**3.1 Modeling**

176 Modern Metrology Concerns

The terminating device may not react in the same way for microwave and DC power absorption, so the effective efficiency *ηe* is used to perform the correction. In equation (1), microwave power *PHF* absorbed by the terminating device is calculated by dividing the

> *DC <sup>e</sup> HF P P*

Since the effective efficiency is independent of the mismatch correction, the calibration

The calibration factor *K* is generally used at the time of calibration transfer from a reference standard to an unknown microwave power sensor. It is the focus in the following sections. The measurement uncertainty is a non-negative parameter characterizing the dispersion of the effective efficiency *ηe* or the calibration factor *K* being attributed to the standards. The uncertainty is evaluated using "law of propagation of uncertainty" following "Guide to the expression of Uncertainty in Measurement" (GUM) (JCGM 100:2008). Evaluation of measurement and calibration uncertainty by Monte Carlo Method (MCM) is to use Monte Carlo simulation in the uncertainty evaluation of output quantities based on "uncertainty probability distribution propagation" (JCGM 101:2008). The following sections will cover

The value of a primary standard is disseminated to a secondary standard through calibration or comparison. Then the reference standard and working power sensors will be calibrated for use. The measurement results through such relations as unbroken chain of calibrations establish the metrological traceability, each contributing to the measurement uncertainty. The traceability is illustrated in Fig. 1. Here for reference purpose we deliberately provide not only the hierarchy, but also the uncertainties typically related. The

real uncertainties depend on the frequency band and each laboratory conditions.

Fig. 1. Dissemination of primary standard to end user- Traceability Chart

*K - <sup>e</sup>* <sup>2</sup> 

(1)

(1 ) (2)

ડ

as follows:

factor *K* is used to describe both the effective efficiency *ηe* and mismatch

substituted DC power *PDC* by the effective efficiency *ηe*:

both methods for the measurement uncertainty evaluations.

The calibration of RF and microwave power sensor is to transfer the effective efficiency or calibration factor from a primary standard to a secondary standard or from a secondary standard to a reference standard or from a reference standard to a power sensor. The parameter transfer is through comparison, or calibration one against the other. The simplest and the most obvious method to calibrate a power sensor against a reference standard is to connect each in turn to a stable power source, as illustrated in Fig. 2.

Fig. 2. Calibration of power sensor by the method of simple direct comparison transfer

Generally power from a source, *Pi,* with reflection coefficient ડ*<sup>G</sup>*, incident to a load with reflection coefficient ડ*<sup>L</sup>* , can be expressed as follows (Agilent, 2003; Engen, 1993; Mial, 2007):

$$P\_i = P\_{Z\_0} \times \frac{1}{|1 \cdot \Gamma\_G \Gamma\_L|^2}$$

and a reflected power *P<sup>r</sup>*

$$P\_r = P\_{Z\_0} \times \frac{\left\| \begin{array}{c} \Gamma\_L \\ \hline \end{array} \right\|^2}{\left\| \begin{array}{c} \Gamma\_G \Gamma\_L \end{array} \right\|^2}$$

For Fig. 2, the power dissipated to the reference power standard *PStd* , can be derived as

$$P\_{Std} = P\_i \text{ - } P\_{r\_r Std} = P\_{Z\_0} \times \frac{1 \text{ - } \left\| \begin{array}{c} \Gamma\_{Std} \end{array} \right\|^2}{\left\| \begin{array}{c} 1 \ -\Gamma\_G \Gamma\_{Std} \end{array} \right\|^2}$$

And the power dissipated to the power sensor to be calibrated (DUT), *PDUT*, is as

$$P\_{DUT} = P\_i - P\_{r, DUT} = P\_{Z\_0} \times \frac{\mathbf{1} \cdot \left| \begin{array}{c} \Gamma\_{DUT} \end{array} \right|^2}{\left| \mathbf{1} \cdot \Gamma\_G \Gamma\_{DUT} \right|^2}$$

where *PZ0* is the power available to a load with characteristic impedance *Z0*. When idealized source with available power *Pa* has no internal impedance *Z0*, *PZ0 =Pa (1-|*ડ*G|2)*.

