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Unformatted text preview: ' ECE 710 Broadband Filters Chapter 7 Patrick Roblin $ Department of Electrical & Computer Engineering The Ohio State University Columbus, OH 43210 & OHIO S ATE T UNIVERSITY T.H.E Microwave Laboratory 0 The Ohio State University % ' Lecture # 23 $ 1. General introduction to lter synthesis 2. Insertion loss method 3. Filter prototypes 4. Scaling of prototypes & OHIO S ATE T UNIVERSITY T.H.E Microwave Laboratory 1 The Ohio State University % ' Filter synthesis $ We have seen so far how narrowband bandstop or bandpass lters can be constructed using resonators. For narrowband cases, lumped element models of the distributed network were good enough to get good performance predictions. For wider bandwidths this will not be the case we need design methods speci cally for distributed circuits. Design methods for lumped element lters exist, and can be adapted in some cases to distributed circuits. & We will start by reviewing the\insertion loss" synthesis, which allows control of the lter bandwidth and performance through a set of \prototype" designs. OHIO S ATE T UNIVERSITY T.H.E Microwave Laboratory 2 The Ohio State University % ' jS21 j2 Insertion loss method $ & !c where !c is the frequency at which 3 dB loss is obtained. For Chebyshev low-pass lters: 1 = 1 + k2T 2 ! N! jS21 j2 c where TN is a Chebyshev polynomial and k is a constant. For ! < !c , an equal ripple response in the range 1 k2 is obtained while for ! > !c higher OHIO losses are obtained. S ATE T T.H.E UNIVERSITY Insertion loss synthesis applies for either lumped element or distributed circuits, so we will focus on lumped designs rst. 1 Filters are designed to achieve desired values of jS21j2 , which determines the insertion loss of the lter. Most common designs produce either maximally at (Butterworth) or Chebyshev responses. For maximally at low-pass lters: 1 = 1 + ! 2N Microwave Laboratory 3 The Ohio State University % ' 2 Synthesis Example of N=3 Butterworth $ & !6 = ;S 6 = S (S )S (S ) jS11 j = 1 ; jS21 j = 11 11 1 + !6 1 ; S 6 1 ; S 6 = 0 admits 6 roots: S1 2 3 4 = 0:5 j 0:866, and S5 6 = 1. As the lter is realizable and passive we select the left hand plane roots to identify S11 (S ): S3 S11 = S 3 + 2S 2 + 2S + 1 One can indeed verify that S 3 + 2S 2 + 2S + 1 = 0 admits the following 3 roots S1 2 = ;0:5 j 0:866, and S5 = ;1. We select the positive sign for S11 and convert S11 to input impedance Zin: 1 + S11 = 2S 3 + 2S 2 + 2S + 1 using Z = 1 OHIO Zin = 1 ; S 0 2S 2 + 2S + 1 11 S ATE T 2 T.H.E UNIVERSITY Let us consider the case of N = 3 with Z0 = 1 and !c = 1 Hz. 1 GT = jS21j2 = 1 + !6 The lter is loss less: thus we have jS11 j2 + jS21j2 = 1 resulting in: Microwave Laboratory 4 The Ohio State University % ' Cauer Expansion 1 Zin,1 Y in,2 Z in,3 $ Port 2 g1 V G g3 g2 g 4 Port 1 & Dividing the higher order polynomial by the lower polynomial we get: 2S 3 + 2S 2 + 2S + 1 = Z + R1 = S + S + 1 Zin 1 = 2S 2 + 2S + 1 ext 1 D 2S 2 + 2S + 1 We have extracted an inductor L = 1 H since Zext 1 = LS = S . D = 2S 2 + 2S + 1 = Y + R2 = 2S + 1 Let us divide D by R1: Yin 2 = ext 2 R1 S+1 R1 S+1 We have extracted a capacitor C = 2 F since Yext 3 = 2S Let us now divided R1 by R2 : Zin 3 = R1 = S + 1 = S + 1 R2 1 OHIO S ATE T We have now extracted an inductor L = 1 H and resistor R = 1 . T.H.E UNIVERSITY Lowpass prototype Microwave Laboratory 5 The Ohio State University % ' Alternate Implementation $ S11 = S 3 + 2S 2 + 2S + 1 ;S 3 1 1 g1 g3 g2 Port 2 If we select the negative sign solution a di erent prototype will result: g2 g 4 VG Port 1 VG Port 1 g1 g3 Port 2 g 4 & OHIO S ATE T UNIVERSITY T.H.E Microwave Laboratory 6 The Ohio State University % ' 1 VG Port 1 More on Prototypes g1 g2 g3 g4 g5 Port 2 $ gN+1 & For the maximally at lters gN +1 is always 1, so the source and load impedances are equal. However, for the Chebyshev lters with an even number of sections, the source and load impedances are not equal. In this case an impedance transformer can be used if needed. Even number section Chebyshev lters are usually avoided. Book also provides plots of the attenuation versus the frequency deviation j!