Stepped-Impedance Low-Pass Filters

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1 4/23/27 Stepped Impedance Low Pass Filters 1/14 Stepped-Impedance Low-Pass Filters Say we know te impedance matrix of a symmetric two-port device: = Regardless of te construction of tis two port device, we can model it as a simple T-circuit, consisting of tree impedances: Port 1 Port In oter words, if te two series impedances ave an impedance value equal to, and te sunt impedance as a value equal to, te impedance matrix of tis Tcircuit is: = Tus, any symmetric two-port network can be modeled by tis T-circuit! Jim Stiles Te Univ. of Kansas Dept. of EECS

2 4/23/27 Stepped Impedance Low Pass Filters 2/14 For example, consider a lengt of transmission line (a symmetric two-port network!): Recall (or determine for yourself!) tat te impedance parameters of tis two port network are: = = j cot β = = j csc β Wit a little trigonometry, ICBST : j tan β = 2 Furtermore, if β is small: sin β β cos β 1 tan β β were β is expressed in radians. Tus, j β 2 and also: Jim Stiles Te Univ. of Kansas Dept. of EECS

3 4/23/27 Stepped Impedance Low Pass Filters 3/14 = = j β csc j β Tus, an electrically sort ( β 1) transmission line can be approximately modeled wit a T-circuit as: j β 2 j β 2 Port 1 j β Port 2 Now, consider also te case were te caracteristic impedance of te transmission line is relatively large. We ll denote tis large caracteristic impedance as. Note te sunt impedance, value j β. Since te numerator ( ) is relatively large, and te denominator ( j β ) is small, te impedance sunt device is very large. So large, in fact, tat we can approximate it as an open circuit! for β1 and j β So now we ave a furter simplification of our model: Jim Stiles Te Univ. of Kansas Dept. of EECS

4 4/23/27 Stepped Impedance Low Pass Filters 4/14 j β 2 j β 2 Port 1 Port 2 Te remaining impedances are now in series, so te circuit can be furter simplified to: j β Port 1 Port 2 Te equivalent circuit for transmission line wit sort electrical lengt ( β 1) and large caracteristic impedance ( ). Now, consider te case were te caracteristic impedance of te transmission line as a relatively low value, denoted as, were. Note te series impedance, values j ( β ) 2. Since bot and β are small, te product of te two is very small. So small, in fact, tat we can approximate it as a sort circuit! Jim Stiles Te Univ. of Kansas Dept. of EECS

5 4/23/27 Stepped Impedance Low Pass Filters 5/14 β j for β1 and 2 So now we ave anoter simplification of our model: Port 1 j β Port 2 Wic of course furter simplifies to: Port 1 j β Port 2 Te equivalent circuit for transmission line wit sort electrical lengt ( β 1) and small caracteristic impedance ( ). Q: But, wat does all tis ave to do wit constructing a low-pass filter? A: Plenty! Recall tat a lossless low-pass filter constructed wit lumped elements consists of a circuit ladder of series inductors and sunt capacitors! Jim Stiles Te Univ. of Kansas Dept. of EECS

6 4/23/27 Stepped Impedance Low Pass Filters 6/14 Q: So? A: Look at te two equivalent circuits for an electrically sort transmission line. Te one wit large caracteristic impedance as te form of a series inductor, and te one wit small caracteristic impedance as te form of a sunt capacitor! I.E.: jx = j β and: j X = j ωl are identical if: j β = jωl β = ωl Tus, te series inductance of our transmission line lengt is: Jim Stiles Te Univ. of Kansas Dept. of EECS

7 4/23/27 Stepped Impedance Low Pass Filters 7/14 L = ω β Q: Yikes! Our inductance appears to be a function of frequency ω. I assume we simply set tis value to cutoff frequency ω, just like we did for Ricard s transformation? c A: Nope! We can simplify te result a bit more. Recall tat β = ω v p, so tat: β L = = ω v p In oter words, te series impedance resulting from our sort transmission line is: jω = v p Q: Wow! Tis realization seems to give us a result tat precisely matces an inductor at all frequencies rigt? A: Not quite! Recall tis result was obtained from applying a few approximations te result is not exact! Moreover, one of tese approximations was tat te electrical lengt of te transmission line be small. Tis obviously cannot be true at all frequencies. As te signal frequency increases, so does te electrical lengt our approximate solution will no longer be valid. Jim Stiles Te Univ. of Kansas Dept. of EECS

