**4.1 Benefit**

The coaxial splitter based calibration setup is illustrated in Fig. 4. By using coaxial splitter in the direct comparison transfer method of RF and microwave power sensor calibration, the measurement accuracy improves from source mismatch effect and the load / device under test (DUT) mismatch effects. The explanations are as follows:

a. Improving source match is achieved by holding effective source output power constant. When a leveling loop or ratio sensor is employed, port 2 of the power splitter becomes the effective source output. *ΓEG* is the equivalent source reflection coefficient rather than *ΓG*, as illustrated in Fig. 5(a). It solves the measurement trouble of *ΓG* for signal generator by obtaining passive component splitter S-parameters. The well known equivalent source reflection coefficient equation for power splitter

$$
\Gamma\_{EG} = \mathcal{S}\_{22} \text{ - } \mathcal{S}\_{21} \frac{\mathcal{S}\_{32}}{\mathcal{S}\_{31}} \tag{27}
$$

will be further discussed in section 6.

186 Modern Metrology Concerns

A leveling circuit or ratio measurement can avoid the troubles. Resistive power splitter or directional coupler is normally used as the intermediate component and constructing the

**3.3.3 Splitter vs divider vs Tee: Scattering parameter matrix and suitable applications**  Before going further analyses, it is time to say a few words about the power splitter (Johson, 1975), power divider and Tee since even some experienced engineers misuse them and not clear of the differences among them fundamentally. The physical and mathematical

As illustrated in Fig. 3 for a) power divider; b) power splitter; and c) Tee, there are three

The three resisters in power divider are *Z0/3* each; for 50 ohm system, they are 16-2/3 Ω.

The power splitter has fixed input at port one; power divider and Tee inputs are

Substantially the scattering parameter matrix or S-parameter matrix are different for these three 3-port components as shown under the physical structures in Fig. 3; in which the S-

Power splitter is used in leveling or ratio measurement; power divider is used for simple power division. Tee is used in low frequency case and in those non-critical measurements.

Fig. 3. Three 3-port components and their corresponding scattering parameter matrix

**4. Coaxial splitter based power sensor calibration by direct comparison** 

test (DUT) mismatch effects. The explanations are as follows:

The coaxial splitter based calibration setup is illustrated in Fig. 4. By using coaxial splitter in the direct comparison transfer method of RF and microwave power sensor calibration, the measurement accuracy improves from source mismatch effect and the load / device under

resistors for power divider; two resistors for power splitter; and no resistors for Tee.

leveling circuit.

expressions are given here.

exchangeable (bi-directional).

**transfer 4.1 Benefit** 

Two resistors in power splitter are *Z0* each.

parameter matrixes are for ideal cases.

Fig. 4. Coaxial splitter based power measurement setup


The source mismatch effect can be read through the monitoring arm, and so do the load / DUT mismatch effects. The system setup and mathematical model is established based on the complex reflection coefficients. Analyses have been provided through their S-parameter matrix to perform full mismatch corrections (Weidman, 1996; Juroshek, 2000; Ginley, 2006; Crowley, 2006; Shan et al., 2008; Shan et al., 2010b). Here we provide models for different cases in the application.

Since the splitter is not perfect, we seek to characterize them so that the mismatch error can be minimized by mathematical correction. The computation involved is not trivial, but the benefits of the technique are considerable, in that accurate measurements are made across a broad band without the need for mechanical adjustment.

Fig. 5. Benefits by inserting coaxial power splitter

#### **4.2 Modeling**

Models of the three cases are considered for the system setup illustrated in Fig. 4.

