### I. Introduction

*ɛ*

*and loss tangent*

_{r}*tanδ*) of the line standard. Because PCB manufacturing exhibits high tolerance and an accurate evaluation of the dielectric properties of the materials used to fabricate a multilayer PCB is challenging, the derived characteristic impedance has significant uncertainty.

*S*-parameters of a device under test (DUT). This is because the

*S*-parameters of a DUT measured with respect to TRL calibration are referenced to the actual characteristic impedance of TRL line,

*Z*

*, which, in general, is not exactly 50 Ω. Therefore, the*

_{line}*S*-parameters of the DUT are often required to be referenced to an idealized reference impedance,

*Z*

*(which is often 50 Ω), and so the*

_{ref}*S*-parameters are renormalized from

*Z*

*to*

_{line}*Z*

*[3, 4].*

_{ref}*S*-parameters was demonstrated by employing the method described in [10]. The impact was shown as the worst-case deviation from the reference simulated

*S*-parameters.

*S*-parameters due to the uncertainty in the dimensions and dielectric properties of the TRL calibration line standard using two extraction methods: (i) a method based on repeated TRL calibrations with a randomly perturbed line standard and (ii) a method combining impedance renormalization using the equations given by Stumper [11]. Both methods employ 3D EM Monte Carlo simulation provided by the Advanced Design System (ADS) circuit simulator [12] to obtain the uncertainties in the characteristic impedance and

*S*-parameters of the TRL line standard. This study shows how to combine impedance renormalization with Stumper’s equations. Using these methods, we evaluate the uncertainty in the measured

*S*-parameters of the DUT fabricated on the multilayer PCB.

*S*-parameters based on 3D EM Monte Carlo simulation. The simulation results are presented in Section IV. The

*S*-parameters of the DUT, measured using a vector network analyzer (VNA), and the evaluation of the uncertainty in these measured

*S*-parameters, are presented in Section V.

### II. Sources of Uncertainty

*W*

*), its thickness (*

_{SL}*T*

*), thicknesses of the dielectric material (*

_{SL}*H*

_{1},

*H*

_{2}), and the dielectric properties (

*ɛ*

*,*

_{r}*tanδ*); the uncertainties in all these quantities propagate to the uncertainty in the characteristic impedance and the uncertainty in the

*S*-parameters of the DUT, subsequently.

*W*

*, we corrected the values by measuring the angle between the cut plane and the plane perpendicular to the striplines. All 26 lines on the PCB are measured, and the mean and standard deviation of the measured values were calculated. This information is summarized in Table 1 [13–15]. Using the mean values of the measured dimensions, an average value for the characteristic impedance (*

_{SL}*W*

*,*

_{SL}*T*

*,*

_{SL}*H*

_{1},

*H*

_{2},

*ɛ*

_{r}_{,}

_{M}_{1},

*ɛ*

_{r}_{,}

_{M}_{2},

*tanδ*

_{M}_{1}, and

*tanδ*

_{M}_{2}).

*H*

_{2}contains three layers of material, which are varied proportionally during the simulations, for simplicity. The number of Monte Carlo trials is selected as 1,000 for each simulation, which results in a computation time of approximately 10 hours.

### III. 3D EM Monte Carlo Simulation

*S*-parameters are then renormalized from an assumed initial reference impedance of

*Z*

_{0}) of the TRL Line standard (either mean or perturbed) is obtained from the following equation [9, 17]:

*S*-parameters are the simulated values for TRL line and

*Z*

*is the system reference impedance of the simulator, which is generally 50 Ω. This equation is applicable only to uniform lines.*

_{sys}*S*-parameters (

*δS*

_{ij}_{,}

*) using the deviation of the*

_{DUT}*S*-parameters of TRL line (

*δs*

*) obtained by randomly varying the dimensions and dielectric properties, as well as the Stumper’s equations shown below [11]:*

_{ij}*L*= exp (−

*γl*),

*l*is the TRL line length and

*γ*is its propagation constant obtained as a by-product during the TRL calibration process, Γ

*is the reflection coefficient of the TRL reflect standard, and*

_{refl}*S*

*are the DUT*

_{ij}*S*-parameters. To obtain corresponding expressions for

*δS*

_{22,}

*and*

_{DUT}*δS*

_{21,}

*, index 1 is replaced by 2 and vice versa in Eqs. (2) and (3).*

_{DUT}*S*-parameter uncertainty increases as the length of the line standard approaches

*n*times λ/2, where

*n*is an integer. In this study, a single line is used as the Monte Carlo simulation takes a long time. To obtain a lower uncertainty at low frequencies, more lines with longer length can be used.

*S*-parameters and

*Z*

_{0}of this “mean” line, which are assigned as

*S*-parameters from

*N*results (we used

*N*= 1,000 in this study). By calculating the standard deviations of the

*N*results, we obtain the uncertainty in the DUT

*S*-parameters. Each of the

*N*trials involves five separate trials, as listed in Table 3, with only one or two parameters being varied at a time. This allows the evaluation of the uncertainty caused by each contribution to be evaluated, separately.

