## An example of how uncertainties combine

This post discusses one aspect of an uncertainty calculation for a radio frequency (RF) power measurement. It builds on the earlier post Formulating uncertainty (3) in which a simple voltmeter model was described.

We consider a measurement setup that is often used in calibration laboratories. A signal is applied to the input to a power splitter (see lazy about differentiation) and a pair of power sensors are connected to the two output ports.

The sensors convert the RF signal into a voltage that is measured with a voltmeter (not shown).

To measure the power at one port, two voltage measurements are required: one with the power switched on, the other with the power off. The port power is then
$P_\mathrm{sens} = \frac{(V_\mathrm{off}-V_\mathrm{on})(V_\mathrm{off}+V_\mathrm{on})} {R_\mathrm{std}}$
where $$R_\mathrm{std}$$ is the resistance of a precision standard resistor inside the power sensor circuit. $$P_\mathrm{sens}$$ is the power registered by the sensor.1

However, the ratio of port powers has to be calculated to benefit from using a splitter. So we are interested in
$R_\mathrm{sens} = \frac{P_\mathrm{sens \cdot 1}}{P_\mathrm{sens \cdot 2}} \;.$
Assuming different sensor electronics (and hence different resistors) at each port
$R_\mathrm{sens} = \frac{R_\mathrm{std\cdot 2}}{R_\mathrm{std\cdot 1}} \frac{(V_\mathrm{off\cdot 1}-V_\mathrm{on\cdot 1})(V_\mathrm{off\cdot 1}+V_\mathrm{on\cdot 1})}{(V_\mathrm{off\cdot 2}-V_\mathrm{on\cdot 2})(V_\mathrm{off\cdot 2}+V_\mathrm{on\cdot 2})} \;.$
So, what is the uncertainty in the power ratio?

### Reusing a model

The voltmeter model developed earlier will serve us here too. But a class for power sensors will by keeping an uncertain number for the precision resistor and defining the power calculation.
class Sensor(object):
def __init__(self,R,u_R,tag=None):
"""Initialise a new sensor object"""
label = "R"
if tag is not None: label += "_%s" % tag
self.R = ureal(R,u_R,label=label)

def power(self,v_off,v_on):
"""Return an estimate of power"""
return (v_off-v_on)*(v_off+v_on)/self.R

Using this class and the Meter class defined here, we can proceed as follows.

First define objects for a voltmeter and two sensors
m = Meter(u_e_gain=3E-6,u_e_zero=1E-6,u_e_ran=1E-7,tag='m1')
s1 = Sensor(R=200,u_R=2E-4,tag='1')
s2 = Sensor(R=200,u_R=2E-4,tag='2')

Then convert raw voltage readings into uncertain numbers, obtain power estimates at each port and finally calculate the power ratio
V1, V2, = m.voltage(2.6,tag='1'), m.voltage(2.4,tag='2')
V3, V4, = m.voltage(2.55,tag='3'), m.voltage(2.41,tag='4')

P1 = s1.power(V1,V2)
P2 = s2.power(V3,V4)

X = P1/P2



### The results

First, the uncertainty budget of the power ratio
print summary(X)
for cpt in rp.budget(X,trim=0):
print "%s: %G" % cpt

The output is
1.4400922, u=4.9E-06, df=inf
e_ran_3: 2.69706E-06
e_ran_4: 2.40904E-06
e_ran_1: 1.947E-06
e_ran_2: 1.65899E-06
R_1: 1.44009E-06
R_2: 1.44009E-06
e_zero_m1: 4.64546E-09
e_gain_m1: 5.0822E-21

Random errors associated with the voltage measurements dominate this estimate.