Case 1 – obtain *ηDUT* from *ηStd*:

$$\eta\_{\rm DUT} = \eta\_{\rm Std} \times P\_{\rm RATIO} \times MM = \eta\_{\rm Std} \times \frac{P\_{\rm DUT}}{P\_{\rm Std}} \frac{P\_{3\rm Std}}{P\_{3\rm DJT}} \times \frac{1 \cdot \left| \Gamma\_{\rm Std} \right|^2}{1 \cdot \left| \Gamma\_{\rm DJT} \right|^2} \frac{\left| 1 \cdot \Gamma\_{\rm DJT} \Gamma\_{\rm EG} \right|^2}{\left| 1 \cdot \Gamma\_{\rm Std} \Gamma\_{\rm EG} \right|^2} \tag{28}$$

The *Γ*Std, *Γ*DUT and *Γ*EG are the complex value reflection coefficients for the standard, DUT and equivalent signal generator respectively as indicated in Fig. 4. The power ratio is different from equation (3), which is caused by using splitter with monitoring arm. And here *ΓEG* is used instead of *ΓG*, different from simple direct comparison transfer. The value of *ΓEG* is obtained by equation (27).

Case 2 – obtain *KDUT* from *KStd*:

$$K\_{DLT} = K\_{Std} \times \frac{P\_{DLT}}{P\_{Std}} \frac{P\_{3Std}}{P\_{3DLT}} \times \frac{\left| \mathbf{1} \cdot \Gamma\_{DLT} \Gamma\_{EG} \right|^2}{\left| \mathbf{1} \cdot \Gamma\_{Std} \Gamma\_{EG} \right|^2} \tag{29}$$

Case 3 – obtain *KDUT* from ηStd:

$$K\_{DIT} = \eta\_{Std} \times \frac{P\_{DIT}}{P\_{Std}} \frac{P\_{3Sld}}{P\_{3DIT}} \times \left(1 \cdot \left| \begin{array}{c} \Gamma\_{Std} \end{array} \right|^2 \right) \times \frac{\left| \begin{array}{c} \mathbf{1} \cdot \Gamma\_{DIT} \Gamma\_{EG} \end{array} \right|^2}{\left| \mathbf{1} \cdot \Gamma\_{Std} \Gamma\_{EG} \right|^2} \tag{30}$$

Case 3 is the practical application calibration equation in national metrology institutes. The obtained value from a primary standard is effective efficiency *ηStd*, the customer DUT calibration factor requests *KDUT* .

In the following sections, we separately provide both Cartesian and Polar representations for practical solution and application.

#### **4.3 Polar model**

The models for the above three cases in terms of magnitude and phase is derived and expressed as follows:

Models of the three cases are considered for the system setup illustrated in Fig. 4.

*P P - K K*

*P P - K -*

3

*DUT Std Std*

*DUT Std*

*P P - - P MM*

The *Γ*Std, *Γ*DUT and *Γ*EG are the complex value reflection coefficients for the standard, DUT and equivalent signal generator respectively as indicated in Fig. 4. The power ratio is different from equation (3), which is caused by using splitter with monitoring arm. And here *ΓEG* is used instead of *ΓG*, different from simple direct comparison transfer. The value of

3

*DUT Std DUT EG*

*Std DUT Std EG*

*DUT Std DUT EG*

*Std DUT Std EG*

*P P -*

*P P -*

3 2

Case 3 is the practical application calibration equation in national metrology institutes. The obtained value from a primary standard is effective efficiency *ηStd*, the customer DUT

In the following sections, we separately provide both Cartesian and Polar representations

The models for the above three cases in terms of magnitude and phase is derived and

3

3

3

*DUT Std Std DUT EG*

2 2

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

2 2

2

(30)

2

(28)

*Std DUT DUT Std EG*

2

2

*P P - -*



Fig. 5. Benefits by inserting coaxial power splitter

*DUT Std RATIO Std*

**4.2 Modeling** 

Case 1 – obtain *ηDUT* from *ηStd*:

 

*ΓEG* is obtained by equation (27). Case 2 – obtain *KDUT* from *KStd*:

Case 3 – obtain *KDUT* from ηStd:

calibration factor requests *KDUT* .

