The multislope system is basically a very precise 'time' meter. Every incoming parameter is translated into a voltage first.

The convertor translates this level into a time.

Base operation: you have an integrator using a capacitor and resistor and an opamp. If you apply a fixed voltage at the input the capacitor will charge linearly over time ( that's the opamp straightening the charge curve ).

The capacitor value is not accurately known. Neither is the resistor value, but we don't care.

The input of the comparator can be switched to four possible values: ground, input, a +10 volt reference and a -10 volt reference. The absolute value of the reference is unimportant. As long as it is temperature stable and does not drift. We will calibrate all that stuff away.

To calibrate this thing. : apply a known voltage at the input, lets make it easy: +10 volts.. Start the stopwatch and wait for a known amount of time.this is called runup.( count up) This is a matter of starting a digital timer with many many bits. The more bits, the more accurate you can measure. So you clock fast ,or you integrate long... (That's why in 61/2 digit mode the measurement takes longer than in 31/2 digit mode. They simply integrate longer to be able to work with larger numbers. Why? We'll see in a moment)

So, we have charged the capacitor for a known time with a known voltage. What the exact charge across it is? We don't give a rat's ass.

Now we switch the integrator input to ground and we start the stopwatch again. The capacitor now discharges. A precision comparator compares the output of the integrator with the ground level. When the capacitor voltage has reached 0 the comparator flips and signals the digital system to stop counting. Rundown is complete.

To make it clear : we have charged from a KNOWN source using a KNOWN time (a fixed number of clockticks) . we have discharged with a KNOWN source (GND) for a MEASURED time.( we have counted how long it took. ).

The number now obtained in conjunction with the runup value( which is a constant) is now used as a fraction. This fraction tells us what the integrator 'time' gain is. Boom. The whole integrator is calibrated. We dont care about resistor or cap. We have a number. ( number of ticks per volt. )

Now we will calibrate our 'own' sources. We again do a runup charging the capacitor from the external source , compensating for the first number. So now we do know the voltage across the capacitor!( it is a 'time' that directly relates to a voltage. Calibrated four our particular r and c in the integrator )

Now we switch the integrator to our internal 'to be assumed 10 volts' and we let the thing rundown again. We get another number. Apply fractional math and we know the exact value of the source. we now how far our source is off from the applied 'standard'

Switch over to our 'assumed -10 volt' repeat. Another number.

So now we know three things :

A first number which is a voltage/time relation for our particular integrator. We don't directly need this one but it was necessary for the other two.

We know how many volts per second we can charge from the 10 volt ref , and we know how many volts per second we can charge (discharge) using the second reference.

Calibration complete.

Apply unknown value at input. Runup from known internal source for a given time( the one we obtained during cal) , switch to unknown. Rundown until we hit zero. Make fraction : we know have the fraction of input vs/ known voltage as a fraction of two numbers.

The rest is a matter of calculating with very large numbers. The larger the numbers are the more digits behind the comma you get. The more digits behind the comma the more precise you can calculate and the more digits you can show on screen.

Since the whole hoopla is time driven this means more digits = longer time needed...

So the actual standard used in the machine is 'time'. Slap in a crystal oscillator... Grin. All problems solved. We don't care about absolute resistor or capacitor values. Its all calculated away.

Theses machines actually compensate their whole internal circuit path this way. They store a set of numbers for every range of every unit possible. So internal switch resistance, relay contacts, pcb traces .. It's all compensated. Simply by 2 numbers that form a fraction.

Now, why do we need a negative 10 volt ?

Well if you measure a negative input voltage.. Duh ! Before runup they compare input voltage with ground and the precision comparator tells them if it is a positive or negative input ( above or below ground). So you switch to the other reference in that case for runup.

But. Agilent would not be Agilent if they stopped there.

Since we know how many volts per second these calibrated sources add or subtract from the integrator... We can shorten measurement time.

Runup is still a fixed time. No escaping that one. But rundown can be 'assisted' if this is taking too long. Lets pull some charge out by injecting some current from the known references. So the fraction becomes unknown volts/second from input +known volts per second from reference You don't need to be a math wiz to figure that one out.

That is the multislope ii algorithm. Multislope iii goes even further. They continuously switch currents around. The attempt is to keep the capacitor discharged. If no input is applied they switch +10 volt for a known time, then -10 volt for a known time. The comparator should toggle if the sources have not drifted ( so the machine can do a confidence test! Aha ! Didn't see that one coming did you? )

When an unknown voltage is applied it upsets this balance. So now the charge/discharge ration from the known sources needs to alter. So they still shuffle 'time around' but now as a fraction of the internal sources (which are again time) . That is the 'fast mode' in these machines.

There is some other trickery they can do. One of the hp bench briefs explains the whole shebang in detail better than i can.

One of the tricks involves a real ADC as we know it. If after a certain time the comparator has not toggled they digitize the remainder using that one. Its a 10 bit adc. Also used as a confidence test. If what that thing gives does not match , within reason, with the multislope output : there is a problem in the meter. The sources drifted, or it is damaged.