3B. RF/Microwave Filters

3B. F/Microwave Filters The information in this work has been obtained from sources believed to be reliable. The author does not guarantee the accuracy or completeness of any information presented herein, and shall not be responsible for any errors, omissions or damages as a result of the use of this information. August 007 006 by Fabian Kung Wai Lee eferences []. E. Collin, Foundations for microwave engineering, nd Edition 99, McGraw-Hill. [] D. M. Pozar, Microwave engineering, nd Edition 998, John Wiley & Sons.* (3 rd Edition 005, John-Wiley & Sons is now available) Other more advanced references: [3] W. Chen (editor), The circuits and filters handbook, 995, CC Press.* [4] I. Hunter, Theory and design of microwave filters, 00, The Instutitution of Electrical Engineers.* [5] G. Matthaei, L. Young, E.M.T. Jones, Microwave filters, impedance-matching networks, and coupling structures, 980, Artech House.* [6] F. F. Kuo, Network analysis and synthesis, nd edition 966, John-Wiley & Sons. * ecommended August 007 006 by Fabian Kung Wai Lee

.0 Basic Filter Theory August 007 006 by Fabian Kung Wai Lee 3 Introduction An ideal filter is a linear -port network that provides perfect transmission of signal for frequencies in a certain passband region, infinite attenuation for frequencies in the stopband region and a linear phase response in the passband (to reduce signal distortion). The goal of filter design is to approximate the ideal requirements within acceptable tolerance with circuits or systems consisting of real components. August 007 006 by Fabian Kung Wai Lee 4

Categorization of Filters Low-pass filter (LPF), High-pass filter (HPF), Bandpass filter (BPF), Bandstop filter (BSF), arbitrary type etc. In each category, the filter can be further divided into active and passive types. In active filter, there can be amplification of the of the signal power in the passband region, passive filter do not provide power amplification in the passband. Filter used in electronics can be constructed from resistors, inductors, capacitors, transmission line sections and resonating structures (e.g. piezoelectric crystal, Surface Acoustic Wave (SAW) devices, and also mechanical resonators etc.). Filter Active filter may contain transistor, FET and Op-amp. LPF HPF BPF Active Passive Active Passive August 007 006 by Fabian Kung Wai Lee 5 Filter s Frequency esponse () Frequency response implies the behavior of the filter with respect to steady-state sinusoidal excitation (e.g. energizing the filter with sine voltage or current source and observing its output). There are various approaches to displaying the frequency response: Transfer function H() (the traditional approach). Attenuation factor A(). S-parameters, e.g. s (). Others, such as ABCD parameters etc. August 007 006 by Fabian Kung Wai Lee 6 3

Filter Frequency esponse () Low-pass filter (passive). V () A Filter H() V () Z L H() Transfer function ( ) ( ) V ( ) V Complex value H = (.a) Arg(H()) c A()/dB 50 40 30 0 0 3 0 c eal value = V A 0Log 0 V Attenuatio n c ( ) ( ) (.b) August 007 006 by Fabian Kung Wai Lee 7 Filter Frequency esponse (3) Low-pass filter (passive) continued... For impedance matched system, using s to observe the filter response is more convenient, as this can be easily measured using Vector Network Analyzer (VNA). Z c V s a b Z c Z c Z c Filter Z c Z c 0dB 0log s () Arg(s ()) Transmission line is optional c b b s = s = a a a 0 = a= 0 Complex value August 007 006 by Fabian Kung Wai Lee 8 4

Filter Frequency esponse (4) Low-pass filter (passive) continued... 50 40 A()/dB Passband Transition band 30 0 0 3 0 c Stopband Cut-off frequency (3dB) V () A Filter H() V () Z L August 007 006 by Fabian Kung Wai Lee 9 Filter Frequency esponse (5) High-pass filter (passive). H() Transfer function 50 40 A()/dB Passband 30 0 0 c 3 0 c Stopband August 007 006 by Fabian Kung Wai Lee 0 5

Filter Frequency esponse (6) Band-pass filter (passive). Band-stop filter. A()/dB A()/dB 40 40 30 0 30 0 0 3 0 o 0 3 0 o H() Transfer function H() Transfer function o o August 007 006 by Fabian Kung Wai Lee Basic Filter Synthesis Approaches () Image Parameter Method (See [4] and []). Z o Z o Filter Z o Z o Z o Z o Consider a filter to be a cascade of linear -port networks. Synthesize or realize each -port network, so that the combine effect gives the required frequency response. The image impedance seen at the input and output of each network is maintained. Z o esponse of a single network H () H () H n () Z o The combined response August 007 006 by Fabian Kung Wai Lee 6

Basic Filter Synthesis Approaches () Insertion Loss Method (See []). Filter Z o H() Use CLM circuit synthesis theorem ([3], [6]) to come up with a resistive terminated LC network that can produce the Z approximate response. o Approximate ideal filter response With polynomial function: Ideal Approximate with rational polynomial function s n + a s n ( ) n + + a s+ a H s = K L o s n + s n bn + L+ b s+ bo We can also use Attenuation Factor or s for this. Z o August 007 006 by Fabian Kung Wai Lee 3 Our Scope Only concentrate on passive LC and stripline filters. Filter synthesis using the Insertion Loss Method (ILM). The Image Parameter Method (IPM) is more efficient and suitable for simple filter designs, but has the disadvantage that arbitrary frequency response cannot be incorporated into the design. August 007 006 by Fabian Kung Wai Lee 4 7

