Home > ANSYS HFSS 教學 > Frequency Dependent Materials

 

This article is available from 2014. It is intended to introduce the characteristic of frequency dependent material：a dielectric material with constant permittivity and loss tangent will cause non-causality issue.

1. Introduction

2. Djordjevic-Sarkar Model

3. Debye Model

4. Check and Enforce Causality

5. Dielectric Model in SIwave

6. Dielectric Model in HFSS and Q3D

7. 問題與討論

   7.1 What is loss tangent?

   7.2 Is there another way to check causality?

   7.3 How to plot epsilon and loss tangent in HFSS?

   7.4 Does loss tangent increase or decrease with frequency?

   7.5 How does the "Auto. use causal materials" affect materials?  (重要)

   7.6 Step2.2中，為何頻率在w_B=1GHz以上，e_r並沒有維持在e_¥，而繼續變小?

8. Reference

1. Introduction 

過去可能您使用的介電係數(real part of relative permittivity, dielectric constant, D_k)與介質損耗(loss tangent, the ratio of the imaginary to the real, D_f)都是常數。比方PCB板廠提供的板材資訊是FR4 D_k=4.2、D_f=0.018，但較專業的板廠會告訴你D_k=4.2、D_f=0.018@1GHz。後面這表示法就很清楚的"明示"了：D_k與D_f是隨頻率變化的，板廠提供的數據只是基於1GHz所量到的特性。

如果做寬頻的電磁模擬(for SI or EMC)，沒有輸入loss tangent (D_f=0)，那表示介質材料是ideal (lossless無損材料)，那當然不對。但即使你輸入了D_k與D_f，而以常數表示(不是一個隨頻率變化的函數)，那也不對，這會引起數值模擬上所謂"因果性"問題(non-causal issue)。

因果性又分"原始因果性"(primitive causality)與"相對因果性"(relativistic causality)，前者意指：系統的輸出出現(response)時間不可能比系統輸入發生時間早。後者則更嚴謹，意指：系統的輸出出現時間不可能比系統輸入到達輸出端的時間還早。我們以前說過，non-passivity與non-causality是最常引發模擬不收斂的兩大原兇，所以使用模擬軟體的過程中，如何讓材料的寬頻特性能被正確的modeling非常重要。

HFSS\Q3D\SIwave中，使用Djordjevic-Sarkar與Debye Model這兩種模型來表示frequency dependent material with causality，而這兩者也有差別。

關於D_k、D_f與causality的關係，Eric Bogatin有很完整的描述[1]：

D_k、D_f must be causal, and so must have this relationship between their real and imaginary parts. This is described by the Kramers-Kronig relationship. This means that "to know the real part of the dielectric constant is to know the imaginary part." If we make the assumption that the real part decreases with the log of the frequency, then we can build a simple analytical expression for the dissipation factor D_f. (由於D_k=permittivity實部，D_f=permittivity虛部/實部；滿足causal關係的D_k、D_f表示： 一種實部已知時，虛部也就被確定了的關係, Kramers-Kronig relationship) [1]

We find that if the D_k drops with the log of the frequency, the D_f will be roughly proportional to the slope of D_k and log F. (D_f與D_k隨頻率衰減的斜率大致呈比例 ，即loss越大，eplison e_r隨頻率衰減越快)

It's the dispersion that causes increased rise time degradation, above and beyond what we expect from just the losses. If your simulator does not include this frequency dependent dielectric constant, you will under estimate the rise time and actual performance may be worse than you predict. (如果模擬時沒有考慮causality，那實際的 the rise time and performance可能會比模擬值worse)

2. Djordjevic-Sarkar Model 

2.1 D-S model is good for SI application. It is a causal frequency-dependent dielectric model developed specifically to model the FR-4 epoxy resin material. It is also useful for many other low-loss insulator materials.

• D-S model需要輸入高低兩個頻點(corner frequency, w_A, w_B)

• 低頻的相對介電系數e_r一定比optical permittivity e_¥高

• 相對介電係數e_r在(w_A, w_B)是線性遞減的特性，之外則是常數

• e_r在線性區衰減的斜率與loss tangent成比例，loss tangent越大，e_r就 越快速下降

• Loss tangent在(w_A, w_B)頻率範圍內幾乎不變。如果DC conductivity=0，loss tangent在(w_A,w_B)外會快速衰減到0，如果DC conductivity¹0，loss tangent在DC會無窮大 。

在HFSS材料庫中，99%都是conductivity與loss tangent二擇一有值 (bakelite電木除外, 但它的導電率也是極小1e-9)

實務應用上，半導體基質(substrate) SiO2的相對介電係數e_r 大約 11，電導率bulk conductivity 大約 10 S/m

2.2 以HFSS 2014實際建一個e_r=4.4@1GHz的FR4 D-S model測試

R15以後多的[High-frequency Corner]，指的就是w_B，這值一般使用者不需更動

D-S model replaces constant relative permittivity and bulk conductivity as a frequency-dependent equation, and sets loss tangent=0. 真正的loss tangent會靠permittivity與causal關係而推出來.