As shown in equation (1), the effective efficiency of a power sensor is a ratio. *PDC* is DC or low frequency equivalent power, generating the same heat effect as the high frequency power being measured. For calibration of a power sensor, that is, to transfer the effective efficiency of a reference standard, *ηStd* , to a power sensor to be calibrated (DUT), *ηDUT*, it can be derived:

*DC DUT DC DUT DUT Z DUT RF DUT G DUT DC DUT Std G DUT Std DC Std DC Std DC Std DUT G Std RF Std std Z G std P <sup>P</sup> - <sup>P</sup> P P - - - P P P - - <sup>P</sup> - <sup>P</sup> -* 0 0 , <sup>2</sup> , <sup>2</sup> 2 2 , , 2 2 , , , <sup>2</sup> , 2 1| | |1 | 1 | | |1 | 1 | | |1 | 1| | |1 | 

So the calibration equation of effective efficiency for the method of Fig. 2 is expressed as:

$$\eta\_{\rm DLT} = \eta\_{Std} \times \frac{P\_{\rm DC,DIT}}{P\_{\rm DC,Std}} \times \frac{\mathbf{1} \cdot \left| \Gamma\_{Std} \right|^2}{\mathbf{1} \cdot \left| \Gamma\_{DUT} \right|^2} \frac{\left| \mathbf{1} \cdot \Gamma\_G \Gamma\_{DUT} \right|^2}{\left| \mathbf{1} \cdot \Gamma\_G \Gamma\_{Std} \right|^2} \tag{3}$$

where *ηDUT* is the effective efficiency of DUT sensor.

This calibration equation involves three factors:

$$
\eta\_{DUT} = \eta\_{Std} \times P\_{RATIO} \times M\_1 M\_2
$$

*ηStd* is the effective efficiency of a standard sensor. It comes from a national metrology institute, a calibration laboratory, or a manufacturer with traceability. *PRATIO* is an equivalent DC or low frequency power ratio, depending on the system setup. *M1M2* is the mismatch factor.

Similarly, calibration equation of calibration factor for the method of Fig. 2 is expressed as follows, considering equation (2):

$$K\_{DITT} = K\_{Std} \times \frac{P\_{DC,DITT}}{P\_{DC,Std}} \times \frac{\left|1 \cdot \Gamma\_G \Gamma\_{DITT}\right|^2}{\left|1 \cdot \Gamma\_G \Gamma\_{Shd}\right|^2} \tag{4}$$

And the calibration factor transfer equation can also be expressed as three factors:

$$\mathcal{K}\_{DUT} = \mathcal{K}\_{S\!\!\!M} \times P\_{R\!\!\!A\!\!T\!\!M\!\!O} \times M^{\!\!\!A}$$

Notice that *M* is the mismatch factor for the calibration factor calibration transfer.

#### **3.2 Uncertainty evaluation**

From calibration transfer equation of effective efficiency (3) and that of calibration factor (4), if DUT power sensor has identical reflection coefficient as that of reference power standard, which means �*DUT =* �*Std*, each of them absorbs exactly the same amount of power from the source. Then by power ratio measurement it transfers the effective efficiency, *ηe*, or calibration factor, *K*, from standard to DUT, to complete the calibration.

But the actual reflection coefficients of power sensors being compared usually differ significantly from one another (refer to Table 1). The signal generator reflection coefficient cannot dismiss easily. The complex items inside (3) and (4), i.e. the mismatch factors, cannot be neglected in accurate power sensor calibration with small uncertainty.

*DUT RF DUT G DUT DC DUT Std G DUT Std DC Std DC Std DC Std DUT G Std*

*P - - P - -*

*DUT Std RATIO P MM*1 2

*ηStd* is the effective efficiency of a standard sensor. It comes from a national metrology institute, a calibration laboratory, or a manufacturer with traceability. *PRATIO* is an equivalent DC or low frequency power ratio, depending on the system setup. *M1M2* is the mismatch

Similarly, calibration equation of calibration factor for the method of Fig. 2 is expressed as

*DC DUT G DUT*

<sup>2</sup> , <sup>2</sup> , |1 | |1 |

*Std*, each of them absorbs exactly the same amount of power from the

(4)

*DC Std G Std*

*P -*

*K KP M DUT Std RATIO*

From calibration transfer equation of effective efficiency (3) and that of calibration factor (4), if DUT power sensor has identical reflection coefficient as that of reference power standard,

source. Then by power ratio measurement it transfers the effective efficiency, *ηe*, or

But the actual reflection coefficients of power sensors being compared usually differ significantly from one another (refer to Table 1). The signal generator reflection coefficient cannot dismiss easily. The complex items inside (3) and (4), i.e. the mismatch factors, cannot

<sup>2</sup> 2 2 , , 2 2 , , ,

*DC DUT Std G DUT*

2 2 , 2 2 ,

1 | | |1 | 1 | | |1 |

(3)

*DC Std DUT G Std*


1 | | |1 |

*P P - - - P P P - -*

2

So the calibration equation of effective efficiency for the method of Fig. 2 is expressed as:

*DC DUT*

1| |

1| | |1 |

 

,

*P*

*G std*

 

*<sup>P</sup> - K K*

And the calibration factor transfer equation can also be expressed as three factors:

Notice that *M* is the mismatch factor for the calibration factor calibration transfer.

calibration factor, *K*, from standard to DUT, to complete the calibration.

be neglected in accurate power sensor calibration with small uncertainty.