A different picture emerges if we look at just one of the power estimates
print summary(P1)
for cpt in rp.budget(P1,trim=0):
print "%s: %G" % cpt

The output is dominated by uncertainty in the residual meter gain error
0.005000000, u=3.2E-08, df=inf
e_gain_m1: 3E-08
e_ran_1: 6.76E-09
e_ran_2: 5.76E-09
R_1: 5E-09
e_zero_m1: 2E-09

print summary(V1)
for cpt in rp.budget(V1,trim=0):
print "%s: %G" % cpt

We see that the random errors were dominated by both the residual meter gain and offset uncertainties.
2.6000000, u=7.9E-06, df=inf
e_gain_m1: 7.8E-06
e_zero_m1: 1E-06
e_ran_1: 2.6E-07


### In conclusion

One of the advantages of object-oriented programming is that code written in one context can often be reused in another. GTC can capture the benefits from this. In this post we have reused the Meter class definition.

This post has also illustrated how to combine classes of objects in data processing: a Meter object was used here to create uncertain numbers for the four voltage estimates and these uncertain numbers were then directly used to obtain estimates of power by the two Sensor objects. The use of classes makes data processing code easier to read, understand and maintain.

Finally, the way in which different sources of uncertainty propagate into the power ratio is not simple. A full analysis using calculus would be quite a lot more work than the GTC calculation here. With GTC, if we want to, we can track the relative importance of influence quantities through the calculation, from one intermediate step to the next. However, there is no need to audit all these details. GTC ensures that uncertainties are correctly propagated.

1. There are other factors that need to be considered to estimate the signal generator power. In this post, we are only interested in the power level registered by the sensor

## Calculating derivatives

While preparing a workshop about measurement uncertainty the other day, I decided to look for discussion topics in an Application Note about an instrument designed for high frequency measurements (at radio and microwave frequencies) .

I found a section where the authors made a reasonable sounding decision, choosing between two alternative circuit configurations, one called a power splitter the other called a power divider.

I decided to check their assumptions, which eventually led me to calculate some partial derivatives with GTC.

#### The splitter and divider networks

The power splitter has two resistors, each of impedance $$Z_0$$.

As this diagram shows, there are three "ports" (pairs of lines). The input port is on the left and the two output ports are on the right.

The power divider has three resistors, each of impedance $$\frac{Z_0}{3}$$.

There are also three ports. The input is shown here on the left and the two outputs are on the right.

#### Ideal behaviour and sensitivity to errors

Now, when these networks are connected to an "ideal" circuit, the two lines of each output port are effectively connected together by an impedance of $$Z_0$$. In that case, the impedance at the input port can be calculated and is equal $$Z_0$$ too.

That is an ideal configuration, and both the power splitter and power divider behave the same way.

However, the Application Note claimed that if the terminating impedance at an output port was slightly different from $$Z_0$$, the resulting change of impedance seen at the input port would be smaller for a splitter than for a divider.

In other words, the power splitter would be less sensitive to variation of the impedance at the output ports.

This seems reasonable, because the impedances in the splitter are three times bigger of those in the divider. However, I decided to check.

#### Checking the assumption

We need equations for the input impedance $$Z_\mathrm{in}$$, which will depend on $$Z_0$$ and the non-ideal impedance at one port $$Z$$.

For the power splitter, the equation is1
$Z_\mathrm{in} = \frac{2 Z_0 (Z_0 + Z)}{3Z_0 + Z}$
and for the power divider, it is
$Z_\mathrm{in} = \frac{Z_0}{3} + \frac{\frac{4 Z_0}{3}(\frac{Z_0}{3} + Z)}{\frac{4 Z_0}{3} + \frac{Z_0}{3} + Z}$
To find the sensitivity of $$Z_\mathrm{in}$$ to changes, these equations must be differentiated with respect to $$Z$$.

GTC can do these calculations quite easily (I'm a bit out of practice with differentiation).

The following code differentiates the equations above. It first prints out the value of $$Z_\mathrm{in}$$, which we expect to be equal to $$Z_0 = 50$$ in this case. Then the function rp.sensitivity() obtains the first partial derivative of $$Z_\mathrm{in}$$ with respect to $$Z$$.

It is worth noting, that when Z is defined as an uncertain number, in line 14, the uncertainty argument is set to unity, although any number could have been used except for zero. A value of zero would have fooled GTC into thinking that Z was a constant, so rp.sensitivity() would have returned a value of zero.