模擬D-S model的e_r線性特性與step2.1所描述的相同

就算沒有使用D-S model，而是輸入constant D_k=4.4、D_f=0.02並且check "Auto. use causal materials"也會得到同樣的e_r結果。

使用D-S model，輸入constant D_k=4.4、D_f=0.02，不論有無 "Auto. use causal materials"都會得到同樣的loss tangent結果。

在D_k已被定義的條件下，若要滿足causality關係，loss tangent在當下也會被決定[1]。所以上例中，雖然一開始指定constant D_k=4.4、D_f=0.02 @1GHz，但不管是使用D-S model或check "Auto. use causal materials"後，loss tangent值會自動隨頻率變化以符合causality.

3. Debye Model 

3.1 Debye's model is valid for many microwave applications. The model can be set up by using low frequency (f₁) and higher frequency (f₂) measured dielectric constants and loss tangents and the optical dielectric constant e_¥. The accuracy of the method, especially at high frequencies, depends on the accuracy and the consistency of the measurement at frequency f₂. If the latter is not available, the model might be less accurate at high frequencies. (Debye model的高頻準度，與modeling時所輸入的f₂頻點資訊有很大關係，如果無法準確量得在該高頻處的dielectric constants and loss tangent，那Debye model在高頻的準度就較差)

Using Debye's model, causal behavior can be maintained in the transient response.

從Help說明來看，Debye model似乎蠻適合表現材料的極化(polarization)行為 ，或說e_r會隨頻率增加快速衰減的high loss material

HFSS內提供兩種Debye model：Debye model input、Multipole Debye model input，前者是single-pole，後者是multipole。[3]

如果使用single-pole Debye model(只提供高低兩個頻點)，那e_r=e'與loss tangent=(e"/e')將如 下圖所示，呈現一個窄頻段的劇烈變化，並不適合modeling low-loss material as FR4。

3.2 以HFSS 2014實際建一個e_r=4.4@1GHz的FR4 Debye model測試

3.2.1 Single-pole Debye model

從上圖結果可以看出，為何說D-S model比較適合modeling FR4 for SI application。

因為大部份PCB板廠提供的介電材料特性通常是在1GHz，如果拿1G當f₂， 這種條件下用single-pole的Debye mode來modeling FR4的效果很差，如果真的要用Debye model最好是用下一節將介紹的：multipole Debye model。

3.2.2 Multipole Debye model (建議至少六個點)

multipole Debye model看來蠻好用的，可以事先評估(D_k, D_f)經過causal fitting後的結果，比較麻煩的是最好要提供至少六個頻點的(D_k, D_f)。另外也發現，multipole Debye model對於(D_k, D_f)值都會調整，讀者可以比較上圖左右(D_k, D_f)值

4. Check and Enforce Causality 

以下所指的都是"原始因果性"(primitive causality)。

4.1 Check Causality (and avoid non-causal)

4.1.1 For HFSS and Q3D

Use [Tools] \ [Network Data Explorer] NDE to check causality as do it in Designer.

NDE has been available in HFSS and Q3D since R15.

Check "Auto. use causal materials" in [HFSS] \ [Design Settings] \ [Lossy Dielectric] tab, and make sure loss tangent>0

4.1.2 For SIwave

Use the dielectric materials with loss tangent>0, or D-S model dielectric materials for FR4.

SIwave has no NDE so far.

SIwave can check and compensate for primitive causality problems. But SIwave cannot determine the length of the device from its scattering parameters, and so cannot detect violations of relativistic causality.

4.1.3 For Designer

Use [Tools \ Network Data Explorer] NDE to check causality.

Designer state-space algorithm is automatically primitive causal but not relativistic causal.

4.2 Enforce Causality

4.2.1 For HFSS and Q3D

Use [Tools \ Network Data Explorer] NDE to enforce causality as do it in Designer.

4.2.2 For SIwave

4.2.3 For Designer

Use [Tools \ Network Data Explorer] NDE to enforce causality.

State-Space fitting algorithm會確保primitive causal，所以"Enforce causality"默認是不用勾的，除非不勾選時Export的model有問題，那才勾選它

5. Dielectric Model in SIwave 

For the differential pair：trace width=7 mils, space=8 mils, trace length=1000mils, change layer from top to bottom once.

