# AC & DC specifications: Offset error, gain error, CMRR, PSRR, SNR and THD

Hello. And welcome to the
TI Precision Lab introducing AC and
DC Specifications. Precision Labs is a
comprehensive online curriculum for analog engineers. More videos can be found by
going to TI.com/PrecisionLabs. In this video, we'll define
offset error, gain error, common mode rejection ratio, and
power supply rejection ratio. We will also give a
brief introduction to the AC
specifications of signal to noise ratio and total
calculation for offset and gain error. The key to understanding
this is to know that the ADC transfer function
is not perfectly linear. So a linear fit curve is
applied to the function. For this calculation, the most
commonly used type of curve fit is an endpoint linear fit.

With this type of curve fit,
the first and last points on the ADC transfer function
define the straight line. Recall that a straight
line has the equation y equals mx plus b. Also the slope can be calculated
by taking the change in y divided by the change in x. Sometimes this is referred
to as the rise over run. The offset is the
Y-axis intercept. That is, the offset is the
value of the transfer function when x equals 0. This value can be
calculated by rearranging the equation y equals
mx plus b and solving for b where b is the offset.

The gain error is the
percentage difference between the ideal slope
and the measured slope. The gain error and
offset error are often referred to as DC errors,
as they can be measured with DC input signals plot. Let's take a closer
look at offset error. Here we introduce the concept of
common mode rejection and power supply rejection. The common mode voltage
is the average voltage applied to both inputs. As this input changes, it
will introduce an error source that can be modeled as an offset
voltage source on the ADC input VCM error. The magnitude of
this error source can be determined using
the common mode rejection ratio, or CMRR, specification. CMRR is usually
specified in decibels and can be calculated by
taking negative 20 times the log of the change
in common mode error divided by the change
in common mode voltage. This equation can be
rearranged to solve for the change in
common mode error based on the change in
common mode voltage.

Power supply rejection,
or PSRR, also generates an error source in
series with the ADC input. Power supply rejection
error is a function of the change in the
power supply voltage. Variations or noise
on the power supply will reflect back to the
input as an error source. The equation for
power supply rejection is the same form as the
common mode rejection. But in this case, it is based
on power supply variations. Again, this can be rearranged
to solve for the change in power supply rejection error based on
the change in supply voltage. We will take a closer
look at CMRR and PSRR in the next few slides.

This slide shows an example of
an ADC's common mode rejection specification. A simple way to test
common mode rejection is to connect the two
inputs together and sweep the common mode voltage. Remember that
common mode voltage is the average of the
voltage on the two inputs. So when the inputs
are tied together the input signal is the
common mode voltage. In this example, if we want
to sweep the common mode voltage from 5 volts
to 2 and 1/2 volts, the change in common mode
voltage is 2 and 1/2 volts. Substituting these numbers
into the common mode rejection equation, we can see
that the common mode error is 25 microvolts. Power supply rejection
looks at the air introduced by a change in
the power supply voltage.

This shift can be a DC
change in the supply voltage, or it may be a noise signal. For this example, let's
consider a 200-millivolt peak to peak 200-kilohertz
noise signal on the supply. Normally, the specification
listed in the datasheet table is the PSRR for DC changes
in the power supply voltage. For the PSRR over
frequency, a bully plot may be shown in the
characteristic curves section. In this example, we
can find that the PSRR is 58 dB at 200 kilohertz. Using the PSRR Equation
introduced earlier, we can determine the error
introduced by the power supply rejection. Plugging the 200-millivolt
peak to peak and 58 dB into the equation yields a noise
of 252 microvolts peak to peak. Let's move on to the
next specification. This slide shows the general
equation for a data converter's signal to noise ratio or SNR. In general, the
signal to noise ratio is a measurement of how clean
or noise-free a signal is. A high SNR indicates
that the signal is very large in
comparison to the noise, whereas a low SNR
indicates that the noise is high relative to the signal.

For this specification,
both the noise and signal are measured and volts RMS. So you need to take 20
times the log of the ratio to convert it to decibels. The ideal SNR in decibels
can be calculated by taking 6.02 times n plus 1.76
where n is the number of bits of resolution of the ADC. A 10-bit converter, for example,
would have 6.02 times 10 plus 1.76 or 61.96 decibels. This relationship was derived
by integrating the quantization noise and applying the
signal to noise relationship. This relationship is true
for an ideal converter where the only error
source considered is quantization noise. No practical data converter
will ever have a better signal to noise than what is
given by this equation, because practical converters
have other noise sources. Another common AC specification
is total harmonic distortion or THD. In order to
understand THD, it is important to understand
nonlinearity. Nonlinearity is a measurement
of how much a transfer function deviates from
its ideal straight line.

The transfer function shown on
the left-hand side of the slide shows an ideal linear transfer
function and a nonlinear transfer function. The ideal transfer
function follows a straight line in
the form y equals mx plus b, whereas the
nonlinear transfer function will have higher order terms causing
deviations from the line. The nonlinear example shown
is exaggerated to make the nonlinearity easy to see. Notice how the nonlinear
function tracks well for low-input voltage levels
and deviates as the input increases. In short, the gain for
higher-input signals is larger than it should be. This has the effect of
stretching out the top half cycle of the sine wave. This stretching of the top
half cycle is called distortion and will create harmonics
in the frequency spectrum. This slide shows the frequency
spectrum for the digitized sine wave at the right. The harmonics are a result of
the distortion on the top half cycle of the waveform. Harmonic distortion will always
occur at integer multiples of the fundamental frequency.

In this case, the fundamental
is at 1 kilohertz, and there are harmonics
at 2 kilohertz, 3 kilohertz, 4
kilohertz, and so on. Sometimes, it is
useful to differentiate between even and odd
harmonics, as different circuit non-idealities may generate
one type of harmonic. Even harmonics
are even multiples of the fundamental frequency. And odd harmonics are odd
multiples of the fundamental. For example 2 kilohertz
and 4 kilohertz are even harmonics, whereas
3 kilohertz and 5 kilohertz are odd harmonics. If the digitized
signal perfectly tracked the input signal, there
would not be any harmonics. The THD calculation is
given here as a percentage as well as in decibels. The IEEE standard
for ADC testing specifies that nine
harmonics should be used in the THD calculations.

THD is the square
root of the sum of the harmonic voltages squared
divided by the RMS signal voltage squared. This quantity is multiplied by
100 to convert to a percentage, or 20 times the log is taken
to convert to decibels. THD plus N is similar
to THD, except that it includes the total RMS
noise in the calculation. SINAD is short for signal
to noise and distortion. Mathematically, SINAD
is simply the reciprocal of the THD plus N calculation. In decibels, taking
the reciprocal will just change the
sign of the number. Note that SINAD or
THD plus N will always be worse than either
the THD or SNR, because SINAD is really a
combination of the two error sources. That concludes this video. Thank you for watching. Please try the quiz to
check your understanding of this video's content.