.0 Passive LC Filter Synthesis Using Insertion Loss Method August 007 006 by Fabian Kung Wai Lee 5 Insertion Loss Method (ILM) The insertion loss method (ILM) allows a systematic way to design and synthesize a filter with various frequency response. ILM method also allows filter performance to be improved in a straightforward manner, at the expense of a higher order filter. A rational polynomial function is used to approximate the ideal H(), A() or s (). Phase information is totally ignored. Ignoring phase simplified the actual synthesis method. An LC network is then derived that will produce this approximated response. Here we will use A() following []. The attenuation A() can be cast into power attenuation ratio, called the Power Loss atio, P L, which is related to A(). August 007 006 by Fabian Kung Wai Lee 6 8

More on ILM There is a historical reason why phase information is ignored. Original filter synthesis methods are developed in the 90s-60s, for voice communication. Human ear is insensitive to phase distortion, thus only magnitude response (e.g. H(), A()) is considered. Modern filter synthesis can optimize a circuit to meet both magnitude and phase requirements. This is usually done using computer optimization procedures with goal functions. August 007 006 by Fabian Kung Wai Lee 7 Power Loss atio (P L ) Z s V s P L = = P inc PLoad Lossless -port network P A P in P L Γ () Power available from source network Power delivered to Load P = A = PA Γ ( ) Γ ( ) (.a) Z L P L large, high attenuation P L close to, low attenuation For example, a low-pass filter response is shown below: P L (f) High attenuation Low-Pass filter P L f c August 007 006 by Fabian Kung Wai Lee 8 0 Low attenuation f 9

P L and s In terms of incident and reflected waves, assuming Z L =Z s = Z C. b a b V s Z c Lossless -port network P A P in P L Z c a a P PA L = = = PL b b P L = s (.b) August 007 006 by Fabian Kung Wai Lee 9 P L for Low-Pass Filter (LPF) Since Γ () is an even function of, it can be written in terms of as: Γ M ( ) ( ) = M ( ) + N ( ) P L can be expressed as: P L = Γ M = = + ( ) M N M + N Various type of polynomial functions in can be used for P(). The requirement is P() must be either odd or even function. Among the classical polynomial functions are: Maximally flat or Butterworth functions. Equal ripple or Chebyshev functions. Elliptic function. Many, many more. (.) P( ) P( ) This is also known as Characteristic Polynomial [ P( )] M [ ( )] ( ) P = P L = + ( ) August 007 006 by Fabian Kung Wai Lee 0 N (.3a) (.3b) The characteristics we need from [P()] for LPF: [P()] 0 for < c [P()] >> for >> c 0

Characteristic Polynomial Functions Maximally flat or Butterworth: Equal ripple or Chebyshev: CN ( ) Bessel [6] or linear phase: For other types of polynomial functions, please refer to reference [3] and [6]. BN ( s) P ( ) N = c P = εcn, ε = ripple C0( ) = C( ) = C ( ) = C ( ) C ( ) = n ( ) ( ) factor n n [ P( ) ] = B( j ) B( j ) B 0 = B Bn N = order of the Characteristic Polynomial P(), n ( s) = ( s) = s + ( s) = ( s ) B ( s) + s B ( s) n (.4a) n (.4b) (.4c), n August 007 006 by Fabian Kung Wai Lee Examples of P L for Low-Pass Filter () P L of low pass filter using 4th order polynomial functions (N=4) - Butterworth, Chebyshev (ripple factor =) and Bessel. Normalized to c = rad/s, k=. Ideal P ( ) = + k 8 4 L chebyshev + c c.0 4.0 3 PLbt( ) PLcbP ( L ) 00 PLbs( ) 0 Chebyshev k= 4 P ( ) L Butterworth = + k c Butterworth Bessel [ ( ) ( ) ] P = + L( Bessel ) k B j B j 0 0.5.5 If we convert into db 4 3 B( s) = s + + + 0 s 45 s 05 s + 05 this ripple is equal to 05 c c c c 3 db August 007 006 by Fabian Kung Wai Lee

Examples of P L for Low-Pass Filter () P L of low pass filter using Butterworth characteristic polynomial, normalized to c = rad/s, k=. N P L ( Butterworth) = + k c. 0 5 PL(, ). 0 4 PL(, 3) PL(, 4).0 3 PL(, 5) PL(, 6) PL(, 7) 00 0 N=7 Conclusion: The type of N=6 polynomial N=5 function and the order N=4 determine the N=3 Attenuation rate N= in the stopband. 0 0.5.5 August 007 006 by Fabian Kung Wai Lee 3 Characteristics of Low-Pass Filters Using Various Polynomial Functions Butterworth: Moderately linear phase response, slow cut-off, smooth attenuation in passband. Chebyshev: Bad phase response, rapid cut-off for similar order, contains ripple in passband. May have impedance mismatch for N even. Bessel: Good phase response, linear. Very slow cut-off. Smooth amplitude response in passband. August 007 006 by Fabian Kung Wai Lee 4

