# RTI error and RTO error and Calculation for error budget

Document created by analog-archivist on Feb 23, 2016
Version 1Show Document

### Q

What is the difference between an RTI error and an RTO error and how do I
calculate them for my error budget?

### A

The AD8422 is composed of a preamplifier stage that applies differential gain,
and a subtractor stage that removes the common-mode voltage. The subtractor is
in a fixed gain of 1 for a differential signal (noise gain =2), whereas the
preamplifier stage gain changes with programmed gain. Because the errors of the
output section are multiplied by a fixed gain, this section is often the
principal error source at low circuit gains. When the in-amp is operating at
higher gains, the gain of the input stage is increased. As the gain is raised,
errors contributed by the input section are multiplied, while output errors are
not. So, at high gains, the input stage errors dominate. In order to combine
the error sources from both stages, the errors are often modeled as a single
error source in series with the input, or Referred to the Input (RTI).
Equivalently the errors can be modeled as a single error source in series with
the output, or Referred to the Output (RTO).

An example of this is the Offset Voltage. For the AD8422BRZ, the maximum offset
voltage is composed of the input offset (VOSI) of 25µV and the output offset
(VOSO) of 150µV. For example, these specifications can be used to calculate the
total offset at a gain of 10.
Total RTI Error = V¬OSI + (VOSO/G) = 25µV + (150µV/10) = 40µV

Total RTO Error = G*V¬OSI + VOSO = 10*25µV + 150µV = 400µV

Note that the two error numbers (RTI vs. RTO) are different: the RTO numbers
are 10 times larger, and logically they should be, as at a gain of 10, the
error at the output of the in-amp should be 10 times the error at its input.

Noise is calculated in a similar way, except that the noise of the two sections
adds as the root sum of squares.

Total RTI noise = √eNI2 + (eNO/G)2

Total RTO noise = √(G*eNI)2 + eNO2

In a gain of 10, AD8422 RTI Voltage Noise spectral density is: √(8nV/√Hz2 +
(80nV/√Hz / 10)2) = 11.3nV/√Hz RTI.

Note that gain errors, which are multiplicative rather than additive, do not
follow this pattern. For example, if there is a 1% error on each stage of a two
stage amplifier, there is approximately 2% error, regardless of the gain of
each stage, as follows: (G1*1.01) * (G2*1.01) = G1*G2*1.0201.