=!c j ; 1 for the maximally at (p. 450) and Chebyshev (p. 453) prototypes versus the number of sections. Notice these prototypes assumed that Z0 = 1 and also result in \breakpoint" frequencies !c of 1. OHIO We need \scaling" techniques to obtain more general lters. S ATE T T.H.E UNIVERSITY Lowpass prototype Microwave Laboratory 7 The Ohio State University % ' 1 g1 V G Scaling of Prototypes Z0 g3 Port 2 $ L1 L3 Port 2 Port 1 g2 g V 4 G Port 1 C2 Z0 It turns out we can make a low-pass prototype with source (and usually load) impedance Z0 instead of 1 just by multiplying the original inductances by Z0 and dividing the original capacitances by Z0 . It turns out we can change the breakpoint frequency from 1 to !c just by dividing the original inductances by !c and dividing the original capacitances by !c . 0 Thus, the new inductance L0k and capacitance Ck values in an impedance and frequency scaled low pass lter are & It is also possible to generate high pass, band pass, and band-stop lters from suitable transformations of the low-pass prototype. 8 0 0 L0k = Z!gk Ck = Zgk where gk are the prototype values. ! c 0 c OHIO S ATE T UNIVERSITY T.H.E Microwave Laboratory The Ohio State University % ' High pass lters $ A high pass lter is obtained by transforming ! in the original low-pass lter to ;!c =!. Note then high frequencies in the new lter will correspond to low frequencies in the original. This transformation winds up switching inductors and capacitors in the loss pass prototype. Makes sense because we want to obtain a high pass response. Values for an impedance and frequency scaled high pass lter are then C 0 = 1 = 1 and L0 = Z0 = Z0 k & k !C Z0!c Lk Z0!c gk !c gk ck It is typically found that equal ripple lters provide a sharper cuto compared to maximally at lters, at the expense of the ripple in the passband. OHIO S ATE T UNIVERSITY T.H.E Microwave Laboratory 9 The Ohio State University % ' V G Transformation from Lowpass to Highpass Prototypes 1 g1 Port 1 $ gN+1 1/g N+1 g3 g2 g4 g5 Port 2 Lowpass prototype 1 1/g 1 V G 1/g 3 1/g 2 1/g 5 1/g 4 Port 2 Port 1 & Highpass prototype T.H.E The band-pass & band-stop lter transforms require a little more work... OHIO S ATE T UNIVERSITY Microwave Laboratory 10 The Ohio State University % ' 0 0 Band-pass lters $ Because band pass lters have a two-sided response, the transform from a low-pass prototype replaces the original ! with 1 (!=! ; ! =!) ; where = !2!0!1 , !2 and !1 de ne the upper and lower passband frequencies, and !0 = p!1 !2 . This transform maps !0 to 0 in the original low pass prototype, and !1 and !2 to 1, the breakpoints of the original lter. This transformation can be achieved only if the original inductors and capacitors are each transformed into LC circuits. For a frequency and impedance scaled band-pass lter, inductors in the original prototype are replaced by a series LC circuit, with values & Z ! 0 Lk = gk ! 0 and Ck = Z ! g or fk = 2 0 and Qk = 2gk 0 00k 0 OHIO S ATE T UNIVERSITY T.H.E Microwave Laboratory 11 The Ohio State University % ' VG Transformation from Lowpass to Bandpass Prototypes $ gN+1 Capacitors in the original prototype are replaced by a shunt LC circuit with ! g Z 0 L0k = ! 0g and Ck = Z gk ! or fk = 2 0 and Qk = 2 k 0 1 k 0 0 g1 Port 1 g3 g2 g4 g5 Port 2 Z0 Lowpass prototype fr2 QE2 V G & Port 1 fr1 QE1 fr3 QE3 Bandpass prototype Port 2 Z 0 T.H.E OHIO S ATE T UNIVERSITY Microwave Laboratory 12 The Ohio State University % ' Lowpass to Band-stop Transformation $ (!=!0 ; !0 =!);1 is To obtain a band-stop response, the transformation ! used. This replaces the original series inductors with a parallel LC circuit with values ! 0 L0k = Z0! gk and Ck = Z !1 g or fk = 2 0 and Qk = g 2 0 00 k k and the original shunt capacitors with a series LC circuit with values ! 