Low-Pass Prototype Design () A lossless linear, passive, reciprocal network that can produce the insertion loss profile for Low-Pass Filter is the LC ladder network. Many researchers have tabulated the values for the L and C for the Low-Pass Filter with cut-off frequency c = ad/s, that works with source and load impedance Z s = Z L = Ohm. This Low-Pass Filter is known as the Low-Pass Prototype (LPP). As the order N of the polynomial P increases, the required element also increases. The no. of elements = N. L =g L =g 4 C =g C =g 3 L = g N+ g 0 = L =g L =g 3 C =g C =g 4 L = g N+ Dual of each other August 007 006 by Fabian Kung Wai Lee 5 Low-Pass Prototype Design () The LPP is the building block from which real filters may be constructed. Various transformations may be used to convert it into a high-pass, band-pass or other filter of arbitrary center frequency and bandwidth. The following slides show some sample tables for designing LPP for Butterworth and Chebyshev amplitude response of P L. See Chapter 3 of Hunter [4], on how the LPP circuits and the tables can be derived. August 007 006 by Fabian Kung Wai Lee 6 3

Table for Butterworth LPP Design N g g g3 g4 g5 g6 g7 g8 g9.0000.0000.44.44.0000 3.0000.0000.0000.0000 4 0.7654.8478.8478 0.7654.0000 5 0.680.680.0000.680 0.680.0000 6 0.576.44.938.938.44 0.576.0000 7 0.4450.470.809.0000.809.470 0.4450.0000 8 0.390..669.965.965.669. 0.390.0000 Taken from Chapter 8, Pozar []. See Example. in the following slides on how the constant values g, g, g 3 etc. are obtained. August 007 006 by Fabian Kung Wai Lee 7 Table for Chebyshev LPP Design ipple factor 0log 0 ε = 0.5dB N g g g 3 g 4 g 5 g 6 g 7 0.6986.0000.409 0.707.984 3.5963.0967.5963.0000 4.6703.96.366 0.849.984 5.7058.96.5408.96.7058.0000 6.754.479.6064.337.4578 0.8696.984 ipple factor 0log 0 ε = 3.0dB N g g g 3 g 4 g 5 g 6 g 7.9953.0000 3.03 0.5339 5.8095 3 3.3487 0.77 3.3487.0000 4 3.4389 0.7483 4.347 0.590 5.8095 5 3.487 0.768 4.538 0.768 3.487.0000 6 3.5045 0.7685 4.606 0.799 4.464 0.6033 5.8095 August 007 006 by Fabian Kung Wai Lee 8 4

5 August 007 006 by Fabian Kung Wai Lee 9 Table for Maximally-Flat Time Delay LPP Design N g g g3 g4 g5 g6 g7 g8 g9.0000.0000.5774 0.46.0000 3.550 0.558 0.9.0000 4.0598 0.56 0.38 0.04.0000 5 0.9303 0.4577 0.33 0.090 0.078.0000 6 0.8377 0.46 0.358 0.364 0.480 0.0505.0000 7 0.7677 0.3744 0.944 0.378 0.778 0.04 0.0375.0000 8 0.75 0.3446 0.735 0.97 0.867 0.387 0.0855 0.089.0000 Taken from Chapter 8, Pozar []. August 007 006 by Fabian Kung Wai Lee 30 Example. - Finding the Constants for LPP Design () ( ) ( )( ) ( ) C L j LC V C j L j V L j V s s C j s C j V + + + + + + + = = = + + ( ) ( ) ( ) ( ) + + = = C L LC V L s V P 8 s A V P = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) + + + = + + = = = + + 4 4 8 8 LC C L LC V V P P L LC C L C L LC P s s L A and Thus Therefore we can compute the power loss ratio as: [P()] V s C L jl V s /jc V Consider a simple case of nd order Low-Pass Filter:

Example. - Finding the Constants for LPP Design () P L can be written in terms of polynomial of : 4 4 ( ) ( ) ( ) [ ] L C LC LC = + + + = + a a PL + 4 For Butterworth response with k=, c = : [ ] 4 4 = + = + 0 + P L ( Butterworth) = + Comparing equation (E.) and (E.): a = LC = LC = a = 0 ( L + C) (E.3) 4 LC = ( L + C) Setting = for Low-Pass Prototype (LPP): = Thus from equation (E.4): ( L C) L = C LC = 0 August 007 006 by Fabian Kung Wai Lee 3 (E.) (E.) ( L + C) L + C LC = 0 C =. 44 LC = = 0 Using (E.3) LC = C = L = C.44 (E.4) Compare this result with N= in the table for LPP Butterworth response. This direct brute force approach can be extended to N=3, 4, 5 Example. Verification () AC AC AC Start=0.0 Hz Stop=.0 Hz Step=0.0 Hz Vin V_AC = Ohm SC Vac=polar(,0) V Freq=freq L L L=.44 H = Vout C C C=.44 F = Ohm August 007 006 by Fabian Kung Wai Lee 3 6