8 4/23/27 Stepped Impedance Low Pass Filters 8/14 us, tis realization is accurate only for low frequencies recall tat was likewise true for Ricard s transformations! Q: Low-frequencies? How low is low? A: Well, for our filter to provide a response tat accurately follows te lumped element design, our approximation sould be valid for frequencies up to (and including!) te filter cutoff frequency ω c. A general rule-of-tumb is tat a small electrical lengt is defined as being less tan π 4 radians. Tus, to maintain tis small electrical lengt at frequency ω c, our realization must satisfy te relationsip: ω L π β c = < c 4 Note tat tis criterion is difficult to satisfy if te filter cutoff frequency and/or te inductance value L tat we are trying to realize is large. Our only recourse for tese callenging conditions is to increase te value of caracteristic impedance. Q: Is tere some particular difficulty wit increasing? Jim Stiles Te Univ. of Kansas Dept. of EECS

9 4/23/27 Stepped Impedance Low Pass Filters 9/14 A: Could be! Tere is always a practical limit to ow large (or small) we can make te caracteristic impedance of a transmission line. For example, a large caracteristic impedance implemented in microstip/stripline requires a very narrow conductor widt W. But manufacturing tolerances, power andling capability and/or line loss (line resistance R increases as W decreases) place a lower bound on ow narrow we can make tese conductors! However, assuming tat we can satisfy te above constraint, we can approximately realize a lumped inductor of inductance value L by selecting te correct caracteristic impedance and line lengt of our sort transmission line: L = v p Q: For Ricard s Transformation, we first set te stub lengt to a fixed value (i.e., = λ c 8), and ten determined te specific caracteristic impedance necessary to realize a specific inductor value L. I assume we follow te same procedure ere? A: Nope! Wen constructing stepped-impedance low-pass filters, we typically do te opposite! Jim Stiles Te Univ. of Kansas Dept. of EECS

10 4/23/27 Stepped Impedance Low Pass Filters 1/14 1) First, we select te value of, making sure tat te sort electrical lengt inequality is satisfied for te largest inductance value L in our lumped element filter: > 4 ω π c L Tis caracteristic impedance value is typically used to realize all inductor values L in our low-pass filter, regardless of te actual value of inductance L. 2) Ten, we determine te specific lengts n of te transmission line required to realize specific filter inductors values L n : n v p = L n Q: Wat about te sunt capacitors? A: Almost forgot! Recall te low-impedance transmission line provided a sunt impedance tat matced a sunt capacitor: Jim Stiles Te Univ. of Kansas Dept. of EECS

11 4/23/27 Stepped Impedance Low Pass Filters 11/14 I.E.: j β and: j ωc are identical if: j j ωc β ωc = 1 β = Tus, te sunt capacitance of our transmission line lengt is: β C = ω But again using te fact tat β = ω v : p C = v p Jim Stiles Te Univ. of Kansas Dept. of EECS

12 4/23/27 Stepped Impedance Low Pass Filters 12/14 And tus te sunt reactance of our transmission line realization is: v p j = ω Altoug tis again appears to provide exactly te same beavior as a capacitor (as a function of frequency), it is likewise accurate only for low frequencies, were β < π 4. Tus from our realization equality: β = ωc We can conclude tat for our approximations to be valid at all frequencies up to te filter cutoff frequency, te following inequality must be valid: β π = ω C < 4 c c Note tat for difficult design cases were ω c and/or C is very large, te line caracteristic impedance must be made very small. Q: I suppose tere is likewise a problem wit making very small? Jim Stiles Te Univ. of Kansas Dept. of EECS

13 4/23/27 Stepped Impedance Low Pass Filters 13/14 A: Yes! In microstrip and stripline, making small means making conductor widt W very large. In oter words, it will take up lots of space on our substrate. For most applications te surface area of te substrate is bot limited and precious, and tus tere is generally a practical limit on ow wide we can make widt W (i.e., ow low we can make ). However, assuming tat we can satisfy te above constraint, we can approximately realize a lumped capacitor of inductance value C by selecting te correct caracteristic impedance and line lengt of our sort transmission line: C = v p Te design rules for sunt capacitor realization are tus: 1) First, we select te value of, making sure tat te sort electrical lengt inequality is satisfied for te largest capacitance value C in our lumped element filter: < π 4ω c C Jim Stiles Te Univ. of Kansas Dept. of EECS

14 4/23/27 Stepped Impedance Low Pass Filters 14/14 Tis caracteristic impedance value is typically used to realize all capacitor values C in our low-pass filter, regardless of te actual value of capacitance C. 2) Ten, we determine te specific lengts n of te transmission line required to realize specific filter capacitor values C n : = ( v ) C n p n An example of a low-pass, stepped-impedance filter design is provided on page of your book (but of course, you already knew tat rigt?). Jim Stiles Te Univ. of Kansas Dept. of EECS

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