0 L0k = ! Z0g and Ck = Z gk or fk = 2 0 and Qk = g 2 0 k 0 !0 k & Insertion loss behaviors of these lters can be obtained by substituting the same transformed frequencies into the low pass insertion loss equations. OHIO S ATE T UNIVERSITY T.H.E Microwave Laboratory 13 The Ohio State University % ' VG Transformation from Lowpass to Bandstop Prototypes 1 g1 Port 1 $ gN+1 g3 g2 g4 g5 Port 2 Lowpass prototype 1 fr1 QE1 fr3 QE3 fr2 QE2 fr5 QE5 fr4 QE4 VG Port 1 Port 2 gN+1 & Bandstop prototype OHIO S ATE T UNIVERSITY T.H.E Microwave Laboratory 14 The Ohio State University % ' Lecture # 24 $ Stepped Impedance Filters Commensurate Filters Impedance and Frequency Scaling Kuroda Identities Coupled line realization Example & OHIO S ATE T UNIVERSITY T.H.E Microwave Laboratory 15 The Ohio State University % ' Stepped impedance Filters $ & We now have learned how to design a lumped element lter to achieve a given spec. using the insertion loss method (i.e. scaling of prototypes.) However lumped elements are not easy to realize at high frequencies, so we need ways to implement these lters using distributed elements. We saw on Problem Set 1 how a short piece of high impedance line can model a series inductance, while a short piece of low impedance line can model a shunt capacitance. We can thus cascade low and high impedance line sections to create an equivalent to our lumped element lter. However, the response will only match at the design frequency. O the design frequency the stepped-impedance lter response may not match the lumped network very well. These techniques are best for narrow bandwidth lters. OHIO See book p. 470-473. S ATE T T.H.E UNIVERSITY Microwave Laboratory 16 The Ohio State University % ' Realization of LC Matching Network with Microstrip $ Example of microstrip realization of a lowpass stepped impedance lter & 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 17 OHIO S ATE T UNIVERSITY T.H.E Microwave Laboratory The Ohio State University % ' Commensurate lters $ Another implementation procedure replaces all inductors with short circuited transmission lines and all capacitors with open circuited transmission lines. Using the Richards' Transformation (i.e. S = j tan ` = j ), the input impedance of our shorted lines will then be SZ0 while the input admittance of an open line will be SY0 . Thus if we simply replace a lumped inductor with inductance L with a shorted line of characteristic impedance L, the response of the distributed circuit versus will be the same as the response of the lumped circuit versus !. Similarly, we can replace lumped capacitors of capacitance C with open circuit lines of characteristic admittance C . & If we de ne the frequency at which the lines are a quarter wavelength long as !r , we can re-write = tan 2!!r . will be unity when ! = !r =2. 18 OHIO S ATE T UNIVERSITY T.H.E Microwave Laboratory The Ohio State University % ' Z in (S) Richard's Transformation: Generalized Inductors and Capacitors Z in (S) $ C Z0 λ r /4 L Short Z0 λ r /4 Open where: Zin = jZ0 tan ` Zin = LS L = Z0 " # Yin = jY0 tan ` Yin = CS C = Y0 We de ne ` = r =4 & 4 is the generalized frequency S=j = tan ` = tan 2 r = tan ! 2 !r OHIO S ATE T UNIVERSITY T.H.E Microwave Laboratory 19 The Ohio State University % ' 1 g1 V G Example $ Z0 L1 L3 Port 2 V G Consider a 3rd order lowpass prototype: g3 Port 2 Port 1 g2 g Port 1 4 C2 Z0 The realization of this lter using commensurate lines would look like: Z0 V Port 1 L1 l= λ r /4 L3 l= λ r /4 g 1 C2 Port 2 Z0 & l= λ r /4 The frequency !r is the frequency at which ` = r =4 and = 1. 20 OHIO S ATE T UNIVERSITY T.H.E Microwave Laboratory The Ohio State University % ' Z0 L1 V G Commensurate lter response $ λr/4 L1 L3 Port 2 ! = tan 2 !r ranges from 0 to in nity as ! ranges from 0 to !r . Response of the distributed lter is then periodic with period 2!r , unlike the lumped circuit. Z0 L3 Port 2 Port 1 C2 Z0 VG Port 1 1 C2 λr/4 Z0 L I (ω ) L I (Ω) Passband Stopband LI Passband Stopband Passband Stopband 3 dB 3 dB ωc Ωc Lump case Line case ω Ω ωc ωr 2ωr 3ω r 4ω r ω & LI (!) = 1 + (!=!c )6 ! LI ( ) = 1 + ( = c )6 21 OHIO S ATE T UNIVERSITY T.H.E Microwave Laboratory The Ohio State University % ' Comparison of Lumped and Distributed Filter Response $ Responses of the distributed LI ( ) = 1=jS21 j2 = 1 + ( = c )6 and lumped element LI (!) = 1=jS21 j2 = 1 + (!=!c )6 \low-pass" Butterworth lters versus are compared below: 50 Lumped Distributed 40 Insertion Loss (dB) 30 N=3, ωr=3 ωc 20 10 & 0 0 ωc/ωr 1 2 ω/ωr 3 4 5 T.H.E The period given above applies to all commensurate lters. 