Example. Verification () Eqn PA=/8 Eqn PL=0.5*mag(Vout)*mag(Vout) PL Eqn PL=PA/PL.5E4.0E4.5E4.0E4 5.0E3 0.0 The power loss ratio versus frequency 0.0 0. 0.4 0.6 0.8.0..4.6.8.0 freq, Hz db(vout/0.5) 5 0-0 -0-30 -40-50 m m freq= 60.0mHz m=-3.056 0.0 0. 0.4 0.6 0.8.0..4.6.8.0 freq, Hz -3dB at 60mHz (milihertz!!), which is equivalent to rad/s August 007 006 by Fabian Kung Wai Lee 33 Impedance Denormalization and Frequency Transformation of LPP () Once the LPP filter is designed, the cut-off frequency c can be transformed to other frequencies. Furthermore the LPP can be mapped to other filter types such as highpass, bandpass and bandstop (see [] and [3] for the derivation and theories). This frequency scaling and transformation entails changing the value and configuration of the elements of the LPP. Finally the impedance presented by the filter at the operating frequency can also be scaled, from unity to other values, this is called impedance denormalization. Let Z o be the new system impedance value. The following slide summarizes the various transformation from the LPP filter. August 007 006 by Fabian Kung Wai Lee 34 7

Impedance Denormalization and Frequency Transformation of LPP () LPP to Low-Pass LPP to High-Pass LPP to Bandpass LPP to Bandstop L Z o L c c LZ o LZ o o ol Z o L Z o o o LZ o Z C o C cc Z o c + o = or (.5a) C o Z o = o Zo oc (.5b) Zo oc C oz o Note that inductor always multiply with Z o while capacitor divide with Z o August 007 006 by Fabian Kung Wai Lee 35 Summary of Passive LC Filter Design Flow Using ILM Method () Step - From the requirements, determine the order and type of approximation functions to used. Insertion loss (db) in passband? Attenuation (db) in stopband? Cut-off rate (db/decade) in transition band? Tolerable ripple? Linearity of phase? Step - Design the normalized low-pass prototype (LPP) using L and C elements. H() L =g L =g 4 C =g C =g 3 L= g N+ 0 August 007 006 by Fabian Kung Wai Lee 36 8

Summary of Passive Filter Design Flow Using ILM Method () Step 3 - Perform frequency scaling and denormalize the impedance. H() 50 79.58nH 0.44pF V s 5.96pF 0.707nH 5.96pF 0.707nH L 50 0 Step 4 - Choose suitable lumped components, or transform the lumped circuit design into distributed realization. All uses microstrip stripline circuit See ef. [4] See ef. [] See ef. [3] August 007 006 by Fabian Kung Wai Lee 37 Filter vs Impedance Transformation Network If we ponder carefully, the sharp observer will notice that the filter can be considered as a class of impedance transformation network. In the passband, the load is matched to the source network, much like a filter. In the stopband, the load impedance is highly mismatched from the source impedance. However, the procedure described here only applies to the case when both load and source impedance are equal and real. August 007 006 by Fabian Kung Wai Lee 38 9

Example.A LPF Design: Butterworth esponse Design a 4th order Butterworth Low-Pass Filter. s = L = 50Ohm, f c =.5GHz. 9 c = π (.5GHz) = 9.448 0 rad/s L =0.7654H L =.8478H Step &: LPP Z = 50 o g 0 = C =.8478F C =0.7654F L = = Z o n Step 3: Frequency scaling and impedance denormalization L =4.06nH L =9.803nH L L = Z n o c C C = n Zoc g 0 =/50 C =3.9pF C =.64pF L = 50 August 007 006 by Fabian Kung Wai Lee 39 Example.B LPF Design: Chebyshev esponse Design a 4th order Chebyshev Low-Pass Filter, 0.5dB ripple factor. s = 50Ohm, f c =.5GHz. 9 c = π (.5GHz) = 9.448 0 rad/s L =.6703H L =.366H Step &: LPP Z = 50 o g 0 = C =.96F C =0.849F L =.984 = Z o n Step 3: Frequency scaling and impedance denormalization L =8.86nH L =.55nH L L = Z n o c C C = n Zoc g 0 =/50 C =.53pF C =.787pF L = 99. August 007 006 by Fabian Kung Wai Lee 40 0

Example. Cont... db(lpf_butterworth..s(,)) db(s(,)) s 5 0-0 -0-30 ipple is roughly 0.5dB Chebyshev 0.0 0.5.0.5.0.5 3.0 freq, GHz Butterworth Computer simulation result Using AC analysis (ADS003C) 0-50 Better phase Linearity for Butterworth LPF in the passband Note: Equation used in Data Display of ADS003C to obtain continuous phase display with built-in function phase( ). Phase_butterworth Phase_chebyshev -00-50 Arg(s-00 ) -50-300 -350 Chebyshev Butterworth Eqn Phase_chebyshev = if (phase(s(,))<0) then phase(s(,)) else (phase(s(,))-360) 0.5.0.5.0.5 0.0 3.0 freq, GHz August 007 006 by Fabian Kung Wai Lee 4 Example.3: BPF Design Design a bandpass filter with Butterworth (maximally flat) response. N = 3. Center frequency f o =.5GHz. 3dB Bandwidth = 00MHz or f =.4GHz, f =.6GHz. Impedance = 50Ω. August 007 006 by Fabian Kung Wai Lee 4