"""
Compare the sensitivity of the input impedance to departures from Z0.

The Application Note, claims that the 2-resistor splitter is less
sensitive at the input port to changes in the output port impedance.
It isn't!!!

"""
Z0 = 50
Z0_3 = Z0/3.0

# This is the impedance of the output port.
# The second parameter can be anything other than zero.
Z = ureal(Z0,1)

# Splitter expression for input impedance
Z_in = 2*Z0*(Z + Z0)/(Z + 3*Z0)

#
print value(Z_in)
print "sensitivity of Z_in to Z = %G" % rp.sensitivity(Z_in,Z)
print

# Divider expression for input impedance
num = 4*Z0_3 * (Z0_3 + Z)
den = 4*Z0_3 + (Z0_3 + Z)
Z_in = Z0_3 + num/den

print value(Z_in)
print "sensitivity of Z_in to Z = %G" % rp.sensitivity(Z_in,Z)

#### And the results are ...

Input impedance Z0 = 50
sensitivity of Z_in to Z = 0.25

Input impedance Z0 = 50
sensitivity of Z_in to Z = 0.25

In both cases, the value of derivative of $$Z_\mathrm{in}$$ with respect to $$Z$$ is $$\frac{1}{4}$$.

So the Application Note was wrong!

It does not matter whether you use a divider or a splitter, the effect of an imperfect termination at the output ports is the same.

I felt obliged to check this result by hand before posting, but it was certainly a lot harder to remember the calculus rules than to write a little GTC script like the one above.

 These equations can be obtained fairly easily if you know the rules for combining series and parallel impedances. Just remember to connect another $$Z_0$$ impedance across the output port terminals.

## Object-oriented programming and systematic errors

In the previous post (March 3, 2013), I showed how to use GTC when both random and systematic errors contribute to the overall measurement uncertainty.

The key is that an estimate of each measurement error in the model must be associated with one, and only one, uncertain number.

In the last post, we saw that the idea of matching one uncertain number to one error estimate offers a straightforward way of using GTC. However, it becomes hard to keep track of all the different uncertain numbers as problems become more complicated.

In this post, I will show how to define a class of objects to represent the voltmeter used earlier. This class of objects will encapsulate the data and data processing needed by the voltmeter error model, making data processing much more straightforward.

### A simple voltmeter

An equation for the meter reading $$x_\mathrm{dis}$$ in terms of the applied voltage $$V$$ and instrument errors is
$x_\mathrm{dis} = (1 + e_\mathrm{gain} + e_\mathrm{ran}) V + e_\mathrm{zero}$
where, $$e_\mathrm{gain}$$ is the residual error in the gain setting, $$e_\mathrm{zero}$$ is the residual error in the zero offset and $$e_\mathrm{ran}$$ is an error due to system noise.

To carry out data processing, we invert this equation and replace the errors by their estimates (indicated by the hat symbol). In this way we can estimate the applied voltage given a reading
$\widehat{V} = \frac{x_\mathrm{dis} - \widehat{e}_\mathrm{zero}}{1 + \widehat{e}_\mathrm{gain} + \widehat{e}_\mathrm{ran}}$
Remember, $$e_\mathrm{gain}$$ and $$e_\mathrm{zero}$$ are systematic errors: they endure from one measurement to the next. Whereas $$e_\mathrm{ran}$$ changes at every meter reading. So we usually estimate $$e_\mathrm{gain}$$ and $$e_\mathrm{zero}$$ just once, but need a fresh estimate of $$e_\mathrm{ran}$$ for every reading.

I'll now define a Python class to implement the required uncertain-number data processing.

### A voltmeter class

In object-oriented languages like Python, the structure and behaviour of new types of objects can be defined by a 'class'. An object of a given class can have data (that other objects of the same class do not share) and functions that implement certain behaviour (shared among all objects of the same class).