若提供1GHz的Epsilon、Loss Tangent定義，Debye model做寬頻modeling時，高頻會與D-S model有差異；若提供高頻點(8GHz)的Epsilon、Loss Tangent定義，Debye model做寬頻modeling時，低頻會與D-S model有差異，故Debye model不適合在整個寬頻modeling low-loss dielectric material。New version SIwave will not support Debye model, because it is not a good match (accuracy) for SI application.

R18, R17.x新版本的SIwave solver已經不再支持這種model (solve with error message)，UI還可以看到是為了跟舊的project相容所保留。未來你只要在SIwave內用Debye model，它會自動轉成D-S model。

6. Dielectric Model in HFSS and Q3D 

7. 問題與討論

7.1 What is loss tangent? 

Ans：The loss tangent is the ratio at any particular frequency between the real and imaginary parts of the impedance of the capacitor. A large loss tangent means you have a lot of dielectric absorption. If you construct a capacitor C from a lossy dielectric, the dielectric absorption causes the value of C to change with frequency.

7.2 Is there another way to check causality? 

Ans：如[4] p.6~8所示，透過觀察TDR是否在T=0之前有響應

7.3 How to plot epsilon and loss tangent in HFSS? 

Ans：透過輸入數學公式plot，可以做到。有興趣的讀者請逕洽ANSYS原廠

7.4 Does loss tangent increase or decrease with frequency? 

Ans：Loss tangent is almost constant with frequency for the material what you specified. It will a little bit increase at the first, then a little bit decrease. It is physical, and the losses should saturate. [3]

7.5 How does the "Auto. use causal materials" affect materials?   (重要)

HFSS \ Design Settings

眼尖的讀者應該有發現，上圖Design Settings\Lossy Dielectrics的文字已經講明，這裡就是套用D-S model，所以我們在setp2.2做出的結果一點也不意外。但它一下說cause loss tangent greater than zero, 一下又說loss tangent will be zero，這在講甚麼鬼呀?

它第一段文字說的就是step2.2中最後跑出的e_r、loss tangent結果。而它第二段文字說的，就是本文step2.2中所述：D-S model replaces constant relative permittivity and bulk conductivity as a frequency-dependent equation, and sets loss tangent=0.

7.5.1 for dielectric material with constant epsilon and loss tangent both

會改變(覆蓋)原有的相對介電係數與loss tangent (constant, non-causal)，讓它們隨頻率變化，並且是causal。在此同時，C matrix result of LastAdaptive會隨著solution frequency高低不同而略有不同，因為不同求解頻點的epsilon與loss tangent已經不同。

7.5.2 for dielectric material with constant epsilon only (no loss tangent value)

對材料特性沒有任何影響，仍然是non-causal

這點要特別注意，HELP內有提到：If the material itself is not a lossy dielectric, Automatic causal material calculations are not performed。也就是說如果你一開始的材料loss tangent就沒有值(=0)，那Auto. use causal material將不會active

7.5.3 for dielectric material with D-S model

如果介質材料已經使用D-S model，此時會ignore "Auto. use causal materials" setting

7.5.4 for the material with non-zero conductivity and loss tangent both

Conductivity will be ignored if  "Auto. use causal materials" is checked

7.6 Step2.2中，為何頻率在w_B=1GHz以上，e_r並沒有維持 在e_¥，而繼續變小?  

Ans：首先，我們試一下在[Properties at DC]指定值，看來這對於我們想在高頻與低頻處看到平坦的曲線沒什幫助

查詢HELP發現，both of epsilon and loss tangent curves have two transition frequencies, the first around 1kHz and the second around 100GHz.

D-S model定義特性的低高兩頻點分別是w_A@1kHz、w_B@100GHz (measurement frequency 1G *10 times at least)。w_B並非我們在step2.2輸入的1GHz，而是R15以後出現的High-frequency corner那一項。In general, please keep it as 1e+12/(2*p) by default。

所以D-S model在1k~100GHz整個寬頻帶內，都呈現Epsilon隨頻率指數增加而線性下降的特性。反觀"single-pole Debye model"則不然，其lower freq.與upper freq.都是user可直接指定，且只能在一個窄頻帶內讓Epsilon隨頻率指數增加而線性變化。

8. Reference 

[1] Eric Bogatin, "Frequency Dependent Material Properties：So What?", Speech Video 2010, 1hourse 20min, DesignCon2010.

[2] Eric Bogatin, "Which one is better? Comparing Options to Describe Frequency Dependent Losses", DesignCon2013.

[3] Electric Properties of Dielectrics, p.3, 7-9 台大吳瑞北教授SI課程

[4] In-Situ De-embedding (ISD) p.6~8 from AtaiTec Corp.

[5] Permittivity