22 OHIO S ATE T UNIVERSITY Microwave Laboratory The Ohio State University % ' Z0 L1 VG Port 1 Commensurate lter design $ λr/4 L1 L3 Port 2 Notice that our original low pass lter can be used as a low pass, band-stop, or high pass lter depending on which portion of the spectrum we use. Z0 L3 Port 2 C2 Z0 V G Port 1 1 C2 λr/4 Z0 L I (ω ) L I (Ω) Passband Stopband LI Passband Stopband Passband Stopband & 3 dB ωc Ωc Lump case Line case ω Ω 3 dB ωc ωr 2ωr 3ω r 4ω r OHIO S ATE T UNIVERSITY T.H.E ω Microwave Laboratory 23 The Ohio State University % ' c 1 Frequency and Impedance Scaling $ ! Pozar gives the design equation for the case where !c = 1 !r ( c = tan 2!rc = 1) 2 is the lter \breakpoint". However, this is not the general case. In general, for distributed lters we should perform frequency scaling of the low pass prototypes using c rather than !c . 0 L0k = Z0gk and Ck = Zgk 0 c using g3 c = tan !c 2!r g5 Port 2 g1 VG Port 1 g2 g4 gN+1 & Lowpass prototype OHIO S ATE T UNIVERSITY T.H.E Microwave Laboratory 24 The Ohio State University % ' Realizability Issue $ Port 1 The circuit can be directly realized with coaxial and coplanar waveguides using a complex layout Z0 λr/4 L1 L3 VG C2 λr/4 Z0 Port 2 1111111111111111111111111111111111111111 0000000000000000000000000000000000000000 1111111111111111111111111111111111111111 0000000000000000000000000000000000000000 1111111111111111111111111111111111111111 0000000000000000000000000000000000000000 1111111111111111111111111111111111111111 0000000000000000000000000000000000000000 1111111111111111111111111111111111111111 0000000000000000000000000000000000000000 1111111111111111111111111111111111111111 0000000000000000000000000000000000000000 1111111111111111111111111111111111111111 0000000000000000000000000000000000000000 1111111111111111111111111111111111111111 0000000000000000000000000000000000000000 1111111111111111111111111111111111111111 0000000000000000000000000000000000000000 1111111111111111111111111111111111111111 0000000000000000000000000000000000000000 1111111111111111111111111111111111111111 0000000000000000000000000000000000000000 1111111111111111111111111111111111111111 0000000000000000000000000000000000000000 1111111111111111111111111111111111111111 0000000000000000000000000000000000000000 1111111111111111111111111111111111111111 0000000000000000000000000000000000000000 1111111111111111111111111111111111111111 0000000000000000000000000000000000000000 1111111111111111111111111111111111111111 0000000000000000000000000000000000000000 1111111111111111111111111111111111111111 0000000000000000000000000000000000000000 1111111111111111111111111111111111111111 0000000000000000000000000000000000000000 1111111111111111111111111111111111111111 0000000000000000000000000000000000000000 1111111111111111111111111111111111111111 0000000000000000000000000000000000000000 1111111111111111111111111111111111111111 0000000000000000000000000000000000000000 1111111111111111111111111111111111111111 0000000000000000000000000000000000000000 1111111111111111111111111111111111111111 0000000000000000000000000000000000000000 1111111111111111111111111111111111111111 0000000000000000000000000000000000000000 & However series stubs cause problems in microstrip or striplines. It would be nice to be able to replace these by shunt stubs. OHIO S ATE T UNIVERSITY T.H.E Microwave Laboratory 25 The Ohio State University % ' C u.e. Z0 L (Ω) (Ω) u.e. Z0 Kuroda identities $ u.e. Z0 n n−1 Z (Ω) 0 n n= 1+C Z 0 u.e. n Z0 (Ω) n−1 n Z0 n=1 + The Kuroda identities provide several equivalents for distributed networks. Equivalence is proven simply by showing the ABCD matrices are the same for both cases. There are several equivalences which can be proved: see book p. 464-465. Two that are most useful for our interests are: L Z0 & The \inductors" and \capacitors" illustrated in the Kuroda identities are in fact short-circuited or open-circuited transmission lines. These are OHIO distributed network identities. S ATE T T.H.E UNIVERSITY Microwave Laboratory 26 The Ohio State University % ' Port 1 C u.e. Z0 Proof Port 2 $ Port 1 u.e. Z0 n n−1 Z (Ω) 0 n Port 2 n= 1+C Z 0 2 3 4 A B 5= p 1 The ABCD matrix on left side is: 1 ; S 2 CS 1 2 41 Z0 CD The ABCD matrix on right side is: 2 3 4 A B 5= p 1 CD & 0 The two networks are equivalent if: Z0 = Z0 + L, C + Y0 = Y00 and CZ0 = LY00 . 