**4.7 Measurement system for 2.4mm connection and comparison**

The measurement setup for 2.4mm connector (up to 50 GHz) is illustrated in photo in Fig. 8. The calibration result is illustrated in Fig. 9. For uncertainty evaluation, the parameters used in the MCM are listed in Table 5. And the two columns in Table 6 compare the results obtained from GUM and MCM for selected frequency points.

Using Monte Carlo method, we generate a graphical approximation in the form of a histogram of the probability density function of the output quantity, represented in Fig. 10(a) with the frequency at 50 GHz. The uncertainty differences from the two methods for different frequency range are illustrated in Fig. 10(b). In this case, the uncertainties obtained by different methods are quite close. They can be used as the verification of the measurement uncertainty evaluations.

Fig. 8. 2.4mm connection realization of splitter based power sensor calibration by direct comparison transfer

Fig. 9. 50 MHz to 50 GHz using the direct comparison transfer. The expanded uncertainties are less than 4% for the frequency range at *k*=2.




Fig. 10. Comparison of uncertainty analyses by GUM and MCM

#### **5. Generic models for power sensor calibration**

#### **5.1 Modeling with signal flow graph**

For the sake of general purpose of application and analyses, in this section we discuss the three port structure expressed in signal flow graph. A flow graph method is particularly helpful in understanding a complex network relying on S-parameter matrix. Its correspondence with the physical behavior of the circuits allows simplification through well-founded approximations with no loss of physical insight (Bryant, 1993).

Fig. 11. (a) Generic model signal flow graph for a 3-port structure. (b) Three two-port networks removed from Generic model

Referring to Fig. 11(a), the central part is signal flow graph for a general three port components, most commonly used are splitter and coupler. Three two-port networks, G, A and B, are added for the generic analysis. The signal source connected to port 1 is denoted by ܾ௦.

$$\frac{a\_X}{a\_Y} = \frac{a\_X}{b\_s} \times \frac{b\_S}{a\_Y} = \frac{k\_2}{k\_3} \left[ \frac{S\_{21}(1 - \Gamma\_r \cdot \mathbb{S}\_{33}) + \mathbb{S}\_{31} \cdot \Gamma\_r \cdot \mathbb{S}\_{23}}{\mathbb{S}\_{31}(1 - \Gamma\_d \cdot \mathbb{S}\_{22}) + \mathbb{S}\_{21} \cdot \Gamma\_d \cdot \mathbb{S}\_{32}} \right] = \left(\frac{k\_2}{k\_3}\right) \left(\frac{\mathbb{S}\_{21}}{\mathbb{S}\_{31}}\right) \left(\frac{1 - \Gamma\_r \left(\mathbb{S}\_{33} - \frac{\mathbb{S}\_{31} \cdot \mathbb{S}\_{23}}{\mathbb{S}\_{21}}\right)}{1 - \Gamma\_d \left(\mathbb{S}\_{22} - \frac{\mathbb{S}\_{21} \cdot \mathbb{S}\_{31}}{\mathbb{S}\_{31}}\right)}\right)$$

$$= \left(\frac{k\_2}{k\_3}\right) \left(\frac{\mathbb{S}\_{21}}{\mathbb{S}\_{31}}\right) \left(\frac{1 - \Gamma\_r \Gamma\_{a3}}{1 - \Gamma\_d \Gamma\_{a2}}\right) \tag{37}$$

$$P\_m = \|\boldsymbol{b}\_i\|^2 \times \mathsf{K} \tag{38}$$

$$\frac{P\_{DUT}}{P\_{3DUT}} = \frac{\left\|\left.\boldsymbol{b}\_{2DUT}\right\|^2 \times \mathbf{K}\_{DUT}}{\left\|\left.\boldsymbol{b}\_{3DUT}\right\|^2 \times \mathbf{K}\_{3DUT}}\right\|}$$

$$K\_{DUT} = K\_{3DUT} \times \frac{P\_{DUT}}{P\_{3DUT}} \times \frac{|\left|\boldsymbol{b}\_{3DUT}\right|^2}{|\left|\boldsymbol{b}\_{\gamma DIT}\right|^2} \tag{39}$$

$$K\_{Std} = K\_{3Std} \times \frac{P\_{Std}}{P\_{3Std}} \times \frac{|b\_{3Std}|^2}{|b\_{2Std}|^2} \tag{40}$$

$$K\_{DUT} = K\_{Std} \times \frac{p\_{DUT}}{p\_{3DUT}} \times \frac{|b\_{3DUT}|^2}{|b\_{2DUT}|^2} \times \frac{p\_{3Std}}{p\_{Std}} \times \frac{|b\_{2Std}|^2}{|b\_{3Std}|^2} \tag{41}$$