Example.3 Cont From table, design the Low-Pass prototype (LPP) for 3rd order Butterworth response, c =. Step &: LPP Z o = g.000h <0 o g.000f g 3.000F g 4 Simulated result using PSPICE c = πf c = fc = = 0.59 Hz π Voltage across g 4 August 007 006 by Fabian Kung Wai Lee 43 Example.3 Cont LPP to bandpass transformation. Impedance denormalization. Step 3: Frequency scaling and impedance denormalization LZ o o o LZ o 50 C o Zo 79.58nH Zo oc 0.44pF = π = π (.4GHz) (.6GHz) fo = f f =.497GHz = = 0.33 o V s 5.96pF 0.707nH 5.96pF 0.707nH L 50 August 007 006 by Fabian Kung Wai Lee 44

Example.3 Cont Simulated result using PSPICE: Voltage across L August 007 006 by Fabian Kung Wai Lee 45 All Pass Filter There is also another class of filter known as All-Pass Filter (APF). This type of filter does not produce any attenuation in the magnitude response, but provides phase response in the band of interest. APF is often used in conjunction with LPF, BPF, HPF etc to compensate for phase distortion. Example of APF response H(f) Arg(H(f)) Nonlinear phase in passband H(f) Arg(H(f)) 0 f f 0 f f BPF APF Z o H(f) Arg(H(f)) Linear phase in passband 0 f f August 007 006 by Fabian Kung Wai Lee 46 3

Example.4 - Practical F BPF Design Using SMD Discrete Components C Ct3 C=Ct_value pf C Ct C=Ct_value pf CPWSub CPWSUB CPWSub H=6.0 mil Er=4.6 Mur= Cond=5.8E+7 T=.38 mil TanD=0.0 ough=0.0 mil S-PAAMETES S_Param SP Start=0. GHz Stop=3.0 GHz Step=.0 MHz Var VA Eqn VA Lt_value=4.8 Ct_value=3.5 Ct_value=.9 C Ct C=Ct_value pf C Ct45 C=Ct_value pf L Lt L=Lt_value nh = Term Term Num= Z=50 Ohm INDQ _0pF_NPO_0603 CPWG L4 C CPW L=5.0 nh Subst="CPWSub" Q=90.0 b8496c39j000 4_7pF_NPO_0603 W=50.0 mil F=800.0 MHz L C G=0.0 mil Mode=proportional to freq L=8.0 mm param=simid 0603-C (. nh +-5%) dc=0. Ohm CPWG CPW b8496c39j000 L3 Subst="CPWSub" param=simid 0603-C (. 4_7pF_NPO_0603 W=50.0 mil nh +-5%) C3 G=0.0 mil L=8.0 mm L Lt L=Lt_value nh = Term Term Num= Z=50 Ohm August 007 006 by Fabian Kung Wai Lee 47 Example.4 Cont BPF synthesis using synthesis tool E-syn of ADS003C August 007 006 by Fabian Kung Wai Lee 48 4

Example.4 Cont db(f_bpf_measured..s(,)) db(s(,)) s /db 0-0 -40-60 0. 0.4 0.6 0.8.0..4.6.8.0..4.6.8 0.0 3.0 freq, GHz phase(f_bpf_measured..s(,)) phase(s(,)) 00 00 0-00 -00 Measured Simulated Measurement is performed with Agilent 8753ES Vector Network Analyzer, using Full OSL calibration Arg(s )/degree 0.0 0. 0.4 0.6 0.8.0..4.6.8.0..4.6.8 3.0 freq, GHz August 007 006 by Fabian Kung Wai Lee 49 3.0 Microwave Filter ealization Using Stripline Structures August 007 006 by Fabian Kung Wai Lee 50 5

3. Basic Approach August 007 006 by Fabian Kung Wai Lee 5 Filter ealization Using Distributed Circuit Elements () Lumped-element filter realization using surface mounted inductors and capacitors generally works well at lower frequency (at UHF, say < 3 GHz). At higher frequencies, the practical inductors and capacitors loses their intrinsic characteristics. Also a limited range of component values are available from manufacturer. Therefore for microwave frequencies (> 3 GHz), passive filter is usually realized using distributed circuit elements such as transmission line sections. Here we will focus on stripline microwave circuits. August 007 006 by Fabian Kung Wai Lee 5 6

Filter ealization Using Distributed Circuit Elements () ecall in the study of Terminated Transmission Line Circuit that a length of terminated Tline can be used to approximate an inductor and capacitor. This concept forms the basis of transforming the LC passive filter into distributed circuit elements. l Z c, β l Z c, β L C Z o Z c, β Z o Z o Z c, β Z c, β Z o August 007 006 by Fabian Kung Wai Lee 53 Filter ealization Using Distributed Circuit Elements (3) This approach is only approximate. There will be deviation between the actual LC filter response and those implemented with terminated Tline. Also the frequency response of distributed circuit filter is periodic. Other issues are shown below. How do we implement series Tline connection? (only practical for certain Tline configuration) Z o Z c, β Connection of physical length cannot be ignored at microwave region, comparable to λ Thus some theorems are used to facilitate the transformation of LC circuit into stripline microwave circuits. Chief among these are the Kuroda s Identities (See Appendix ) Z c, β Z c, β Z o August 007 006 by Fabian Kung Wai Lee 54 7

