5.7 Chebyshev Multi-section Matching Transformer

Transcription

1 3/8/6 5_7 Chebyshev Multisection Matching Transforers / 5.7 Chebyshev Multi-section Matching Transforer Reading Assignent: pp We can also build a ultisection atching network such that Γ f is a Chebyshev function. the function ( ) Chebyshev functions axiize bandwidth, albeit at the cost of pass-band ripple. HO: The Chebyshev Multi-section Matching Transforer Ji Stiles The Univ. of Kansas Dept. of EECS

2 3/8/6 The Chebyshev Matching Transforer / The Chebyshev Matching Transforer An alternative to Binoial (Maxially Flat) functions (and there are any such alternatives!) are Chebyshev polynoials. Chebyshev solutions can provide functions Γ ( ω) with wider bandwidth than the Binoial case albeit at the expense of passband ripple. Γ f f Figure 5.7 (p. 55) Reflection coefficient agnitude versus frequency for the Chebyshev ultisection atching transforers of Exaple 5.7. Ji Stiles The Univ. of Kansas Dept. of EECS

3 3/8/6 The Chebyshev Matching Transforer / It is evident fro the plot above 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 (the plot above shows the locations of the frequencies for ). 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 Ji Stiles The Univ. of Kansas Dept. of EECS

4 3/8/6 The Chebyshev Matching Transforer 3/ ow, the reflection coefficient of a Chebyshev atching network has the for: where θ ω T cos θ Γ ( θ ) The function T ( cos sec ) order. jθ Ae T cos θ jθ Ae T cos sec ( θ θ ) 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 Ji Stiles The Univ. of Kansas Dept. of EECS

5 3/8/6 The Chebyshev Matching Transforer / 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: 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 sec θ ( 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 funtion! cos θ For exaple, a -section Chebyshev atching network will have the for: j θ Γ θ Ae T cosθsecθ ( ) ( ) j θ Ae sec θ cos θ + cos θ sec θ ( sec θ ) ( 3sec θ sec θ ) + + Ji Stiles The Univ. of Kansas Dept. of EECS

6 3/8/6 The Chebyshev Matching Transforer 5/ While the general for of a -section atching transforer is a polynoial with these sae ters: Γ Γ +Γ + Γ j θ e Γ cosθ +Γcos ( ) θ + Γ] j θ e Γ cos θ +Γ cosθ + Γ] jθ ( θ) e cos θ cos( ) θ ] Thus, we can deterine the values of arginal reflection coefficients Γ, Γ, Γ by siply equating the 3 ters of the two previous expressions: e Γ cos θ Ae sec θ cos θ j θ j θ Γ Asec θ θ ( θ ) ( sec θ ) e Γ cos θ Ae sec sec j θ j θ Γ A sec θ cos θ j θ e Γ Ae 3sec sec + Γ A sec sec + j θ ( θ θ ) ( 3 θ θ ) And because it s syetric, we also know that Γ 3 Γ and Γ Γ. Ji Stiles The Univ. of Kansas Dept. of EECS

7 3/8/6 The Chebyshev Matching Transforer 6/ ow, we can again deterine the values of characteristic ipedance Z n, using the iterative (approxiate) relationship: Γ n Z + ln Z n n Q: But what about the value of A? A: 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 ( ) Ji Stiles The Univ. of Kansas Dept. of EECS

8 3/8/6 The Chebyshev Matching Transforer 7/ Q: OK, so θ ω T defines the bandwidth of the Chebychev atching network. What about the ripple value Γ? How can we deterine what its value is? Can we specify it as a design paraeter? A: So any questions! There is a definite relationship between bandwidth θ and ripple agnitude Γ. In fact, we will find that the two values represent a design trade: increase the atching network bandwidth (good!), and you will likewise increase the ripple value Γ (bad!); decrease Γ (good!), and you will decrease the bandwidth(bad!)! Typically, the ripple agnitude Γ is specified by the designer, and then the resulting bandwidth frequencies θ are calculated. Q: What happens if the bandwidth is too sall; are we forced to increase ripple Γ?? A: ot necessarily! We have one other option increase the nuber of atching sections. This can increase bandwidth without odifying ripple agnitude Γ. ow, let s deterine explicitly the relationship between θ and Γ. We know that the agnitude of the reflection coefficient is, by definition, equal to Γ at θ θ : Ji Stiles The Univ. of Kansas Dept. of EECS

9 3/8/6 The Chebyshev Matching Transforer 8/ ( θ θ ) Γ Γ ( ) ( θ θ ) jθ A e T sec cos ( θ θ ) jθ A e T sec cos AT One property of Chebyshev polynoials is that: T ( ) for all values Therefore, we conclude: ( θ θ ) Γ Γ A But recall: A RL Z R + Z T secθ ( ) L And thus: T R Z R + Z A L ( secθ ) L RL Z R + Z Γ L Look at the above expression! It explicitly relates the values θ and Γ. Ji Stiles The Univ. of Kansas Dept. of EECS

10 3/8/6 The Chebyshev Matching Transforer 9/ We can approxiate this relationship as: RL Z RL T ( secθ) ln Γ R + Z Γ Z L Making sense of the above relationship now requires a little hand-waving! It can be shown that ( a phrase professors use while in hand-waving ode!) the general for of the Chebyshev polynoial can also be written as: ( θ θ) ( θ θ) T sec cos cosh cosh sec cos for all values θ outside the passband of the atching network. ote the value θ is ost definitely outside the passband, and thus according to the above expression: ( θ ( θ ) ) ( θ ) T sec cos T sec ( sec θ ) cosh cosh Inserting this result into the expression on the top of this page, we find: RL T ( sec θ ) ln cosh cosh ( sec θ ) Γ Z Ji Stiles The Univ. of Kansas Dept. of EECS

11 3/8/6 The Chebyshev Matching Transforer / Thus, for a given value of θ (as well as ), we find that: Γ R L ln Z cosh cosh sec θ ( ) Or, we can rearrange this equation and deterine θ for a specified value of Γ (and ): R L ± Z sec θ cosh cosh ln Γ ote that there are two solutions θ for this equation one value of θ will be less than π (defining the lower passband frequency), while the other 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 of θ is π π, then the upper value of θ will be π + π. Q: So?? Ji Stiles The Univ. of Kansas Dept. of EECS

12 3/8/6 The Chebyshev Matching Transforer / 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 : λ where λ v p ω. Ji Stiles The Univ. of Kansas Dept. of EECS

13 3/8/6 The Chebyshev Matching Transforer / 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