Here is a class definition for the voltmeter

class Meter(object):
def __init__(self,u_e_gain,u_e_zero,u_e_ran=0,tag=None):
"""Initialise a new voltmeter object"""

self.u_e_ran = u_e_ran

tag = "_%s" % tag if tag is not None else ""
label = 'e_gain%s' % tag
self.e_gain = ureal(0,u_e_gain,label=label)

label = 'e_zero%s' % tag
self.e_zero = ureal(0,u_e_zero,label=label)

self.u_e_ran = u_e_ran

def voltage(self,x,tag=None):
"""Return an uncertain number for the voltage"""

tag = "_%s" % tag if tag is not None else ""
if self.u_e_ran != 0:
label = 'e_ran%s' % tag
e_ran = ureal(0,self.u_e_ran,label=label)
else:
e_ran = 0

return (x - self.e_zero)/(1 + self.e_gain + e_ran)

When objects of the Meter class are created, their data is initialised by the __init__ function. We see here that uncertain numbers for the residual gain and zero setting errors are created during initialisation. The standard uncertainty associated with random noise is also saved.

The voltage function uses the object-data to return an uncertain number for an applied voltage estimate. When voltage is called, an uncertain number associated with the random error is created first. This is combined with the uncertain numbers already defined to estimate the applied voltage.

### Resistance measurement with one meter

As in the previous post, I will calculate an estimate of the unknown resistance $$R_\mathrm{x}$$ from measurements of the voltage ratio $$V_\mathrm{x} / V_\mathrm{s}$$ and an estimate of $$R_\mathrm{s}$$

To do this, I create an object of the Meter class by supplying numbers for the standard uncertainty parameters: u_e_gain, u_e_zero and u_e_ran:
m = Meter(u_e_gain=3E-6,u_e_zero=1E-6,u_e_ran=1E-7)
Then use the meter object to obtain a pair of uncertain numbers for the voltage readings
Vs = m.voltage(5.0100,tag='Vs')
Vx = m.voltage(4.9885,tag='Vx')


The code to calculate the unknown resistance and display the results (apart from the Meter class definition) now looks like this
m = Meter(u_e_gain=3E-6,u_e_zero=1E-6,u_e_ran=1E-7)

Rs = 1000 + ureal(0,1E-3,label='Rs')

Vs = m.voltage(5.0100,tag='Vs')
Vx = m.voltage(4.9885,tag='Vx')

Rx = Rs * Vx/Vs

print "Rx = %G, u(Rx) = %G" % (value(Rx), uncertainty(Rx))
for cpt in reporting.budget(Rx,trim=0):
print "%s: %G" % cpt

we obtain (as before)
Rx = 995.709, u(Rx) = 0.00100562
Rs: 0.000995709
e_ran_Vs: 9.95709E-05
e_ran_Vx: 9.95709E-05
e_zero: 8.5657E-07
e_gain: 0

This is easier to read, write and understand than the code I posted earlier.

### Resistance measurement with two meters

As might be expected, if two physically different meters are used to measure the different voltages, then two different Meter objects should be created to do the data processing.

So with just a few simple changes to the code we have
m1 = Meter(u_e_gain=3E-6,u_e_zero=1E-6,u_e_ran=1E-7,tag='m1')
m2 = Meter(u_e_gain=3E-6,u_e_zero=1E-6,u_e_ran=1E-7,tag='m2')

Vs = m1.voltage(5.0100,tag='Vs')
Vx = m2.voltage(4.9885,tag='Vx')

Rs = 1000 + ureal(0,1E-3,label='Rs')

Rx = Rs * Vx/Vs
print "Rx = %G, u(Rx) = %G" % (value(Rx), uncertainty(Rx))
for cpt in reporting.budget(Rx,trim=0):
print "%s: %G" % cpt

which gives the same results as we obtained last time
Rx = 995.709, u(Rx) = 0.0043516
e_gain_m1: 0.00298713
e_gain_m2: 0.00298713
Rs: 0.000995709
e_zero_m2: 0.000199601
e_zero_m1: 0.000198744
e_ran_Vs: 9.95709E-05
e_ran_Vx: 9.95709E-05


This simple and intuitive behaviour is due to the way that uncertain numbers are created during object initialisation. When one Meter object is initialsed, a pair of uncertain numbers are created; but when two objects are initialsed, each has its own pair of uncertain numbers representing independent estimates of the different gain and zero setting errors in each meter.