More on Approximating L and C with Terminated Tline: ichard s Transformation l tan( βl) Z L in Z c, β ( βl) = jl jl Zin = jzc tan = = (3..a) Zc = L Here instead of fixing Z c and tuning l to approach an L or C, we allow Z c to be variable too. Yin = jy tan l = j C = l c ( β ) Z C in tan( βl) = Z c, β (3..b) Yc = = C Z For LPP design, a further requirement is c that: λ tan ( β l) = c = tan π l l c = (3..c) λ c = 8 August 007 006 by Fabian Kung Wai Lee 55 jc Wavelength at cut-off frequency Example 3. LPF Design Using Stripline Design a 3rd order Butterworth Low-Pass Filter. s = L = 50Ohm, f c =.5GHz. Step & : LPP Z o = g.000h g.000f g 3.000H g 4 Z c =.000 Z c =.000 Step 3: Convert to Tlines = 0.500.000 Length = λ c /8 for all Tlines at = rad/s August 007 006 by Fabian Kung Wai Lee 56 Z c =0.500 8

Example 3. Cont Step 4: Add extra Tline on the series connection and apply Kuroda s nd Identity. Z =.0 Z = l β Y c = 0.5 n Z Extra Tline Z c =.0 Z c =.000 Z c =.000 Extra Tline Z c =.0 n Z = β Z n = + Z = + = Similar operation is performed here Z c =0.500 Length = λ c /8 for all Tlines at = rad/s August 007 006 by Fabian Kung Wai Lee 57 Example 3. Cont After applying Kuroda s nd Identity. Z c =.0 Z c =.0 Z c =.000 Z c =0.500 Z c =.000 Length = λ c /8 for all Tlines at = rad/s Since all Tlines have similar physical length, this approach to stripline filter implementation is also known as Commensurate Line Approach. August 007 006 by Fabian Kung Wai Lee 58 9

Example 3. Cont Step 5: Impedance and frequency denormalization. 50 Here we multiply all impedance with Z o = 50 Z c =00 Z c =00 50 Z c =00 Z c =5 Length = λ c /8 for all Tlines at f = f c =.5GHz Z c =00 Microstrip line using double-sided F4 PCB (ε r = 4.6, H=.57mm) Z c /Ω λ/8 @.5GHz /mm W /mm 50 3.45.85 5.77 8.00 00 4.3 0.6 We can work out the correct width W given the impedance, dielectric constant and thickness. From W/H ratio, the effective dielectric constant ε eff can be determined. Use this together with frequency at.5 GHz to find the wavelength. August 007 006 by Fabian Kung Wai Lee 59 Example 3. Cont Step 6: The layout (top view) August 007 006 by Fabian Kung Wai Lee 60 30

Example 3. Cont Simulated results S-PAAMETES MSub S_Param MSUB SP MSub Start=0. GHz H=.57 mm Stop=4.0 GHz Er=4.6 Step=5 MHz Mur= Cond=.0E+50 MTEE MTEE Hu=3.9e+034 mil Tee Tee3 T=0.036 mm Subst="MSub" Subst="MSub" TanD=0.0 W=.85 mm W=0.6 mm ough=0 mil W=0.6 mm W=0.6 mm W3=0.6 mm W3=8.00 mm MTEE Tee Subst="MSub" W=0.6 mm W=.85 mm W3=0.6 mm Term Term Num= Z=50 Ohm L L L=5.305 nh = m freq=.500ghz m=-6.09 L L C L=5.305 nh C = C=4.44 pf Term Term Num= Z=50 Ohm MLIN TL Subst="MSub" W=.85 mm Term L=5.0 mm Term Num= Z=50 Ohm MLIN TL3 Subst="MSub" W=0.6 mm L=4.3 mm MLOC TL6 Subst="MSub" W=0.6 mm L=4.3 mm MLIN TL4 Subst="MSub" W=0.6 mm L=4.3 mm MLOC TL5 Subst="MSub" W=8.0 mm L=.77 mm MLIN TL Subst="MSub" W=.85 mm L=5.0 mm MLOC TL7 Subst="MSub" W=0.6 mm L=4.3 mm Term Term Num= Z=50 Ohm db(butter_lpf_lc..s(,)) db(s(,)) 0-0 -0-30 m 0.5.0.5.0.5 3.0 3.5 0.0 4.0 August 007 006 by Fabian Kung Wai Lee 6-40 freq, GHz Conclusions for Section 3. Further tuning is needed to optimize the frequency response. The method just illustrated is good for Low-Pass and Band-Stop filter implementation. For High-Pass and Band-Pass, other approaches are needed. August 007 006 by Fabian Kung Wai Lee 6 3

3. Further Implementations August 007 006 by Fabian Kung Wai Lee 63 ealization of LPF Using Step- Impedance Approach A relatively easy way to implement LPF using stripline components. Using alternating sections of high and low characteristic impedance tlines to approximate the alternating L and C elements in a LPF. Performance of this approach is marginal as it is an approximation, where sharp cutoff is not required. As usual beware of parasitic passbands!!! August 007 006 by Fabian Kung Wai Lee 64 3