0 Solving for Z0 and L we get using n = 1 + Z0 C : Z00 = Z0 and L = n ; 1 Z0 n n 27 Microwave Laboratory (Ω) 3 32 0 5 4 1 Z0 S 5 =p 1 S 1 1 ; S2 2 4 1 Z0 S (C + Y0 )S CZ0 S 2 + 1 3 5 1 ; S2 2 0 6 1 Z0S 4 S Z0 0 1 32 3 7 4 1 LS 5 = p 1 5 01 1;S 3 2 0 6 1 (Z0 + L)S 7 5 4 S LS2 2 Z0 0 Z0 0 +1 OHIO S ATE T UNIVERSITY T.H.E The Ohio State University % ' Z0 V Port 1 g Using the Kuroda identities $ L3 To use the Kuroda identities we usually need to add additional Z0 unit elements before or after the lter. L1 λ r /4 λ r /4 u.e. Z0 1 C2 u.e. Z0 Port 2 Z0 λ r /4 λ r /4 λ r /4 The identities can then be applied to get rid of series elements Z0 & V g Port 1 1 C1 u.e. Z1 1 C2 u.e. Z2 1 C3 Port 2 Z0 T.H.E λ r /4 λ r /4 λ r /4 λ r /4 λ r /4 OHIO S ATE T UNIVERSITY Microwave Laboratory 28 The Ohio State University % ' V G A Simple Normalized Example 1 1 2 1 1 $ 1 1 = n−1 2 n Z0 1 1 VG 1 u.e. 1 L= 1 2 u.e. Z 0 =1 n=1+ 1 VG L =2 Z0 u.e. 2 2 n Z 0= λ r /4 1 2 u.e. 2 & 111111111111111111111111111111111111 000000000000000000000000000000000000 111111111111111111111111111111111111 000000000000000000000000000000000000 111111111111111111111111111111111111 000000000000000000000000000000000000 111111111111111111111111111111111111 000000000000000000000000000000000000 111111111111111111111111111111111111 000000000000000000000000000000000000 Microstrip λ r /4 111111111111111111111111111111111111 000000000000000000000000000000000000 111111111111111111111111111111111111 000000000000000000000000000000000000 111111111111111111111111111111111111 000000000000000000000000000000000000 111111111111111111111111111111111111 000000000000000000000000000000000000 111111111111111111111111111111111111 000000000000000000000000000000000000 OHIO S ATE T UNIVERSITY T.H.E Microwave Laboratory 29 The Ohio State University % ' Z0 V g Results for a general 3rd order Circuit $ L3 Port 2 For the general case using the Kuroda identities we get: Z0 L1 VG Port 1 C2 Z0 Port 1 1 C1 u.e. Z1 1 C2 u.e. Z3 1 C3 Port 2 Z0 λ r /4 λ r /4 λ r /4 λ r /4 λ r /4 & 1 L In the above we have Z1 = Z3 = nZ0 , C = n;0 , and n = 1 + Z0 nZ assuming L1 = L3 = L. OHIO S ATE T UNIVERSITY T.H.E Microwave Laboratory 30 The Ohio State University % ' Example $ Design a 3 element Butterworth low pass lter with a 3 dB point of 2 GHz. The lter will be implemented in 50 ohm microstrip line sections that are one quarter wavelength long at 6 GHz. Choose the series L, shunt C, series L prototype. For this design, !r = 2 6(109), but !c = 2 2(109). We need to impedance and frequency scale ! the prototype using Z0 = 50 and with c = tan 2!rc = tan =6. The resulting values for the L's and C 0s are L1 = L3 = 86:6025, C2 = 0:0693. Now use the Kuroda identities to replace the series stubs. We nd unit elements of impedance 136:6025 Ohms and shunt \cap" values of 1=78:8675. & Thus we have shunt open stubs with characteristic impedances 78:8675 Ohms ( rst and third), and 14:4338 Ohms. In between are line sections of impedance 136:6025 Ohms. All sections will be a quarter wavelength long at 6 GHz. OHIO S ATE T UNIVERSITY T.H.E Microwave Laboratory 31 The Ohio State University % ' 20 15 Insertion Loss (dB) 10 Example $ Resulting insertion loss for our design: 5 & 0 0 1 2 f/fr 3 4 OHIO S ATE T UNIVERSITY T.H.E Microwave Laboratory 32 The Ohio State University % ' where Couple Line Realization $ 2 Microstrip Z 0e & Z 0o Sometimes the impedances needed using the Kuroda identies are not realizable and alternative implementation is required. 10 Identities (see handout posted) for coupled line realization: λ r /4 1 1111111111 0 0000000000 u.e. L 11 1111111111 00 0000000000 1111111111 0000000000 Z0 1 2 1111111111 0000000000 1111111111 0000000000 1 1111111111 0000000000 λ r /4 q Z0e = Z0 + L + L(Z0 + L) & Z Z0o = 2Z 0Z0eZ ; 0e 0 OHIO S ATE T UNIVERSITY T.H.E Microwave Laboratory 33 The Ohio State University % ' u.e. 1 Z0 λ r /4 L 2 Proof λ r /4 1 0 1 1111111111 0 0000000000 0000000000 1111111111 1111111111 0000000000 1111111111 0000000000 0000000000 1111111111 Microstrip Z 0e & Z 0o $ 1 2 & Converting the S-parameters we found for a coupled line to Z-parameters we get: j 1 Z11 = Z22 = Z33 = Z44 = S Z0e + Z0o = ; 2 (Z0e + Z0o) cot 2 1 j Z12 = Z21 = Z33 = Z44 = S Z0e ; Z0o = ; 2 (Z0e ; Z0o) cot 2 p 1 ; S 2 Z0e ; Z0o = ; j (Z ; Z ) 1 Z13 = Z31 = Z34 = Z43 = S 2 2 0e 0 sin p 1 ; S 2 Z0e + Z0o = ; j (Z + Z ) 1 Z14 = Z41 = Z23 = Z32 = S 2 2 0e 0 sin 34 OHIO S ATE T UNIVERSITY T.H.E Microwave Laboratory The Ohio State University % ' (Z0e +Z0o )S Proof II $ 01 1 ; S2 2 3 + 4 1 (Z0 2 L)S 5 S Z0 LS Z0 Using the boundary condition V2 = 0 and I3 = 0 we get: (Z0e +Z0o )2S 32 3 2 3 2 2Z Z p 2Z0e Z0o 1;S 2 0e 0o I1 4 V1 5 = 6 (Z0e+Z0o )S 2 2 (Z0e +Z20o)S 5 4 2Z0e Z0o p1;S S (Z0e;Z0o ) +4Z0e Z0o 7 4 5 I4 V4 2 32 3 4 1 Z0S 5 4 1 LS 5 = p 1 2 S Z0 The ABCD matrix on right side is: 2 3 4 A B 5= p 1 CD 1;S 1 +1 2Z 3 p Z0 1;S 2 0 S 5 ) Z ] = 4 pS 2 2 +Z Z0 1;S LS 0 S S & The two networks are equivalent if: 2Z0e Z0o and L = (Z0e ; Z0o )2 Z0 = Z + Z 2(Z0e + Z0o ) 0e 0o 35 OHIO S ATE T UNIVERSITY T.H.E Microwave Laboratory The Ohio State University % ' 1 V G Example for a Normalized Circuit 1 V G $ 1 1 1 2 n−1 Z 0 n 1 2 1 1 2 u.e. 1 u.e. 1 1 C= 1 u.e. Z 0 =1 n=1+ C Z 0 =2 1 VG u.e. 1 1 2 2 u.e. 1 2 Z0 u.e. =1 n 2 & 11 00 1111111111 0000000000 1 1111111111 0 0000000000 1111111111 1111111111 0000000000 0000000000 111111111111111111111111111111111111 000000000000000000000000000000000000 111111111111111111111111111111111111 000000000000000000000000000000000000 111111111111111111111111111111111111 000000000000000000000000000000000000 11111111111 00000000000 Microstrip 1 11111111111 0 00000000000 1 0 OHIO S ATE T UNIVERSITY T.H.E Microwave Laboratory 36 The Ohio State University % ' Lecture # 25 $ Bandstop lters as a prototype for Bandpass lters Realizability Issues Horton & Wenzel Filters Coupled Line Filters & OHIO S ATE T UNIVERSITY T.H.E Microwave Laboratory 37 The Ohio State University % ' VG Bandstop lters as a prototype for Bandpass lters $ Port 2 Like the lowpass prototype is the basis for realizing bandstop lters, similarly the highpass propotype is the basis for realizing bandpass lters. 1 1/g 1 Port 1 1/g 3 1/g 2 1/g 5 1/g 1/g 4 N+1 Highpass prototype Commensurate lters are realized by replacing ! by = tan 2 !!r . & This is equivalent to replacing inductors by shorted stubs and capacitors for open stubs. All the stub length are ` = r =4 (Note: c = fr r ). T.H.E As a consequence the lter has a periodic response. The lter response OHIO will repeat every 2!r . S ATE T UNIVERSITY Microwave Laboratory 38 The Ohio State University % ' Z0 V G Response of a Commensurate Bandstop Filter $ 1 C3 Port 2 ! = tan 2 !r ranges from 0 to in nity as ! ranges from 0 to !r . Response of the distributed lter is then periodic with period 2!r , unlike the lumped circuit. λr/4 Z0 C1 Port 1 1 C1 C3 Port 2 L2 Z0 VG Port 1 L2 λr/4 L I (ω ) L I (Ω) LI Z0 Stopband Passband Stopband Passband Stopband Passband 3 dB 3 dB ωc Ωc Lump case Line case ω Ω ωc ωr 2ωr 3ω r 4ω r ω & LI (!) = 1 + (!c=!)6 ! LI ( ) = 1 + ( c= )6 39 OHIO S ATE T UNIVERSITY T.H.E Microwave Laboratory The Ohio State University % ' Realizability Issue $ Port 1 The circuit can be directly realized with coaxial and coplanar waveguides using a complex layout Z0 λr/4 L1 L3 VG C2 λr/4 Z0 & However series stubs cause problems in microstrip or striplines and there are no Kuroda identity to transform series open stubs in shunt stubs. OHIO S ATE T UNIVERSITY T.H.E Port 2 1111111111111111111111111111111111111111 0000000000000000000000000000000000000000 1111111111111111111111111111111111111111 0000000000000000000000000000000000000000 111111111111111111 000000000000000000 111111111111111111 000000000000000000 1111111111111111111111111111111111111111 0000000000000000000000000000000000000000 111111111111111111 000000000000000000 111111111111111111 000000000000000000 111111111111111111 000000000000000000 111111111111111111 000000000000000000 1111111111111111111111111111111111111111 0000000000000000000000000000000000000000 111111111111111111 000000000000000000 111111111111111111 000000000000000000 1111111111111111111111111111111111111111 0000000000000000000000000000000000000000 111111111111111111 000000000000000000 111111111111111111 000000000000000000 1111111111111111111111111111111111111111 0000000000000000000000000000000000000000 1111111111111111111111111111111111111111 0000000000000000000000000000000000000000 1111111111111111111111111111111111111111 