$$\frac{a\_2}{a\_3} = \frac{a\_X}{a\_Y} = \left(\frac{k\_2}{k\_3}\right) \left(\frac{S\_{21}}{S\_{31}}\right) \left(\frac{1 - \Gamma\_r \Gamma\_{d3}}{1 - \Gamma\_d \Gamma\_{d2}}\right) \tag{42}$$

$$K\_{DUT} = K\_{Std} \times \frac{p\_{DUT}}{p\_{3DUT}} \times \frac{p\_{3Std}}{p\_{Std}} \times \left| \frac{k\_{2Std}}{k\_{2DUT}} \right|^2 \times \left| \frac{1 - \Gamma\_{DUT} \Gamma\_{e2}}{1 - \Gamma\_{Std} \Gamma\_{e2}} \right|^2 \tag{43}$$

$$\frac{P\_{DUT}}{P\_{3Std}} = \frac{|\,\,b\_{2DUT}\,\,\vert^2 \times \mathcal{K}\_{DUT}}{|\,\,b\_{3Std}\,\,\vert^2 \times \mathcal{K}\_{3Std}\,\,\vert}$$

$$K\_{DUT} = K\_{3Std} \times \frac{P\_{DUT}}{P\_{3Std}} \times \frac{|\left|b\_{3Std}\right|^2}{\left|b\_{2DUT}\right|^2} \tag{44}$$

$$K\_{DUT} = K\_{Std} \times \frac{p\_{DUT}}{p\_{3DUT}} \times \left| \frac{k\_{3Std}}{k\_{2DUT}} \right|^2 \times \left| \frac{S\_{31}}{S\_{21}} \right|^2 \times \left| \frac{1 - I\_{DUT} \Gamma\_{\ell 2}}{1 - \Gamma\_{3Std} \Gamma\_{\ell 3}} \right|^2 \tag{45}$$

$$\Gamma\_{EG} = S\_{22} - S\_{21}\frac{S\_{32}}{S\_{31}}$$

$$\Gamma\_{EG} = \frac{S\_{32}}{S\_{31}}$$

$$\Gamma\_{EG} = \frac{S\_{32}}{S\_{31}}$$

$$\Gamma\_{EG} = \frac{S\_{32}}{S\_{31}}$$

$$\Gamma\_{EG} = \frac{S\_{32}}{S\_{31}}$$

$$\Gamma\_{EG} = \frac{S\_{32}}{S\_{31}}$$

$$\Gamma\_{EG} = \frac{S\_{32}}{S\_{31}}$$

$$\Gamma\_{EG} = \frac{S\_{32}}{S\_{31}}$$

$$\Gamma\_{EG} = \frac{S\_{32}}{S\_{31}}$$

$$\Gamma\_{EG} = \frac{S\_{32}}{S\_{31}}$$

$$\Gamma\_{EG} = \frac{S\_{32}}{S\_{31}}$$

$$\Gamma\_{EG} = \frac{S\_{32}}{S\_{31}}$$

$$\Gamma\_{EG} = \frac{S\_{32}}{S\_{31}}$$

$$\Gamma\_{EG} = \frac{S\_{32}}{S\_{31}}$$

$$\Gamma\_{EG} = \frac{S\_{32}}{S\_{31}}$$

$$\Gamma\_{EG} = \frac{S\_{32}}{S\_{31}}$$

$$\Gamma\_{EG} = \frac{S\_{32}}{S\_{31}}$$

$$\Gamma\_{EG} = \frac{S\_{32}}{S\_{31}}$$

$$\Gamma\_{EG} = \frac{S\_{32}}{S\_{31}}$$

$$\Gamma\_{EG} = \frac{S\_{32}}{S\_{31}}$$

$$\Gamma\_{EG} = \frac{S\_{32}}{S\_{31}}$$

32

Fig. 12. Rectangular waveguide power sensor calibration system setup

One possible way of determining ડ*EG* is to measure the splitter's complex S-parameters and calculate its value from equation (27). The measurement uncertainty in the S-parameters is then evaluated (EURAMET, 2011). Reference (Ridler and Salter, 2001) presented the law of propagation of uncertainty using matrix notation, treating the complex quantities with real and imaginary evaluation.

Since the calculation method from S-parameters is sensitive to small measurement errors, several different measurement methods were proposed, such as the "passive open circuit" and "active open circuit" method (Moyer, 1987), and the direct calibration method (Juroshek,1997). In the direct calibration method, the splitter is connected through ports 1 and 3 to a VNA. This effectively gives a new one port VNA at splitter port 2. This new VNA is calibrated using a one port calibration algorithm, e.g. short-open-load. The ડ*EG* is then obtained as one of the three one-port VNA error terms. References (Rodriguez, 2000; Yhland & Stenarson, 2007) assessed the measurement uncertainty and traceability in power splitter effective source reflection coefficient. Reference (Furrer, 2007) compared direct calibration method with the calculation method. It seems that the similar results were obtained.
