## Dual Slope ADC

The output of the integrator on the dual slope ADC, Vc, as shown above,
has the same functional form as the integrator output on the voltage to
time and voltage to frequency ADC's, that is:

Vc = (-Vin)t/tau

where tau = RC

Starting with switch S in the Vin position, the capacitor begins to
charge as the integration starts. As Vc becomes less than zero and continues
to decrease, the comparator output is high, allowing the clock to increment
the mod 2^{n} counter. When the MSB switches from one to zero so
that the counter is once again all zeros, the flip-flop output goes high,
switching S to the -Vr position. Thus Vc begins to increase from its
negative value, and when it crosses the zero point, the comparator output
goes low, stopping the clock signal from reaching the counter. The counter
now shows the number of clocks since S was switched to -Vr. This number is
proportional to Vin with proportionality constant 2^{n}/Vr, as shown
below.

The time that it took from the beginning of the integration when switch S
was at Vin to when the counter returned to its zero value is

t = 2^{n}Tclock

The expression for Vc after S has switched to -Vr is

Vc = -Vin(2^{n})Tclock/tau + Vr(t' - 2^{n}Tclock)/tau

where 2^{n}Tclock < t' < 2(2^{n})Tclock

After S has been at -Vr long enough to make Vc = 0, then the previous
equation becomes:

0 = -Vin(2^{n})Tclock/tau + Vr(t' - 2^{n}Tclock)/tau

t' - 2^{n}Tclock = Vin(2^{n})Tclock/Vr

t' = (Vin/Vr + 1)2^{n}Tclock

This is the total time since S was switched to Vin. The time since S was
switched to -Vr is

t' - 2^{n}Tclock = Vin(2^{n})Tclock/Vr
**Thus the number of clocks since S was switched to -Vr is
Vin(2 ^{n})/Vr** which is recorded on the counter.

Notice that the number of clocks since S was switched to -Vr is directly
proportional to the input analog voltage Vin, and also that there is no tau
in the expression, so this method is the most accurate of the integrating
ADC's. The total time for one conversion is (Vin/Vr + 1)2^{n}
clocks.