*DC DUT DUT Z*

0

<sup>2</sup> ,

*<sup>P</sup> - <sup>P</sup>*

factor.

*<sup>P</sup> - <sup>P</sup> -*

*DUT Std*

where *ηDUT* is the effective efficiency of DUT sensor.

This calibration equation involves three factors:

follows, considering equation (2):

**3.2 Uncertainty evaluation** 

�*DUT =* �

which means

 

*RF Std std Z*

0

*DUT Std*

<sup>2</sup> ,

In recognition of the standards adopted internationally, the GUM (JCGM 100:2008) is considered as the basic technique to evaluate the uncertainty of measurement. The method proposed in the GUM is based on the law of propagation of uncertainty which is essentially the first-order Taylor series approximation of calibration equation, such as equation (4) of calibration factor. The partial differentiation of the output estimate with respect to the input estimates, which is termed sensitivity coefficient, is interpreted as a description of how the output estimate varies with changes in the values of the input estimates. The following equation is the definition of combined standard uncertainty *uc(y)* which includes both type A (*ua(x)*) and type B (*ub(x)*) uncertainties when the mathematical model is of the form *y = f(x1, x2,…)*:

$$\mu\_c(\mathbf{y}) = \sqrt{\mu\_a^2(\mathbf{x}) + \sum\_{i=1}^{N} \left(\frac{\partial f}{\partial \mathbf{x}\_i}\right)^2 \mu\_b^2(\mathbf{x}\_i)}\tag{5}$$

Considering that the setup in Fig. 2 is the simple direct comparison transfer method, we present the uncertainty evaluation equations in different implementations, aiming at providing laboratories more realistic choices. The realistic magnitudes for reflection coefficients of instruments are used for the calculation and comparison. The specifications of reflection coefficients of the instruments quoted from different products at some frequency points are listed in Table 1.


Table 1. Typical reflection coefficients of power sensors and signal generators

#### **3.2.1 The simplest evaluation of measurement uncertainty**

The simplest way in the calibration transfer of calibration factor from standard to DUT sensor is to simplify equation (4) to the following equation:

$$K\_{DIT} = K\_{Std} \times \frac{P\_{\underline{U}}}{P\_S} = \text{Ratio Factor} \tag{6}$$

In which only the ratio factor is considered and set the mismatch factor *M* equal to 1. The real values of reflection coefficients are considered for the uncertainty budget only. In this case, the mismatch factor *M* for calibration factor *K*DUT is expressed as:

$$\mathbf{M} = \frac{\mathbf{M}\_{\mathbf{U}}}{\mathbf{M}\_{\mathbf{S}}}$$

where *MU* and *MS* are given by:

$$\mathbf{M}\_{\mathrm{II}} = \left| \mathbf{1} \cdot \Gamma\_{\mathrm{G}} \Gamma\_{\mathrm{D}LT} \right|^2 \quad \mathbf{M}\_{\mathrm{S}} = \left| \mathbf{1} \cdot \Gamma\_{\mathrm{G}} \Gamma\_{\mathrm{S}\mathrm{H}} \right|^2$$

The associated uncertainties are calculated with the following equations (Agilent, 2003; Shan et al., 2010a):

$$\mu\left(M\_X\right) = \sqrt{2\left\|\Gamma\_G\right\|\left\|\Gamma\_X\right\|}; \qquad \chi = Std,\\
\text{DUT} \tag{7}$$

According to GUM for the law of propagation of uncertainties, the sensitivity coefficients are partial differentiations with respect to the individual variables in equation (6):

$$\begin{aligned} \frac{\partial \mathcal{K}\_{DUT}}{\partial \mathcal{K}\_{Std}} &= \frac{P\_{\mathcal{U}}}{P\_{\mathcal{S}}} = \frac{\mathcal{K}\_{DUT}}{\mathcal{K}\_{Sdd}} \\\\ \frac{\partial \mathcal{K}\_{DUT}}{\partial P\_{\mathcal{U}}} &= \frac{\mathcal{K}\_{Sdd}}{P\_{\mathcal{S}}} = \frac{\mathcal{K}\_{DUT}}{P\_{\mathcal{U}}} \\\\ \frac{\partial \mathcal{K}\_{DUT}}{\partial P\_{\mathcal{S}}} &= -\frac{\mathcal{K}\_{Sdd} \times P\_{\mathcal{U}}}{P\_{\mathcal{S}}^2} = (-1) \frac{\mathcal{K}\_{DUT}}{P\_{\mathcal{S}}} \end{aligned} \tag{8}$$