Equivalent Circuit of a Transmission Line Section T-network equivalent circuit Z - Z Z - Z Ideal lossless Tline l Z = βl ( ) [ ] Z cos ( βl ) Z = jzc sin( βl ) sin( βl ) ( ( )) sin ( ( )) [ ( ( )) ] cos βl βl jzc βl = jz sin c βl sin cos = jz c tan Positive reactance βl ( ( )) ( ( )) Positive susceptance Z = Z = jzc cot Z c β Z = Z = jzc cosec β ( l) ( l) β β µ ε ε = o e o ε k e o (3..a) (3..b) (3..c) August 007 006 by Fabian Kung Wai Lee 65 Approximation for High and Low Z C () When βl < π/, the series element can be thought of as inductor and the shunt element can be considered a capacitor. X β l Z Z = = Z tan = B = sin( β l) c Z Zc For βl < π/4 and Z c =Z H >> : X ZH β l B 0 For βl < π/4 and Z c =Z L : X 0 B β l Z Z - Z Z - Z L Z When Z c >> βl < π/4 X Z H βl jx/ jx/ jb When Z c βl < π/4 B Y L βl August 007 006 by Fabian Kung Wai Lee 66 33

Approximation for High and Low Z C () Note that βl < π/ implies a physically short Tline. Thus a short Tline with high Z c (e.g. Z H ) approximates an inductor. cl l L = Z β H (3..a) A short Tline with low Z c (e.g. Z L ) approximates a capacitor. ccz l C = β L (3..b) The ratio of Z H /Z L should be as high as possible. Typical values: Z H = 00 to 50Ω, Z L = 0 to 5Ω. August 007 006 by Fabian Kung Wai Lee 67 Example 3. - Mapping LPF Circuit into Step Impedance Tline Network For instance consider the LPF Design Example.A (Butterworth). Let us use microstrip line. Since a microstrip tline with low Z c is wide and a tline with high Z c is narrow, the transformation from circuit to physical layout would be as follows: L =4.06nH L =9.803nH g 0 =/50 C =3.9pF C =.64pF L = 50 August 007 006 by Fabian Kung Wai Lee 68 34

Example 3. - Physical ealization of LPF Using microstrip line, with ε r = 4., d =.5mm: W/d d/mm W/mm ε e Z c = 5Ω 0.0.5 5.0 3.68 Z c = 50Ω.0.5 3.0 3. Z c = 0Ω 0.36.5 0.6.83 β 9 L = εel k o = εel π f c 3.3356 0 = 60.307 s β H = 9 εeh k o = εeh π f c 3.3356 0 = 53.58 s L =4.06nH, L =9.083nH, C =3.9pF, C =.64pF. August 007 006 by Fabian Kung Wai Lee 69 Example 3. - Physical ealization of LPF Cont L l c = = 6.5mm ZH βh C Z l c L = = 9. mm β L l3 =5. 0mm l4 = 3. 8mm Verification: β = 0.39 < π H l = 0.7854 4 β l = 0.490 < π L = 0.7854 4 β 3 = 0.905 > π H l = 0.7854 4 β l4 = 0.0 < π L = 0.7854 4 Nevertheless we still proceed with the implementation. It will be seen that this will affect the accuracy of the -3dB cutoff point of the filter. 3.0mm l l l 3 l 4 50Ω line 50Ω line To 50 Ω Load 5.0mm 0.6mm August 007 006 by Fabian Kung Wai Lee 70 35

Example 3. - Step Impedance LPF Simulation With ADS Software () Transferring the microstrip line design to ADS: Microstrip line substrate model Microstrip line model Microstrip step junction model August 007 006 by Fabian Kung Wai Lee 7 Example 3. - Step Impedance LPF Simulation With ADS Software () m freq=.40ghz db(s(,))=-3.05 0 m -5 db(s(,)) -0-5 -0-5 0.0 0.5.0.5.0.5 3.0 3.5 4.0 freq, GHz August 007 006 by Fabian Kung Wai Lee 7 36

Example 3. - Step Impedance LPF Simulation With ADS Software (3) However if we extent the stop frequency for the S-parameter simulation to 9GHz... Parasitic passbands, artifacts due to using transmission lines. m freq=.40ghz db(s(,))=-3.05 0 m -5 db(s(,)) -5-5 0 3 4 5 6 7 8 9 freq, GHz August 007 006 by Fabian Kung Wai Lee 73 Example 3. - Verification with Measurement The -3dB point is around.47ghz! The actual LPF constructed in year 000. Agilent 870D Vector Network Analyzer is used to perform the S-parameters measurement. August 007 006 by Fabian Kung Wai Lee 74 37