Hence the definition of a Meter class is able to capture the creation of uncertain numbers we need in a natural and intuitive way that matches the model of the measurement system.

## Systematic errors

In the previous post (Feb 10, 2013), I showed how GTC lets you build up an uncertainty calculation step by step. In this post, I will explain how to deal with systematic errors.

### A simple voltmeter

Last time, I used an equation for the reading displayed by the meter $$x_\mathrm{dis}$$ in terms of the actual applied voltage $$V$$ and instrumental errors
$x_\mathrm{dis} = (1 + e_\mathrm{gain}) V + e_\mathrm{zero}$
where, $$e_\mathrm{gain}$$ is the residual error in the gain setting and $$e_\mathrm{zero}$$ is the residual error in the zero offset. I did not include a term for random noise, because that was estimated from a sample of readings. In this post we will add a term for random noise to the model instead. So the model becomes
$x_\mathrm{dis} = (1 + e_\mathrm{gain} + e_\mathrm{ran}) V + e_\mathrm{zero}$
Now, $$e_\mathrm{gain}$$ and $$e_\mathrm{zero}$$ are systematic errors: they endure from one measurement to the next. So, for example, the contribution to uncertainty from random errors can be reduced by averaging readings from the same meter, but not the uncertainty contributions associated with $$e_\mathrm{gain}$$ and $$e_\mathrm{zero}$$.

How can this be handled with GTC?

Simple.

We associate one, and only one, uncertain number with the estimate of any quantity in the meter model. So, there will be one uncertain number for the estimate of $$e_\mathrm{gain}$$ and one for the estimate of $$e_\mathrm{zero}$$, regardless of how many readings are made with the voltmeter.

There will be a different uncertain number for each reading representing the random noise $$e_{\mathrm{ran}\cdot i}$$, because the error changes from one reading to the next.

### A voltage ratio measurement

We will consider a simple measurement in which a resistance value $$R_\mathrm{x}$$ is estimated by measuring a voltage ratio.

The circuit is shown in this figure,

where a single meter is used to measure the potential difference across each resistor. We will just assume that the current through the circuit does not vary.

With $$R_\mathrm{s}$$ known, $$R_\mathrm{x}$$ can be calculated
$R_\mathrm{x} = R_\mathrm{s}\frac{V_\mathrm{x}}{V_\mathrm{s}}$
Suppose that we measure $$V_\mathrm{x} = 4.9885 \, \mathrm{V}$$ and  $$V_\mathrm{s} =5.0100 \, \mathrm{V}$$. Suppose also that $$R_\mathrm{s} = 1000.000 \; u(R_\mathrm{s}) = 1 \times 10^{-3} \, \Omega$$ and that the relative standard uncertainty in the meter gain is  $$u(e_\mathrm{gain}) = 3 \times 10^{-6} \; \mathrm{V}/ \mathrm{V}$$, that the relative standard uncertainty due to meter noise is $$u(e_\mathrm{ran}) = 1 \times 10^{-7} \; \mathrm{V}/ \mathrm{V}$$ and the standard uncertainty in the zero setting is $$u(e_\mathrm{zero}) = 1 \times 10^{-6} \; \mathrm{V}$$.

Common sense tells us that the uncertainty due to gain error cancels when the voltage ration is calculated. So, one might simply ignore that term. The zero setting uncertainty will not have very much effect on the combined uncertainty $$R_\mathrm{s}$$ either.

However, I am not going to make those assumptions. Instead, I will show that GTC arrives at the correct result anyway, with no need to think about simplifying the mathematical model.