In which the expression is prepared for relative uncertainty representation since the combined standard uncertainty *uc(y)* can be expressed as an estimated relative combined

variance 2 2 2 1 ( ) ( ) *<sup>N</sup> <sup>c</sup> <sup>i</sup> i i i u y u x <sup>P</sup> y x* when the mathematical model is of the form 1 2 1 2 ... *P P PN Y cX X XN .* 

The uncertainty budget is listed in Table 2 at frequency 18 GHz with type N connector. The calculations of mismatch uncertainties are based on Table 1 best and worst specifications.

It can be seen from Table 2 that the simplest evaluation of measurement uncertainty method is not an accurate method to obtain a small uncertainty. But it is useful for calibration laboratories with simple measurement set up as illustrated in Fig. 2. When this uncertainty value meets the demand, it is acceptable for industrial applications. In some evaluation experiments, it is also a practical method. Note that the value and uncertainty of a reference standard comes from the calibration of national metrology institutes or other calibration laboratories if the laboratory provides the calibration service using the method.


The associated uncertainties are calculated with the following equations (Agilent, 2003; Shan

According to GUM for the law of propagation of uncertainties, the sensitivity coefficients

*K P K K PK* 

*DUT U DUT Std S Std*

*DUT Std DUT USU K K K P PP*

*DUT Std U DUT S S S*

when the mathematical model is of the form

*K KP K - - P P <sup>P</sup>*<sup>2</sup> ( 1)

In which the expression is prepared for relative uncertainty representation since the combined standard uncertainty *uc(y)* can be expressed as an estimated relative combined

The uncertainty budget is listed in Table 2 at frequency 18 GHz with type N connector. The calculations of mismatch uncertainties are based on Table 1 best and worst specifications.

It can be seen from Table 2 that the simplest evaluation of measurement uncertainty method is not an accurate method to obtain a small uncertainty. But it is useful for calibration laboratories with simple measurement set up as illustrated in Fig. 2. When this uncertainty value meets the demand, it is acceptable for industrial applications. In some evaluation experiments, it is also a practical method. Note that the value and uncertainty of a reference standard comes from the calibration of national metrology institutes or other calibration

Quant. Estim. Standard probability Sensitivity Uncertainty

*X*i *x*i *u*(*x*i) *c*i *u*i*(y) K*<sup>S</sup> 0.9894 0.0012 normal 1.0137 0.0012 *M*<sup>S</sup> 1.0000 0.0098 U-shaped 1.0000 0.0098 *M*<sup>U</sup> 1.0000 0.0195 U-shaped 1.0000 0.0195 *P*<sup>U</sup> 1.0158 0.0018 normal 0.9873 0.0018 *P*<sup>S</sup> 1.0021 0.0004 normal -1.0008 -0.0004 *y=K*<sup>U</sup> 1.0029 0.0219

uncertainty distribution coefficient contribution

laboratories if the laboratory provides the calibration service using the method.

18 GHz based on best specifications

2 2 2 1 ( ) ( ) *<sup>N</sup> <sup>c</sup> <sup>i</sup> i i i*

*u y u x <sup>P</sup> y x*

are partial differentiations with respect to the individual variables in equation (6):

*X Std,DUT u MX GX* 2 ; (7)

(8)

et al., 2010a):

variance

1 2 1 2 ... *P P PN Y cX X XN .* 


Table 2. Uncertainty budget at 18 GHz for simplest evaluation of uncertainty for Fig. 2 measurement setup. Uncertainties *ui* are at one standard deviation. Powers are measured in mW.

#### **3.2.2 Measurement uncertainty improvement with mismatch correction**

To improve the calibration accuracy and uncertainty evaluation, we have to perform a mismatch correction, i.e., the complex reflection coefficients have to be considered in the calibration model. The complex value is either representing in term of magnitude and phase or its real and imaginary components. In the majority of engineering applications, the magnitude and phase representation (Polar coordinates) is generally preferred because this representation bears a direct relationship to physical phenomena affecting the measurement process (Ridler & Salter, 2002). For example, phase relates directly to the electrical path length of a signal, and magnitude relates directly to the signal attenuation. The same cannot be said for the representation of real and imaginary components (Cartesian coordinates). If the scales are used to depict the different representations, the real and imaginary axes in the complex plane extend infinitely (±∞). It is the same as the scale is used to depict all real numbers which is routinely for the arithmetic operations. While the scales are used to represent magnitude and phase each possess a significant difference. The magnitude has a lower bound of zero below which values cannot exist, and phase is represented convertionally on a cyclical scale, either from -180⁰ to +180⁰ or from 0⁰ to 360⁰.