Example 3.3 - ealization of BPF Using Coupled StripLine () Based on the BPF design of Example.3: To source network V s λ o 4 50 5.96pF 79.58nH 0.707nH Admittance inverter tline 0.44pF 5.96pF 0.707nH L 50 An equivalent circuit model for coupled tlines with open circuit at ends. See appendix (using ichard s transformation And Kuroda s identities) J J J 3 J 4-90 o -90 o -90 o -90 o To L λ o 4 Section Section Section 3 Section 4 An Array of coupled microstrip line λ o = wavelength at o August 007 006 by Fabian Kung Wai Lee 75 Example 3.3 - ealization of BPF Using Coupled StripLine () Each section of the coupled stripline contains three parameters: S, W, d. These parameters can be determined from the values of the odd and even mode impedance (Z oo & Z oe ) of each coupled line. Z oo and Z ee are in turn depends on the gain of the corresponding admittance inverter J. From Example.3 And each J n is given by: = π (.4GHz) ( ) Zoe = Zo + JZo + JZ = π.6ghz o J = Zo J = n Zo J N + = Zo π g π gn gn π gn gn + for n =,3,4L N fo = f f =.497GHz = = 0.33 o Zoo = Zo August 007 006 by Fabian Kung Wai Lee 76 W S W ( ( ) ) JZ + ( JZ ) ( ) For derivation see chapter 8, Pozar []. o o d 38

Example 3.3 - ealization of BPF Using Coupled StripLine (3) Section : J = π = 0.00963 g J J J Z o Section : = π = Z o g g 3 = π = Z o g g 4 = π = g g Z o Section 3: Section 4: 3 3 4 0.00969 0.00969 0.00963 Zoe = Zo Zoo = Zo Zoe = Zo Zoo = Zo Zoe3 = 83.403 Zoo3 = 37.588 Zoe4 = 58.53 Zoo4 = 43.680 ( + JZo + ( JZo ) ) J Z + ( J Z ) = 83.403 ( ) = 37. 588 o ( + JZo + ( JZo ) ) J Z + ( J Z ) ( ) = 43. 680 o o o = 58.53 Note: g =.0000 g =.0000 g 3 =.0000 g 4 =.0000 August 007 006 by Fabian Kung Wai Lee 77 Example 3.3 - ealization of BPF Using Coupled StripLine (4) In this example, edge-coupled stripline is used instead of microstrip line. Stripline does not suffers from dispersion and its propagation mode is pure TEM mode. Hence it is the preferred structured for coupled-line filter. From the design data (next slide) for edge-coupled stripline, the parameters W, S and d for each section are obtained. Length of each section is l. v p l = = vp 4 f o ε ε µ r = o o =.463 0 8.463 0 4.5 0 9 8 = 0.04 or 4.0mm ε r = 4. August 007 006 by Fabian Kung Wai Lee 78 39

Example 3.3 - ealization of BPF Using Coupled StripLine (5) Section and 4: S/b = 0.07, W/b = 0.3 Section and 3: S/b = 0.5, W/b = 0.4 By choosing a suitable b, the W and S can be computed. W S b August 007 006 by Fabian Kung Wai Lee 79 Example 3.3 - Coupled Line BPF Simulation With ADS Software () Using ideal transmission line elements: Ideal open circuit Ideal coupled tline August 007 006 by Fabian Kung Wai Lee 80 40

Example 3.3 - Coupled Line BPF Simulation With ADS Software () Parasitic passbands. Artifacts due to using distributed elements, these are not present if lumped components are used..0 0.8 mag(s(,)) 0.6 0.4 0. 0.0.0.5.0.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 0.0 f o freq, GHz August 007 006 by Fabian Kung Wai Lee 8 Example 3.3 - Coupled Line BPF Simulation With ADS Software (3) Using practical stripline model: Stripline substrate model Coupled stripline model Open circuit model August 007 006 by Fabian Kung Wai Lee 8 4

Example 3.3 - Coupled Line BPF Simulation With ADS Software (4).0 0.9 0.8 0.7 Attenuation due to losses in the conductor and dielectric mag(s(,)) 0.6 0.5 0.4 0.3 0. 0. 0.0.0.5.0.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 0.0 freq, GHz August 007 006 by Fabian Kung Wai Lee 83 Things You Should Self-Study Network analysis and realizability theory ([3] and [6]). Synthesis of terminated LCM one-port circuits ([3] and [6]). Ideal impedance and admittance inverters and practical implementation. Periodic structures theory ([] and []). Filter design by Image Parameter Method (IPM) (Chapter 8, []). August 007 006 by Fabian Kung Wai Lee 84 4

Other Types of Stripline Filters () LPF For these delightfully simple approaches see Chapter 43 of [3] HPF: BPF: SMD capacitor August 007 006 by Fabian Kung Wai Lee 85 Other Types of Stripline Filters () More BPF: BSF: More information can be obtained from [], [3], [4] and the book: J. Helszajn, Microwave planar passive circuits and filters, 994, John-Wiley & Sons. August 007 006 by Fabian Kung Wai Lee 86 43

Appendix Kuroda s Identities August 007 006 by Fabian Kung Wai Lee 87 Kuroda s Identities As taken from []. l Z n = + Z Z Z β Z /n Z Z l β l l n Z Note: The inductor represents shorted Tline while the capacitor represents open-circuit Tline. β β Z n n Z Note: the length of all transmission lines is l = λ/8 l l : n Z Z β Z /n β Z n l l n : Z Z β n Z β n Z August 007 006 by Fabian Kung Wai Lee 88 44

THE END August 007 006 by Fabian Kung Wai Lee 89 45