**4.3 Polar model** 

expressed as follows:

for practical solution and application.

$$\eta\_{\rm DJT} = \eta\_{\rm Sdd} \times \frac{P\_{\rm DJT}}{P\_{\rm Sdd}} \frac{P\_{\rm Sdd}}{P\_{\rm Sld}} \times \frac{\mathbf{1} \cdot \left| \Gamma\_{\rm Sdd} \right|^2}{\mathbf{1} \cdot \left| \Gamma\_{\rm DJT} \right|^2} \times \frac{\mathbf{1} + \left| \Gamma\_{\rm DJT} \right|^2 \left| \Gamma\_{\rm EG} \right|^2 \cdot \mathbf{2} \left| \Gamma\_{\rm DJT} \right| \left| \Gamma\_{\rm EG} \right| \cos(\theta\_{\rm DJT} + \theta\_{\rm EG})}{\mathbf{1} + \left| \Gamma\_{\rm Sdd} \right|^2 \left| \Gamma\_{\rm EG} \right|^2 \cdot \mathbf{2} \left| \Gamma\_{\rm Sdd} \right| \left| \Gamma\_{\rm EG} \right| \cos(\theta\_{\rm Sdd} + \theta\_{\rm EG})} \tag{31}$$

$$K\_{DIT} = K\_{Sld} \times \frac{P\_{DIT}}{P\_{Sld}} \frac{P\_{3Sld}}{P\_{3DUT}} \times \frac{1 + |\Gamma\_{DIT}|^2 |\Gamma\_{EG}|^2 \cdot \text{2} \left|\Gamma\_{DIT}\right| \left|\Gamma\_{EG}\right| \cos(\theta\_{DIT} + \theta\_{EG})}{1 + |\Gamma\_{Sld}|^2 |\Gamma\_{EG}|^2 \cdot \text{2} \left|\Gamma\_{Sld}\right| \left|\Gamma\_{EG}\right| \cos(\theta\_{Sld} + \theta\_{EG})} \tag{32}$$

$$K\_{DIT} = \eta\_{Sd} \times \frac{P\_{DIT}}{P\_{Sd}} \frac{P\_{SSt}}{P\_{SDIT}} \times \left(1 \cdot \left|\Gamma\_{Sd}\right|^2\right) \times \frac{1 + \left|\Gamma\_{DIT}\right|^2 \left|\Gamma\_{EG}\right|^2 - 2\left|\Gamma\_{DIT}\right| \left|\Gamma\_{EG}\right| \cos(\theta\_{DIT} + \theta\_{EG})}{1 + \left|\Gamma\_{Sd}\right|^2 \left|\Gamma\_{EG}\right|^2 - 2\left|\Gamma\_{Sd}\right| \left|\Gamma\_{EG}\right| \cos(\theta\_{Sd} + \theta\_{EG})} \tag{33}$$

### **4.4 Uncertainty based on polar representation by Monte Carlo method**

If using GUM to evaluate the uncertainty of measurement, it is based on propagation of uncertainties which is similar to analyses in previous section with more items. The mismatch uncertainty part is similar, just replace �*G* with �E*<sup>G</sup>*. It is seen from previous section that the sensitivity coefficients are quite tedious for complex value involved models although only first order summation of uncertainties are used. The derived sensitivity coefficients with partial differentiations with respect to each variable for the above models (31), (32) and (33) are obtained and have been numerically appreciated by making use of MATLAB. Here we discuss the Monte Carlo simulation Method (MCM). Then we compare the uncertainties by two methods and discuss the findings.

MCM is based on the propagation of distribution proposed by GUM supplement 1 (JCGM, 101:2008) instead of the GUM propagation of uncertainty method. The MCM allows one to get rid of much of the calculation of partial derivatives where analytical expressions are complex. The MCM evaluates measurement uncertainty by setting a probability function to each input quantity in the measurement equation. From a series of numerical calculations, probability density function (pdf) of the output function is obtained and the standard uncertainty is evaluated from this pdf.

The steps for applying MCM are summarized as follows:


The Monte Carlo numerical simulation is performed through program developed using MATLAB software. It allows estimating the measurement uncertainties based on the mathematical models.