With mismatch correction added in computing the calibration factor using equation (4), the calculation of sensitivity coefficients involves the partial differentiations with respect to complex reflection coefficients. In the following sections, we separately provide the derived sensitivity coefficients in both Cartesian and Polar coordinate representations for practical measurement uncertainty solution and application; and also examples for their associated uncertainties are included. The derived sensitivity coefficients in both representations have been numerically appreciated by making use of MATLAB version R2010a, symbolic differentiation in Math Toolbox.

#### **3.2.2.1 Cartesian representation of sensitivity coefficients for equation (4)**

Representing calibration factor *K*DUT of equation (4) with magnitude and phase components, it becomes

$$\mathbf{K}\_{\mathrm{DJT}} = \mathbf{K}\_{\mathrm{Sfd}} \times P\_{\mathrm{RATIO}} \times M = \mathbf{K}\_{\mathrm{Sfd}} \times \frac{P\_{\mathrm{DJT}}}{P\_{\mathrm{Sfd}}} \times \frac{\mathbf{1} + \|\boldsymbol{\Gamma}\_{\mathrm{DJT}}\|^2 \left\|\boldsymbol{\Gamma}\_{\mathrm{G}}\right\|^2 \cdot \mathbf{2} \left\|\boldsymbol{\Gamma}\_{\mathrm{DJT}}\right\| \left\|\boldsymbol{\Gamma}\_{\mathrm{G}} \left|\cos(\boldsymbol{\theta}\_{\mathrm{DJT}} + \boldsymbol{\theta}\_{\mathrm{G}})\right\rangle}{\mathbf{1} + \|\boldsymbol{\Gamma}\_{\mathrm{Sfd}}\|^2 \left\|\boldsymbol{\Gamma}\_{\mathrm{G}}\right\|^2 \cdot \mathbf{2} \left\|\boldsymbol{\Gamma}\_{\mathrm{G}}\right\| \left\|\boldsymbol{\Gamma}\_{\mathrm{G}} \left|\cos(\boldsymbol{\theta}\_{\mathrm{Sid}} + \boldsymbol{\theta}\_{\mathrm{G}})\right\|\right\|}\tag{9}$$

Let *MN* represent the numerator and *MD* denominator of mismatch *M*:

$$M\_{\rm N} = \left| \mathbf{1} \cdot \Gamma\_{D\rm MT} \Gamma\_{G} \right|^{2} = \mathbf{1} + \left| \Gamma\_{D\rm MT} \right|^{2} \left| \Gamma\_{G} \right|^{2} \cdot 2 \left| \Gamma\_{D\rm MT} \right| \parallel \Gamma\_{G} \parallel \cos(\theta\_{\rm DIT} + \theta\_{\rm G})$$

$$M\_{\rm D} = \left| \mathbf{1} \cdot \Gamma\_{S\rm d} \Gamma\_{G} \right|^{2} = \mathbf{1} + \left| \Gamma\_{S\rm d} \right|^{2} \left| \Gamma\_{G} \right|^{2} \cdot 2 \left| \Gamma\_{S\rm d} \right| \parallel \Gamma\_{G} \parallel \cos(\theta\_{\rm S\rm d} + \theta\_{\rm G})$$

$$\begin{array}{llll} \textbf{Let} & A = \left| \begin{array}{c} \Gamma\_{G} \end{array} \right|\_{\prime} & B = \left| \begin{array}{c} \Gamma\_{D\rm MT} \end{array} \right|\_{\prime} & \mathbf{C} = \left| \begin{array}{c} \Gamma\_{S\rm d} \end{array} \right|\_{\prime} & D = \cos(\theta\_{\rm DIT} + \theta\_{\rm G}) & \mathbf{E} = \cos(\theta\_{\rm S\rm d} + \theta\_{\rm G}) & \mathbf{and} \end{array}$$

Then,

$$\begin{aligned} \mathcal{K}\_{\text{DLT}} &= \mathcal{K}\_{\text{Sfd}} \times P\_{\text{RATIO}} \times M\\ &= \text{Ratio Factor} \times M = \text{Ratio Factor} \times \frac{1 + A^2 B^2 \cdot 2ABD}{1 + A^2 C^2 \cdot 2ACE} \end{aligned} \tag{10}$$

According to GUM for the law of propagation of uncertainties, the sensitivity coefficients are partial differentiations with respect to the individual variables in equation (10), total 9 items. The derived sensitivity coefficients are as follows:

1. The sensitivity coefficient for KStd

*Std*

*<sup>P</sup>* .