### 1. Method based on Repeated TRL Calibration

*S*-parameters. Finally, we renormalize the DUT

*S*-parameters to 50 Ω assuming that they are initially normalized to

*N*repetitions of this process, we obtain

*N*sets of DUT

*S*-parameters. By calculating the standard deviations in these N repetitions, we can establish the uncertainties in the line

*Z*

_{0}and the DUT

*S*-parameters.

### 2. Method based on Stumper’s Equations

*S*-parameters are extracted for the perturbed line (

*S*-parameters are extracted with respect to two 50 Ω ports. However, the deviation in the

*S*-parameters of TRL line,

*δs*

*, occurs with respect to the characteristic impedance of TRL line. Therefore, we renormalize the*

_{ij}*S*-parameters from 50 Ω to

*δs*

*. Using Eqs. (2) and (3), we can calculate the deviation in the DUT*

_{ij}*S*-parameters. In the equations, Γ

*can be obtained during TRL calibration, and the DUT*

_{refl}*S*-parameters

*S*

*can use either the simulated values or the measured values, which should both be renormalized to 50 Ω. That is,*

_{ij}*S*

*can be either*

_{ij}*S*-parameter after renormalization from

*N*repetitions of this process, we obtain

*N*sets of deviations in the DUT

*S*-parameters (

*δS*

_{ij}_{,}

*). From the standard deviations of*

_{DUT}*δS*

_{ij}_{,}

*, we obtain the uncertainties in the DUT*

_{DUT}*S*-parameters.

### IV. Simulation Results

*S*-parameters (

*N*Monte Carlo trials. We calculate

*N*characteristic impedances (

*N*sets of

*Z*

_{0}also has a small imaginary part, which is taken into account in the renormalization process.

*W*

*is dominant for the conditions listed in Table 3. This shows that, to evaluate the characteristic impedance with low uncertainty, high accuracy is required when measuring the line width.*

_{SL}*S*-parameters for the DUT (Beatty line), as shown in Fig. 8. Fig. 8 and 8(b) shows the magnitude of

*S***and**

_{11}

*S***, according to the deviation of**

_{12}

*W**, respectively. The black line indicates the result obtained from the mean values of the parameters and the gray lines indicate those obtained from the perturbed values. Finally, the uncertainty caused by TRL line can be obtained by calculating the standard deviation of the 1,000*

_{SL}*S*-parameter results at each frequency. The uncertainties of |

*S*

_{11}| and |

*S*

_{12}| are presented in Fig. 9.

*S*-parameters

*S*

*. Both sets of results agree exactly with each other. If we do not renormalize the*

_{ij}*S*-parameters of TRL line from 50 Ω to

*Z*

_{0}, there is a discrepancy between the two methods.

*s*

_{11}and

*δs*

_{12}, we can verify how each uncertainty contribution affects the uncertainty in the DUT

*S*-parameters. The sensitivity coefficient of

*δs*

_{11}for

*u*(|

*S*

_{12}|) is

*S*

_{22}·

*S*

_{12}/(1−

*L*

^{2}), from Eq. (3), which is plotted in Fig. 10(a). By multiplying the sensitivity coefficient by the uncertainty of

*δs*

_{11}, we obtain the uncertainty contribution for

*δs*

_{11}. Fig. 10(b) shows the uncertainty of

*δs*

_{11}, obtained by the Monte Carlo simulation, and Fig. 10(c) shows the uncertainty contribution for

*u*(|

*S*

_{12}|).

*S*-parameter. As expected from the results of characteristic impedance, the uncertainty caused by the width of the line,

*W**, is the dominant contribution.*

_{SL}### V. Experimental Results

*S*-parameters of the DUT (Beatty line) are shown in Fig. 12(a) and 12(b), respectively. The solid line shows the measured results, and the dotted line shows the simulated results. Because the datasheet for RO4350 does not provide the frequency dependency of permittivity in this specific multilayer structure, this simulation does not consider any change in the values of the permittivity and loss tangent with regard to frequency. This is likely to cause a slight discrepancy between the simulated and measured results shown in Fig. 12(a) and 12(b).

*S*-parameters. First, we obtain the uncertainty of the line,

*δs*

*, using the 3D EM Monte Carlo simulation with 1,000 trials, as discussed in Section III. The uncertainties of |*

_{ij}*S*

_{11}| and |

*S*

_{12}|, with respect to all parameters, are obtained from the measured DUT

*S*-parameters and

*δs*

*s. These uncertainties are presented (as standard uncertainties) in Fig. 12(c) and 12(d). Their null positions are matched with those of the measured*

_{ij}*S*-parameters.

*S*-parameters of the DUT (obtained using the measured mean values) from the values obtained assuming the designed (i.e., intended) values listed in Table 1. We use the measured DUT

*S*-parameters. The deviations are generally larger than the uncertainty, which indicates the need to use the measured values for each fabricated PCB, rather than the designed (i.e., specified) values.

### VI. Conclusion

*S*-parameters. In particular, we obtain the uncertainties in the DUT

*S*-parameters by two methods—repeated TRL calibration and Stumper’s equations—and show that the two methods obtain consistent results. This study demonstrates the use of Stumper’s equations taking into account the renormalization process. We present the uncertainty contributions due to the width and thickness of the strip line, the thickness of the dielectric material, and dielectric properties (permittivity and loss tangent), and confirm that the uncertainty contribution caused by the stripline width is the dominant source of uncertainty.