are SPICE statements already in the file - simply comment out "*"
the voltage sources and remove the comment symbol "*" from the capacitors.
You'll also need to extend your simulation time to 1000
μs or so.
CIRCUIT INSIGHT
Run a simulation of your filter circuits. Next, clear
your plot windows and view the output voltages at V(3) and V(5). Okay now we
are getting somewhere! The response of the simulated and standard filters
look similar. But why does the switched-capacitor filter's output
voltage appear as a stair-stepped waveform? This is the result of sharp
current pulses being delivered to the C2. The capacitor simply does its job
of accepting the total charge delivered and changing its voltage according
(inversely-proportional) to its own capacitance value.
Try increasing or decreasing C2 and C4. When the C2 is made smaller than
10 nF, does the circuit do a better or worse job of approximating the
standard RC circuit?
SWITCHED-CAPACITOR CIRCUITS
As mentioned before, many switched-capacitor circuits available today are
op amp based. We'll take a look at some basic building blocks - like the the
op amp integrator and the simple pole-zero filter - in future topics. In
addition, the switched-capacitor resistor above has its problems - it
can be very sensitive to stray capacitance. We'll also look at an improved
switch arrangement that's more immune to the capacitive strays on an IC.
REFERENCES
*Switched-Capacitor Circuit Design*,
R. Gregorian, et al, Proceedings of the IEEE, Vol 71, No. 8,
August 1983.
SPICE FILE
Download the file
or copy this netlist into a text file with the *.cir
extension.
SWCAP_RES.CIR - SWITCHED-CAPACITOR RESISTOR
*
V1 1 0 DC 5V
*
* SWITCHES AND CAP
S1A 1 2 10 0 SW1
C1 2 0 1000PF IC=0V
S1B 2 3 11 0 SW1
*
V2 3 0 DC 1V
*C2 3 0 10NF
*
* SWITCH CONTROL
VSA 10 0 PULSE(0V 5V 0US 0.1US 0.1US 5US 20US)
VSB 11 0 PULSE(0V 5V 10US 0.1US 0.1US 5US 20US)
*
.MODEL SW1 VSWITCH(VOFF=0 VON=5 RON=100 ROFF=10MEG)
*
*
* STANDARD RESISTOR
V3 4 0 DC 5V
R1 4 5 20K
V4 5 0 DC 1V
*C4 5 0 10NF IC=0V
*
*
* ANALYSIS
.TRAN 0.1US 100US
*
* VIEW RESULTS
.PROBE
.END
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