*Std*

$$\frac{\partial \mathbf{K}\_{DUT}}{\partial \mathbf{K}\_{Std}} = \frac{\mathbf{K}\_{DUT}}{\mathbf{K}\_{Std}} \tag{11}$$

2. The sensitivity coefficient for PDUT

$$\frac{\partial \mathcal{K}\_{DUT}}{\partial P\_{DUT}} = \frac{\mathcal{K}\_{DUT}}{P\_{DUT}} \tag{12}$$

3. The sensitivity coefficient for PStd

$$\frac{\partial \mathcal{K}\_{DUT}}{\partial P\_{Std}} = -\frac{\mathcal{K}\_{DUT}}{P\_{Std}} \tag{13}$$

4. The sensitivity coefficient for |ΓStd|(=C)

$$\frac{\partial K\_{DIT}}{\partial \left| \Gamma\_{Std} \right|} = \frac{K\_{DIT}}{M\_D} \times 2A \text{(AC - E)}\tag{14}$$

5. The sensitivity coefficient for θStd

$$\frac{\partial \mathcal{K}\_{DUT}}{\partial \,\theta\_{Std}} = -\frac{K\_{DUT}}{M\_D} \times \text{2AC}\sin(\theta\_{Std} + \theta\_G) \tag{15}$$

6. The sensitivity coefficient for |ΓDUT|(=B)

$$\frac{\partial K\_{DUT}}{\partial \left| \Gamma\_{DUT} \right|} = \text{Ratio Factor} \times \frac{1}{M\_D} \times \text{2A} \{AB \text{ -} D\} \tag{16}$$

7. The sensitivity coefficient for θDUT

182 Modern Metrology Concerns

*M - N DUT G DUT G DUT G DUT G -* 2 22 |1 | 1 | || | 2| || |cos( )

*M - D Std G Std G Std G Std G -* 2 22 |1 | 1 | || | 2| || |cos( )

Let *A=|ΓG|, B=|ΓDUT|, C=|ΓStd|, D=cos(θDUT+θG), E=cos(θStd+θG),* and

*<sup>A</sup> B - ABD Ratio Factor M Ratio Factor*

According to GUM for the law of propagation of uncertainties, the sensitivity coefficients are partial differentiations with respect to the individual variables in equation (10), total 9

> *DUT DUT Std Std*

*DUT DUT DUT DUT*

*DUT DUT Std Std*

*- A AC - E <sup>M</sup>* 2( ) | |

> *<sup>M</sup>* 2 sin( )

*M* <sup>1</sup> 2( ) | |

*K K K K*

*K K P P*

*K K - P P*

*- AC*

*DUT D*

*<sup>K</sup> Ratio Factor A AB - D*

*DUT DUT Std D*

*K K*

*DUT DUT*

*K K*

*DUT*

*Std D*

*A C - ACE*

(11)

(12)

(13)

(14)

*Std G*

(16)

(15)

2 2 2 2 1 2 1 2 (10)

Let *MN* represent the numerator and *MD* denominator of mismatch *M*:

*DUT*

*Std*

*K KP M DUT Std RATIO*

items. The derived sensitivity coefficients are as follows:

*Std*

*<sup>P</sup>* .

1. The sensitivity coefficient for KStd

2. The sensitivity coefficient for PDUT

3. The sensitivity coefficient for PStd

5. The sensitivity coefficient for θStd

4. The sensitivity coefficient for |ΓStd|(=C)

6. The sensitivity coefficient for |ΓDUT|(=B)

*<sup>P</sup> Ratio Factor K*

Then,

$$\frac{\partial K\_{DIT}}{\partial \theta\_{DIT}} = \text{Ratio Factor} \times \frac{2AB}{M\_D} \text{sin} (\theta\_{DIT} + \theta\_G) \tag{17}$$

8. The sensitivity coefficient for |ΓG|(=A)

$$\frac{\partial \mathcal{E}\_{\text{DUT}}}{\partial \left| \, \Gamma\_{\text{G}} \right|} = \text{Ratio Factor} \times \frac{2B}{M\_{\text{D}}} \times \text{(AB - D)} \cdot \frac{M\_{\text{N}} \times 2\text{C}}{\text{(M}\_{\text{D}}\text{)}^{2}} \times \text{(AC - E)}\tag{18}$$