0000000000000000000000000000000000000000 1111111111111111111111111111111111111111 0000000000000000000000000000000000000000 1111111111111111111111111111111111111111 0000000000000000000000000000000000000000 1111111111111111111111111111111111111111 0000000000000000000000000000000000000000 1111111111111111111111111111111111111111 0000000000000000000000000000000000000000 1111111111111111111111111111111111111111 0000000000000000000000000000000000000000 1111111111111111111111111111111111111111 0000000000000000000000000000000000000000 1111111111111111111111111111111111111111 0000000000000000000000000000000000000000 1111111111111111111111111111111111111111 0000000000000000000000000000000000000000 1111111111111111111111111111111111111111 0000000000000000000000000000000000000000 1111111111111111111111111111111111111111 0000000000000000000000000000000000000000 1111111111111111111111111111111111111111 0000000000000000000000000000000000000000 1111111111111111111111111111111111111111 0000000000000000000000000000000000000000 1111111111111111111111111111111111111111 0000000000000000000000000000000000000000 1111111111111111111111111111111111111111 0000000000000000000000000000000000000000 1111111111111111111111111111111111111111 0000000000000000000000000000000000000000 Microwave Laboratory 40 The Ohio State University % ' LI ( ) = 1 + c 2m 50 40 Insertion Loss (dB) 30 Horton & Wenzel Filters $ Lumped Butterworth Horton & Wenzel Horton & Wenzel introduced modi ed Butterworth & Chebyshev polynomials which allow the synthesis of bandstop/bandpass lters. Butterworth Example: 1+ 1+ 2 c !n 2 with m # of inductors or capacitors n # of unit elements Example of response for n = 2 and m = 1: N=2, M=1 20 ω =2/3 ω c r & 10 0 0 ωc/ωr 1 2 ω/ωr 3 4 5 OHIO S ATE T UNIVERSITY T.H.E Microwave Laboratory 41 The Ohio State University % ' Z in Synthesis Results for n = 2 and m = 1 $ Z0 Consider a Butterworth with n = 2 and m = 1 ! 2m 1 + 2 n with n = 1 inductors or capacitors c LI ( ) = 1 + c 1+ 2 m = 2 unit elements The lters are synthesized in parts using Richard's Synthesis theorem (see Chapter 3) to extract unit elements. 3rd Order example: Z0 Z 01 U.E. L Z 02 U.E. Z0 Z 01 C Z 02 U.E. U.E. Z0 & OHIO T This is a non-redundant design (minimum number of sections possible). S ATE UNIVERSITY Z in T.H.E Microwave Laboratory 42 The Ohio State University % ' Z in =K2 Z L De nition of Ideal Inverters $ Admittance Inverter Elements that are useful in realizing lters are inverters: Impedance Inverter K ZL +90 J YL +90 Y in =J 2 YL Their ABCD matrices are as follow: 3 2 0 jK 5 K inverter: 4 j K 0 2 0 J inverter: 4 ;j ;jJ 0 3 J5 & As indicated in the gures, K inverters produce an impedance inversely proportional to the load impedance with a phase shift of 90 , while J inverter produce an admittance inversely proportional to the load admittance. OHIO T.H.E S ATE T UNIVERSITY Microwave Laboratory 43 The Ohio State University % ' Z0 V G Alternate Highpass Filter Implementation $ Port 2 An alternate realize implementation for the highpass prototype can be realized using ideal inverters: 1 1/g 1 VG Port 1 1/g 3 1/g 2 1/g 5 1/g 1/g 4 N+1 C1 K 01 K12 C2 CN K N,N+1 Port 2 Z0 Port 1 Highpass prototype & A possible implementation uses: C = 1 K = Z0 K i c Z0 01 p g1 = p Z0 i i+1 44 gigi+1 Z and KN N +1 = p 0 gN OHIO S ATE T UNIVERSITY T.H.E Microwave Laboratory The Ohio State University % ' Narrowband Coupled-Line Implementation 1 1 Z 0o θ U. E. Z 0e Z 0o 2 Microstrip 1 Z 0o 2 1 $ 1000000000 0111111111 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 θ Z 0e & Z 0o 2 A coupled line is equivalent to a transmission line anked by two series shorted stub (generatlized capacitor) on both sides We know that a transmission line of length ` = r can be used as an inverter around !r . So this leads to narrowband implementation for our bandpass lter prototype. & But we are interested in a broadband implementation ... OHIO S ATE T UNIVERSITY T.H.E Microwave Laboratory 45 The Ohio State University % ' Approximate Broadband Coupled-Line Implementation of Inverter $ 2 An approximately equivalent to an admittance inverter can be realized using coupled-lines by equating their image image impedances Zi and propagation constant cos at = =2. θ θ Microstrip 1 Z0 