臺灣博碩士論文加值系統

本論文主要提出一從量測之介電常數估計介質損耗之方法。利用量測
所得之介電常數透過克拉莫-克若尼關係式得到介電係數之虛數部，並且針對克
拉莫-克若尼關係式在高頻數據不足時所產生之誤差進行探討與修正。此外本論
文亦整理四種常用之介電係數量測方法，分別為:帶狀線、環形共振器、微帶線
共振器、基板合成波導共振腔，並使用量測數據帶入本文提出之估計方法與其
他四種方法做比較以驗證其可行性。
傳統介電常數與損耗因數之量測，因損耗直接量測困難，必須經由計算金
屬損耗以及輻射損耗獲得介質損耗，然而考慮電路、金屬材質、製程等等…的
不同，計算金屬之衰減係數時常需使用近似值，導致計算金屬損耗產生誤差。
本文將使用克拉莫-克若尼關係式，由介電係數之實部估算虛部，以獲得板材之
損耗因數，而不需考慮金屬之損耗。此外，不同於共振器之量測方法僅能獲得
單一頻率點，也就是共振點之介電係數與損耗，本文提出之方法可計算出一寬
頻結果。然而克拉莫-克若尼關係式必須符合積分範圍從零到無窮，但顯然量測
所得之數據並無法涵蓋所有頻率，因此量測數據需要根據合理的介電係數模型
進行外插，本文參考德拜模型，建立出高頻資訊，相較於直接使用有限數據計
算，可得到較為準確之結果。另一方面，克拉莫-克若尼關係式的積分式存在一
個奇異點，當積分頻率靠近此點，計算值將會發散。但透過分部積分法將克拉
莫-克若尼關係式分解成三個部分之後，將可把其中發散的部分獨立出來(此為
常數)，因此可得知，在數據帶入克拉莫-克若尼關係式之前減去此常數，即可
得到準確的結果。並且此一特性亦可使用德拜模型解釋。
最後，為驗證本文所提出的估計方法，使用帶狀線之介電常數量測結果，
透過克拉莫-克若尼關係式，計算獲得一寬頻損耗後與三種共振腔之量測值做比
較，並討論其誤差比率。

In this thesis, a method which can estimate dissipation factor (Df) from dielectric constant (Dk) is proposed. By using measured dielectric constant, the imaginary part of permittivity can be obtained through Kramers-Kronig relation. Also, the error generated by lacking data at high frequencies in truncated Kramers-Kronig relation would be revised and discussed in detail. Furthermore, four methods, including transmission coefficient extraction of both strip line (SL) and substrate integrated waveguide (SIW) from through, reflection and line (TRL) method , microstrip line ring resonators and open resonators which are commonly applied to measure Dk and losses, are implemented to validate the accuracy and reliability of proposed method, and to verify its feasibility.
Since most of printed circuit board (PCB) based measurements for permittity can only directly obtain the dielectric constant and total losses (with dielectric loss, radiation loss and conductor loss together), researchers must calculate the conductor loss and radiation loss in order to find out dielectric loss. However, in many cases, conductor losses will depend on different circuit schematics, conductor materials, manufacturing progresses and so on. The calculation of conductor losses usually refers to some approximation theories and therefore the dielectric loss cannot be obtain precisely.
Unlike these methods, the proposed method calculates the dielectric loss from dielectric constant through Kramers-Kronig relationsince losses can only slightly affect the results of DK. Besides, it can also measure DF in a broad frequency band, which is different from the measurements with resonators. Nevertheless, there are problems when utilizing Kramers-Kronig relation. First, the Kramers-Kronig relation should have the limits of integration from zero to infinite, but in reality only limited frequency data can be acquired. The truncated KK relation will diverge and include numerous error.
Therefore, the data should be modified to eliminate the divergent terms. Second, the contribution of outband data should be considered or the results will be much smaller at the edge of frequency bands. Therefore, the extrapolation should be applied with a reasonable model to estimate dielectric constant in high frequencies. In our method, muti-debye model is investigated to create data in high frequencies ,and the estimated results can be more precise in comparison to the results adopting truncated data. Last but not the least, , the equation of Kramers-Kronig relation exists a singular point. When integrating through frequency, the singular point causes disperse. However, through integration by part, the equation can be split into three parts. Then, the dispersion part can be isolated (this part would be a constant). Finally, to verify the propose method, the measurement results of strip line is adopted. The dissipation factor of wide frequency band is obtained through Kramers-Kronig relation, and then it is compared with three measurement results from different resonators in order to discuss the errors.

口試委員會審定書 #
中文摘要 i
ABSTRACT ii
CONTENTS iv
LIST OF FIGURES vi
LIST OF TABLES viii
Chapter 1 Introduction 1
1.1 Research Motivation 1
1.2 Literature Review 3
1.3 Contribution 5
1.4 Thesis Organization 6
Chapter 2 Theory 7
2.1 Dielectric constant 7
2.2 Dissipation factor 9
2.3 Causality and Kramers-Kronig relation 11
2.4 Different form of Kramers-Kronig relations 16
2.5 Model of permittivity with dielectric material 17
2.6 Verification of Coding 24
Chapter 3 Method to extracting dielectric loss through Kramers-Kronig relations 27
3.1 Introduction 28
3.2 Operation with simulation data 30
3.3 Model fitting and results 35
3.4 Measured data without satisfying Debye model 41
Chapter 4 Comparison and discussion 43
4.1 Method of Kramers-Kronig relations 43
4.1.1 Introduction 43
4.1.2 Design of strip line structure 43
4.1.3 Measurement results 45
4.2 Traditional methods 48
4.2.1 Ring resonator 48
4.2.2 Substrate integrated waveguide 57
4.2.3 Strip line 66
4.2.4 Open cavity 70
4.3 Comparison and Discussion 75
Chapter 5 Conclusion 78
REFERENECE 79

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