9. The sensitivity coefficient for θ<sup>G</sup>

$$\begin{aligned} \frac{\partial K\_{\text{DIT}}}{\partial \theta\_{\text{DIT}}} &= \text{Ratio Factor} \times \{ \frac{2AB}{M\_D} \sin(\theta\_{\text{DIT}} + \theta\_{\text{G}}) \cdot \frac{M}{M\_D} \times 2AC \sin(\theta\_{\text{Sd}} + \theta\_{\text{G}}) \} \\ &= \text{equation} \text{(17)} + \text{equation} \text{(15)} \end{aligned} \tag{19}$$

#### **3.2.2.2 Polar representation of sensitivity coefficients for equation (4)**

Representing calibration factor *K*DUT of equation (4) with real and imaginary components, the following denotation is used to denote the real and imaginary components of reflection coefficient:

 $a = \text{Re}\{\Gamma\_{\text{Sid}}\} = \Gamma\_{\text{Sid} \cdot \text{Re}\,\prime\,\prime\,\prime}$   $b = \text{Im}\{\Gamma\_{\text{Sid}}\} = \Gamma\_{\text{Sid} \cdot \text{Im}\,\prime\,\prime\,\prime\,\prime} = \text{Re}\{\Gamma\_{\text{DIT}}\} = \Gamma\_{\text{DIT} \cdot \text{Re}\,\prime\,\prime\,\prime\,\prime\,\prime}$   $d = \text{Im}\{\Gamma\_{\text{DIT}}\} = \Gamma\_{\text{DIT} \cdot \text{Im}\,\prime\,\prime\,\prime}$   $e = \text{Re}\{\Gamma\_{\text{G}}\} = \Gamma\_{\text{G-Im}}$   $\text{and } f = \text{Im}\{\Gamma\_{\text{G}}\} = \Gamma\_{\text{G-Im}}$ 

Then the calibration model becomes:

$$\begin{split} K\_{\text{DIT}} &= K\_{\text{Sfd}} \times P\_{\text{RATIO}} \times M1 \\ &= \text{Ratio Factor} \times \frac{1 + 2df \cdot 2ce + c^2e^2 + d^2e^2 + c^2f^2 + d^2f}{1 + 2bf \cdot 2ae + a^2e^2 + b^2e^2 + a^2f^2 + b^2f^2} \end{split} \tag{20}$$

Let *M1N* represent the numerator and *M1D* denominator of mismatch *M1* in real and imaginary format:

$$\begin{aligned} M\mathbf{1}\_N &= \|\mathbf{1} \cdot \boldsymbol{\Gamma}\_{DLT}\boldsymbol{\Gamma}\_G\|^2 = \mathbf{1} + 2df \cdot \mathbf{2}ac + a^2e^2 + d^2e^2 + c^2f^2 + d^2f^2 \\ M\mathbf{1}\_D &= \|\mathbf{1} \cdot \boldsymbol{\Gamma}\_{Std}\boldsymbol{\Gamma}\_G\|^2 = \mathbf{1} + 2bf \cdot \mathbf{2}ac + a^2e^2 + b^2e^2 + a^2f^2 + b^2f^2 \end{aligned}$$

Similarly, according to GUM for the law of propagation of uncertainties, the sensitivity coefficients are partial differentiation with respect to each input quantities in equation (20), total 9 items. The derived sensitivity coefficients are as follows:


$$\frac{\partial \,\,\mathrm{K}\_{\mathrm{DUT}}}{\partial \,\,\mathrm{T}\_{\mathrm{Std\cdot Re}}} = \frac{\mathrm{K}\_{\mathrm{DUT}}}{M \mathbf{1}\_{D}} \times \mathrm{(2e-2ae^{2}-2af^{2})}\tag{21}$$

3. The sensitivity coefficient for Γ*Std-Img* (= *b*)

$$\frac{\partial \, K\_{\text{DIT}}}{\partial \, \Gamma\_{S\text{tf}\text{-}\text{Im}\, \text{g}}} = \frac{K\_{\text{DIT}}}{M \mathbf{1}\_D} \times \{ (2f + 2be^2 + 2bf^2) \} \tag{22}$$

4. The sensitivity coefficient for Γ*DUT-Re* (= *c*)

$$\frac{\partial \, K\_{DIT}}{\partial \, \Gamma\_{DIT-\text{Re}}} = \text{Ratio Factor} \times \frac{2e + 2ce^2 + 2cf^2}{M1\_D} \tag{23}$$