2 1 Z0 J −90 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 θ Z 0e & Z 0o For this equivalence to be true, we need Z0 e = Z0 1 + JZ0 + (JZ0) h h 2 2 i & Z0 o = Z0 1 ; JZ0 + (JZ0) The above equivalence is valid for =2 (near the design frequency.) The electrical length = ` is such that at the lter center frequency !r OHIO the line length is a quarter-wavelength (` = r =4). S ATE T T.H.E UNIVERSITY i Microwave Laboratory 46 The Ohio State University % ' A Little Detour: Concept of Image Impedances $ Consider an 2-port circuit. Z i1 1 AB CD 2 Z i2 Z in1 Z in2 The image impedances at port 1 and 2 are the impedances verifying: Zin1 = Zi1 and Zin2 = Zi2 The following expression can be derived: & AB Zi1 = CD = Z11 and Zi2 = BD = Z22 Y11 AC Y22 Image impedances which results from impedance matching are to be di erentiated from conjugate impedance matching used for maximum power transfer. 47 s s s s OHIO S ATE T UNIVERSITY T.H.E Microwave Laboratory The Ohio State University % ' Image Propagation Constant $ Zi1 = Zi2 = Z0 and ej = ej The image propagation constant is: 1p j = pAD + pBC = p e AD ; BC For example for a transmission line of impedance Z0 and electrical length = ` we have (!) & In general Zi1(!), Zi1(!), and ej uniquely characterize a circuit. are frequency dependent functions which T.H.E OHIO S ATE T UNIVERSITY Microwave Laboratory 48 The Ohio State University % ' θ 1 Z0 Conditions for Approximate Equivalence θ J −90 Z0 2 Microstrip 1 $ Z 0e & Z 0o 2 θ 10000000000 01111111111 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 The image impedance of the coupled lines can be derived from its Z parameters (see previous lecture) to be: 1 q(Z ; Z )2 csc2 ; (Z + Z )2 cot2 = 1 (Z ; Z ) for = Zi = 2 0e 0o 0e 0o 2 0e 0o 2 cos = Z0e + Z0o cos & Z0e ; Z0o The image impedance for the inverter circuit is calculated to be: v u 2 u JZ0 sin2 ; 1=J cos2 2 t Zi = 1=(JZ 2 ) sin2 ; J cos2 = JZ0 for = 2 0 cos = A = J Z0 + 1 sin cos JZ0 49 OHIO S ATE T UNIVERSITY T.H.E Microwave Laboratory The Ohio State University % ' Conditions for Approximate Equivalence II $ h i Equating the image impedance and propagation constant at = 2 we get the following system of equations: 1 (Z ; Z ) = JZ 2 0 2 0e 0o Z0e + Z0o = JZ + 1 0 Z0e ; Z0o JZ0 which can be solved to obtain Z0e and Z0o : Z0 e = Z0 1 + JZ0 + (JZ0) 2 2 & Z0 o = Z0 1 ; JZ0 + (JZ0) h i OHIO S ATE T UNIVERSITY T.H.E Microwave Laboratory 50 The Ohio State University % ' Z0J1 = s Coupled line lter example $ Design a 3 section bandpass lter with a 10% bandwidth centered at 2 GHz for a 50 ohm system. The lter should have a 0.5 dB equal ripple response. First from Pozar Table 8.4, we get the low pass prototype: g1 = g3 = 1:5963, g2 = 1:0967, g4 = 1. = 0:1 for a 10% fraction bandwidth. We apply the design equations: 2g1 = 0:3137 & Z0J2 = 2pg g = 0:1187 12 Z0J3 = 2pg g = 0:1187 Z0J4 = s 23 2g3 g4 = 0:3137 OHIO S ATE T UNIVERSITY T.H.E Microwave Laboratory 51 The Ohio State University % ' Z Z0(1) o Z0(2) e Z0(2) o (1) 0e Coupled Line Filter Layout $ 1111111111 0000000000 1111111111 0000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 00000000000000000000 11111111111111111111 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111 00000000000 11111111111 00000000000 Microstrip 11111111111 00000000000 Next we nd the line even and odd mode impedances from: = = = = & Z0 1 + J1Z0 + (J1Z0) = 70:6054 h i 2 Z0 1 ; J1Z0 + (J1Z0) = 39:2354 h i 2 Z0 1 + J2Z0 + (J2Z0) = 56:6395 h i 2 Z0 1 ; J2Z0 + (J2Z0) = 44:7695 2 h i Third and fourth line values are the same as for lines 2 and 1, respectively. 52 OHIO S ATE T UNIVERSITY T.H.E Microwave Laboratory The Ohio State University % ' 0 −10 2 Coupled line lter Response $ Comparison of prototype with coupled line implementation. Also shown is a Butterworth Horton & Wenzel lter for comparison. Gain |S21| −20 −30 −40 & Theory Real Horton 0.8 −50 0.6 Normalized Frequency f/fr 1 1.2 1.4 OHIO S ATE T UNIVERSITY T.H.E Microwave Laboratory 53 The Ohio State University % ...
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This note was uploaded on 04/30/2011 for the course ECE 710 taught by Professor Roblin during the Spring '11 term at Ohio State.

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