The Chebyshev Matching Transformer

Transcription

1 /9/ The Chebyshev Matching Transforer /5 The Chebyshev Matching Transforer An alternative to Binoial (Maxially Flat) functions (and there are any such alternatives!) are Chebyshev polynoials. Pafnuty Chebyshev 8-89 Chebyshev solutions can provide functions Γ ( ω) with wider bandwidth than the Binoial case albeit at the expense of passband ripple. It is evident fro the plot below that the Chebychev response is far fro axially flat! Instead, a Chebyshev atching network exhibits a ripple in its passband. ote the agnitude of this ripple never exceeds soe axiu value Γ (within the pass-band). The two frequencies at which the value Γ ( ω) does increase beyond Γ define the bandwidth of the atching network. We denote these frequencies ω π f (the plot above shows the locations of the frequencies for ). Ji Stiles The Univ. of Kansas Dept. of EECS

2 /9/ The Chebyshev Matching Transforer /5 Γ Figure 5.7 (p. 55) Reflection coefficient agnitude versus frequency for the Chebyshev ultisection atching transforers of Exaple 5.7. f f ote that the bandwidth defined by f increases as the nuber of sections is increased. ote also that the reflection coefficient in not necessarily zero at the design frequency f!!! Instead, we find: ( f f ) Γ Γ for odd-valued for even-valued Ji Stiles The Univ. of Kansas Dept. of EECS

3 /9/ The Chebyshev Matching Transforer 3/5 ow, Chebyshev transforers are syetric, i.e.: Γ Γ, Γ Γ, etc. Recall we can express the ulti-section function Γ ( θ ) (where θ ωt β ) in a sipler for when the transforer is syetric: where: j θ ( ) e cos cos( ) cos ( n) θ G ( θ) Γ θ Γ θ +Γ θ + +Γn + + G ( θ ) Γ Γ ( ) cos θ for even for odd ow, the reflection coefficient of a Chebyshev atching network has the for: where θ ω T cos θ Γ ( θ ) j θ Ae T cos θ jθ Ae T cos sec ( θ θ ) Ji Stiles The Univ. of Kansas Dept. of EECS

4 /9/ The Chebyshev Matching Transforer /5 The function T ( cos sec ) order. θ θ is a Chebyshev polynoial of T ( x) x T ( x) x 3 T ( x) x 3x 3 T ( x) x x We can deterine higher-order Chebyshev polynoials using the recursive forula: T ( x) xt ( x) T ( x) n n n Inserting the substitution: x cosθ secθ into the Chebyshev polynoials above (and then applying a few trig identities) gives the results shown in equations 5.6 on page 5 of your book: Ji Stiles The Univ. of Kansas Dept. of EECS

5 /9/ The Chebyshev Matching Transforer 5/5 3 ( θ θ) θ θ ( θ θ ) θ ( θ) T cos sec cos sec T cos sec sec + cos ( sec θ ) sec θ cos θ + 3 ( ) ( ) T cos θ sec θ sec θ cos 3θ + 3cos θ 3sec θ cos θ T ( 3 3) sec θ cos 3θ + sec θ sec θ 3 ( cos θsec θ) sec θ ( cos θ + cos θ + 3) sec θ ( cos θ + ) + sec θ cos θ θ sec sec ( θ ) co ( 3sec θ sec θ ) s θ ote that these polynoials have a cos θ ter, a cos ( ) θ ter, cos ( ) θ ter, etc. just like the syetric ulti-section transforer function! cos θ For exaple, a -section Chebyshev atching network will have the for: j θ ( θ) Ae T ( cosθsecθ ) Γ j θ Ae sec θ cos θ sec θ ( sec θ ) co ( 3sec θ sec θ ) + s θ + + While the general for of a -section atching transforer is a polynoial with these sae ters: Ji Stiles The Univ. of Kansas Dept. of EECS