5. The sensitivity coefficient for Γ*DUT-Img* (= *d*)

$$\frac{\partial \, K\_{DIT}}{\partial \, \Gamma\_{DIT-\text{Im}\,g}} = \text{Ratio Factor} \times \frac{2f + 2de^2 + 2df^2}{M1\_D} \tag{24}$$

6. The sensitivity coefficient for Γ*G-Re* (= *e*)

$$\frac{\partial \text{ } \mathbf{K}\_{\text{DITT}}}{\partial \text{ } \Gamma\_{G\text{-Re}}} = \text{Ratio Factor} \times \frac{2c + 2c^2e + 2d^2e + \left(M1\right)\left(2a - 2a^2e - 2b^2e\right)}{M1\_D} \tag{25}$$

7. The sensitivity coefficient for Γ*G-Img* (= *f*)

$$\frac{\partial \, K\_{\text{DLT}}}{\partial \, \sigma \, \Gamma\_{G-\text{Im}\,\mathcal{g}}} = \text{Ratio Factor} \times \frac{2d + 2c^2 f + 2d^2 f \cdot \text{(M1)} \left(2b + 2a^2 f + 2b^2 f\right)}{M \mathbf{1}\_D} \tag{26}$$

#### **3.2.2.3 Example**

With the same data as used in Table 2, the uncertainty budget is listed in Table 3 with mismatch corrections considered in the calibration equation.


) *DUT DUT*

{(2 2 2 } <sup>1</sup>

2 2

*M*

*M*

2 2 2 2

2 2 2 2

*M*

*M*

uncertainty distribution coefficient contribution

1

2 2 2 12 2 2

(26)

2 2 2 12 2 2 1 (25)

22 2 1 (24)

22 2 1 (23)

(22)

2 2

2 2

*<sup>K</sup> <sup>K</sup> <sup>f</sup> be bf <sup>M</sup>*

*DUT - D <sup>K</sup> e ce cf Ratio Factor*

*DUT - g D <sup>K</sup> f de df Ratio Factor*

*G- D*

*G- g D*

18 GHz based on best specifications

*DUT*

*DUT*

*<sup>K</sup> d c f d f- M b a f b f Ratio Factor*

With the same data as used in Table 2, the uncertainty budget is listed in Table 3 with

Quant. Estim. Standard probability Sensitivity Uncertainty

*X*i *x*i *u*(*x*i) *c*i *u*i*(y) K*Std 0.9894 0.0012 normal 0.9996 0.0012 *P*DUT 1.0158 0.0018 normal 0.9737 0.0018 *P*Std 1.0021 0.0004 normal -0.9870 -0.0004 DUTmag 0.0600 0.0120 normal -0.4613 -0.0055 DUTPhase 3.1416 1.5709 normal 0.0000 0.0000 Stdmag 0.0300 0.0060 normal 0.4581 0.0027 StdPhase 3.1416 1.5709 normal 0.0000 0.0000 Gmag 0.2300 0.0460 normal -0.0606 -0.0028 Gphase 3.1416 1.5709 normal 0.0000 0.0000 *y=K*<sup>U</sup> 0.9890 0.0071

*K c c e d e M a- a e- b e*

*Std- g D*

Im

*DUT*

*DUT*

*Ratio Factor*

mismatch corrections considered in the calibration equation.

Im

Re

3. The sensitivity coefficient for Γ*Std-Img* (= *b*)

4. The sensitivity coefficient for Γ*DUT-Re* (= *c*)

5. The sensitivity coefficient for Γ*DUT-Img* (= *d*)

6. The sensitivity coefficient for Γ*G-Re* (= *e*)

Re

7. The sensitivity coefficient for Γ*G-Img* (= *f*)

Im

**3.2.2.3 Example** 


Table 3. Uncertainty budget at 18 GHz for measurement uncertainty improvement with mismatch correction for Fig. 2 measurement setup. Uncertainties *ui* are at one standard deviation. Powers are measured in mW.

Compare the result in Table 3 with that in Table 2, the uncertainty is improved by mismatch correction. In the calculation, the magnitude uncertainty of reflection coefficient is assumed to be 40% of its value and the phase uncertainty is assumed to be 180⁰ and value is *π* for all. In terms of computation cost, it is the same for both Cartesian and Polar coordinate representations. Note that additional uncertainties should also be included such as the connector repeatability, noise, cable flexibility, drift, linearity and frequency error when they are not negligible in practical application.