I will define uncertain numbers for the estimates of $$R_\mathrm{s}$$, $$V_\mathrm{s}$$ and $$V_\mathrm{x}$$ and then work out
$R_\mathrm{x} = R_\mathrm{s} \frac{V_\mathrm{x}}{V_\mathrm{s}}$
Here is the standard resistance
R_s = 1000 + ureal(0,1E-3,label='R_s')

and here are the voltmeter errors
e_zero = ureal(0,1E-6,label='e_zero')
e_gain = ureal(1,3E-6,label='e_gain')

Uncertain numbers representing the individual voltage measurements are then (note an uncertain number for the meter noise is created independently for each reading)
e_ran_1 = ureal(1,1E-7,label='e_ran_1')
V_s = (5.0100 - e_zero) / (e_gain * e_ran_1)

e_ran_2 = ureal(1,1E-7,label='e_ran_2')
V_x = (4.9885 - e_zero) / (e_gain * e_ran_2)

And the unknown resistance is
R_x = R_s * V_x/V_s

The value, the uncertainty and the uncertainty budget of the result are displayed by
print "R_x = %G, u(R_x) = %G" % (value(R_x), uncertainty(R_x))
for cpt in reporting.budget(R_x,trim=0):
print "%s: %G" % cpt

We obtain the following results:
R_x = 995.709, u(R_x) = 0.00100562
R_s: 0.000995709
e_ran_1: 9.95709E-05
e_ran_2: 9.95709E-05
e_zero: 8.5657E-07
e_gain: 0

As expected, the uncertainty due to an error in the voltmeter gain setting is zero and the uncertainty due to an error in the zero offset is also small. The uncertainty of this measurement is dominated by the standard resistor uncertainty, followed in importance by the meter noise.

Now, suppose that two meters of the same type are used to simultaneously measure $$V_\mathrm{s}$$ and $$V_\mathrm{x}$$. This would ensure that the current flowing in the circuit elements was indeed the same, rather than just assuming it to be constant.

Different meters will have different residual gain and zero setting errors. So the calculation must now define uncertain numbers for each meter.

The code now looks like this
R_s = 1000 + ureal(0,1E-3,label='R_s')

# Meter 1
e_zero_1 = ureal(0,1E-6,label='e_zero_1')
e_gain_1 = ureal(1,3E-6,label='e_gain_1')

# Meter 2
e_zero_2 = ureal(0,1E-6,label='e_zero_2')
e_gain_2 = ureal(1,3E-6,label='e_gain_2')

e_ran_1 = ureal(1,1E-7,label='e_ran_1')
V_s = (5.0100 - e_zero_1) / (e_gain_1 * e_ran_1)

e_ran_2 = ureal(1,1E-7,label='e_ran_2')
V_x = (4.9885 - e_zero_2) / (e_gain_2 * e_ran_2)

# Result
R_x = R_s * V_x/V_s

# Display results
print "R_x = %G, u(R_x) = %G" % (value(R_x), uncertainty(R_x))
for cpt in reporting.budget(R_x,trim=0):
print "%s: %G" % cpt

We obtain the following results:
R_x = 995.709, u(R_x) = 0.0043516
e_gain_1: 0.00298713
e_gain_2: 0.00298713
R_s: 0.000995709
e_zero_2: 0.000199601
e_zero_1: 0.000198744
e_ran_1: 9.95709E-05
e_ran_2: 9.95709E-05

This is quite a different result. The combined standard uncertainty $$u(R_\mathrm{x})$$ is a lot bigger and the gain error uncertainties now dominate the uncertainty budget.

### In summary

Every error in a measurement model has to be estimated in the GUM process of calculating uncertainty. Often residual errors are approximately zero, or unity, but these estimates still carry uncertainty.

Using GTC, every estimate of a quantity in the model is associated with one, and only one, uncertain number. When this rule is understood, GTC can be used to handle quite complicated data processing problems.

The next post will revisit this example. I will show then how object-oriented programming techniques can be used to implement the voltmeter model.