6 /9/ The Chebyshev Matching Transforer 6/5 Γ Γ +Γ + Γ jθ ( θ) e cos θ cos( ) θ ] e Γ θ +Γ + Γ j θ e Γ cos θ +Γ cosθ + Γ] ( ) θ ] j θ cos cos Thus, we can deterine the values of arginal reflection coefficients Γ, Γ, Γ by siply equating the 3 ters of the two previous expressions: j θ j θ e Γ cos θ Ae sec θ cos θ Γ Asec θ j θ j θ ( θ ) ( ec θ ) e Γ cos θ Ae sec θ sec cos θ Γ A sec s θ ( θ θ ) ( 3 θ θ ) j θ j θ e Γ Ae 3sec sec + Γ A sec sec + And because it s syetric, we also know that Γ 3 Γ and Γ Γ. ow, we can again (i.e., as we did in the binoial atching network) deterine the values of characteristic ipedance Z n, using the iterative (approxiate) relationship: Ji Stiles The Univ. of Kansas Dept. of EECS

7 /9/ The Chebyshev Matching Transforer 7/5 Z Z exp n+ n Γn Q: But what about the value of A? A: Also using the sae boundary condition analysis that we used for the binoial function, we find fro our transission line knowledge that for any ulti-section atching network, at θ : RL Z Γ ( θ ) R + Z L Likewise, a Chebyshev atching network will have the specific value at θ of: ( θ ) ( ) ( θ ( )) ( sec θ ) j Γ Ae T sec cos AT These two results ust of course be equal, and equating the allows us to solve for A : R Z L A R L + Z T secθ ( ) Here again (just like the binoial case) we will find it advantageous to use the approxiation: Ji Stiles The Univ. of Kansas Dept. of EECS

8 /9/ The Chebyshev Matching Transforer 8/5 R Z R ln R Z Z L L + L So that the value A is approxiately: ( Z L ) ( secθ ) ln R A (A can be negative!) T Q: Gosh, both the values of arginal reflection coefficients Γ n and value A depend explicitly on sec θ. Just what is this value, and how do we deterine it? A: Recall that θ ω T, where T is the propagation tie through one section (i.e., T ) and ω πf defines the v p bandwidth associated with ripple value Γ. Thus, for a given ripple Γ and value, we can find sec θ by solving this equations ( θ θ ) Γ Γ ( ) ( θ θ ) ( θ θ ) jθ A e T sec cos jθ A e T sec cos AT Ji Stiles The Univ. of Kansas Dept. of EECS

9 /9/ The Chebyshev Matching Transforer 9/5 Q: Yikes! The value sec θ disappeared fro this equation; how can we use this to deterine sec θ? A: Don t forget the value A!! Inserting this into the expression: Γ Γ ( θ θ) ( ) ln( R Z L ) ( secθ ) AT T T ( ) One property of Chebyshev polynoials is that: T ( ) for all values Therefore, we conclude: ( θ θ ) Γ Γ ln R T ( Z L ) ( secθ ) And so rearranging: T secθ ln R Z L Γ ( ) ( ) ote that RL, Z and Γ are design paraeters, thus we can use the above to deterine T ( secθ ), and thus sec θ!!! Ji Stiles The Univ. of Kansas Dept. of EECS

10 /9/ The Chebyshev Matching Transforer /5 Q: U, I not at all clear about how one deterines sec θ T secθ. Can you be ore specific?? fro the value ( ) A: Perhaps I should. Recall that the Chebyshev polynoials we use in the design of T cos θ sec θ, this atching transforer were of the for ( ) i.e.: jθ ( θ ) Ae T ( cosθsecθ) Γ ote the value ( ) Chebyshev T ( cos sec ) is the case when θ!! T secθ is just the agnitude of the θ θ when evaluated at cos θ, which ( θ θ ) ( θ ) ( θ) T cos sec T cos sec T sec θ ow, it can be shown that ( a phrase professors use while in hand-waving ode!) for all values θ outside the passband of the atching network, the general for of the Chebyshev polynoial can also be written as: ( θ θ) ( θ θ) T sec cos cosh cosh sec cos ote the value θ ωt eans that the frequency ω. This frequency is ost definitely outside the passband, and thus according to the above expression: Ji Stiles The Univ. of Kansas Dept. of EECS

11 /9/ The Chebyshev Matching Transforer /5 Likewise, ( θ ( θ ) ) ( θ ) T sec cos T sec ( sec θ ) cosh cosh Q: I know I should reeber exactly what function cosh is, but I don t. Can you help refresh y eory? A: The function cosh is the hyperbolic cosine. Recall that cosine (the regular kind) can be expressed as: cos θ e jθ + e jθ Siilarly, the hyperbolic cosine is: e coshx x + e Plotting coshx fro x to x + : x Ji Stiles The Univ. of Kansas Dept. of EECS

12 /9/ The Chebyshev Matching Transforer /5 Hopefully it is apparent to you, fro the above expression for coshx, that it results in a real value that is always greater than or equal to one (provided x is real): coshx for < x < Thus coshx coshx. Q: What about cosh y? A: The function cosh y is the inverse hyperbolic cosine, aka the hyperbolic arccosine (arcosh y). ote that this value is defined only for y, and is specifically: ( ) cosh y ± ln y + y for y ote that there is always two solutions (positive and negative) for the inverse hyperbolic cosine! You will find that ost scientific calculators support the cosh and cosh - functions as well. Anyway, cobining the previous results, we find: RL T ( sec θ ) ln cosh cosh ( sec θ ) Γ Z Therefore: Ji Stiles The Univ. of Kansas Dept. of EECS

13 /9/ The Chebyshev Matching Transforer 3/5 Γ R ( θ ) L ln cosh cosh sec Z ow (finally!) we have a for that can be anipulated algebraically. Solving the above equation forsec θ : R L ± Z sec θ cosh cosh ln Γ Of course, the specific values of θ can be deterined with the sec (i.e., arcsecant) function. ote there are two solutions for sec (one for the plus sign and one for the inus); The two solutions are related as: θ π θ ow, we can convert the values of θ into specific bandwidth frequencies f and f. Since θ ωt, we find: And siilarly: v p ω θ θ T v p ω θ θ T Ji Stiles The Univ. of Kansas Dept. of EECS

14 /9/ The Chebyshev Matching Transforer /5 ote that there are two solutions θ for this equation one value of θ ( θ )will be less than π (defining the lower passband frequency), while the other ( θ ) will be greater than π (defining the upper passband frequency). Moreover, we find that the two values of θ will be syetric about the value π! For exaple, if the lower value θ is π π, then the upper value θ will be π + π. Q: So?? A: This eans that the center of the passband will be defined by the value θ π and the center of the passband is our design frequency ω! In other words, since ωt π : v p π π ω T Thus, we set the center (i.e., design) frequency by selecting the proper value of section length. ote the above expression is precisely the sae result obtained for the Binoial atching network, and thus we have precisely the sae design rule! That design rule is, set the section lengths such that they are a quarter wavelength at the design frequency ω : Ji Stiles The Univ. of Kansas Dept. of EECS

15 /9/ The Chebyshev Matching Transforer 5/5 λ where λ v p ω. Suarizing, the Chebyshev atching network design procedure is:. Deterine the value required to eet the bandwidth and ripple Γ requireents.. Deterine the Chebychev function j θ Γ θ Ae T cosθsecθ. ( ) ( ) 3. Deterine all Γ n by equating ters with the syetric ultisection transforer expression: j θ ( ) e cos cos( ) cos ( n) θ G ( θ) Γ θ Γ θ +Γ θ + +Γn + +. Calculate all Z n using the approxiation: Γ n Z + ln Z n n 5. Deterine section length λ. Ji Stiles The Univ. of Kansas Dept. of EECS