The Scattering Matrix

Transcription

1 2/23/7 The Scatterig Matrix 723 1/13 The Scatterig Matrix At low frequecies, we ca completely characterize a liear device or etwork usig a impedace matrix, which relates the currets ad voltages at each device termial to the currets ad voltages at all other termials. But, at microwave frequecies, it is difficult to measure total currets ad voltages! * Istead, we ca measure the magitude ad phase of each of the two trasmissio lie waves ( z) ad ( z). * I other words, we ca determie the relatioship betwee the icidet ad reflected wave at each device termial to the icidet ad reflected waves at all other termials. These relatioships are completely represeted by the scatterig matrix. It completely describes the behavior of a liear, multi-port device at a give frequecy ω, ad a give lie impedace. Jim Stiles The Uiv. of Kasas Dept. of EECS

2 2/23/7 The Scatterig Matrix 723 2/13 Cosider ow the 4-port microwave device show below: port 1 port 2 port 3 4-port microwave device z = z P port 4 z = z P z z = z 4 4P = z P Note that we have ow characterized trasmissio lie activity i terms of icidet ad reflected waves. Note the egative goig reflected waves ca be viewed as the waves exitig the multi-port etwork or device. iewig trasmissio lie activity this way, we ca fully characterize a multi-port device by its scatterig parameters! Jim Stiles The Uiv. of Kasas Dept. of EECS

3 2/23/7 The Scatterig Matrix 72/13 Say there exists a icidet wave o port 1 (i.e., ), while the icidet waves o all other ports are kow to be zero z = z = z = ). (i.e., ( ) ( ) ( ) port z = z P Say we the measure/determie the voltage of the wave flowig out of port 2, at the port 2 plae (i.e., determie z = z ). ( ) 2P ( = ) z z 1p Say we measure/determie the voltage of the wave flowig ito port 1, at the port 1 plae (i.e., z = z ). determie ( ) 1P port 2 ( = ) z z The complex ratio betwee 1 ( z 1 = z 1 P) ad 2 ( z 2 = z 2 P) is kow as the scatterig parameter S 21 : 2p z = z P S 21 j βz ( = ) e = = = 2P 2 z2 z2p jβz1p ( z = z ) e 1P e ( ) jβ z z 2P 1P ikewise, the scatterig parameters S 31 ad S 41 are: S ( z = z ) ( z = z ) ( ) ( ) 3P 4 4 4P 31 = ad S41 = 1 z1 = z1p 1 z1 = z1p Jim Stiles The Uiv. of Kasas Dept. of EECS

4 2/23/7 The Scatterig Matrix 723 4/13 We of course could also defie, say, scatterig parameter S 34 as the ratio betwee the complex values 4 ( z 4 = z 4 P ) (the wave ito port 4) ad 3 ( z3 = z3p ) (the wave out of port 3), give that the iput to all other ports (1,2, ad 3) are zero. Thus, more geerally, the ratio of the wave icidet o port to the wave emergig from port m is: m ( zm = zmp) Sm = (give that k ( zk ) for all k ) ( z = z ) P Note that frequetly the port positios are assiged a zero value (e.g., z1, z2 ). This of course simplifies the P scatterig parameter calculatio: P S m ( z ) e = = = z j β m m m m j β ( ) e We will geerally assume that the port locatios are defied as z P, ad thus use the above otatio. But remember where this expressio came from! Microwave lobe Jim Stiles The Uiv. of Kasas Dept. of EECS

5 2/23/7 The Scatterig Matrix 723 5/13 Q: But how do we esure that oly oe icidet wave is o-zero? A: Termiate all other ports with a matched load! 2 4-port microwave device Jim Stiles The Uiv. of Kasas Dept. of EECS

6 2/23/7 The Scatterig Matrix 723 6/13 Note that if the ports are termiated i a matched load (i.e., Z = Z ), the ad therefore: = I other words, termiatig a port esures that there will be o sigal icidet o that port! Q: Just betwee you ad me, I thik you ve messed this up! I all previous hadouts you said that if, the wave i the mius directio would be zero: if but just ow you said that the wave i the positive directio would be zero: if Of course, there is o way that both statemets ca be correct! A: Actually, both statemets are correct! You must be careful to uderstad the physical defiitios of the plus ad mius directios i other words, the propagatio directios of waves ad! Jim Stiles The Uiv. of Kasas Dept. of EECS

7 2/23/7 The Scatterig Matrix 723 7/13 For example, we origially aalyzed this case: if I this origial case, the wave icidet o the load is (plus directio), while the reflected wave is (mius directio). Cotrast this with the case we are ow cosiderig: port N-port Microwave Network For this curret case, the situatio is reversed. The wave icidet o the load is ow deoted as (comig out of port ), while the wave reflected off the load is ow deoted as (goig ito port ). As a result, whe! Jim Stiles The Uiv. of Kasas Dept. of EECS

8 2/23/7 The Scatterig Matrix 723 8/13 Perhaps we could more geerally state that for some load : reflected icidet ( z = z ) = ( z = z ) For each case, you must be able to correctly idetify the mathematical statemet describig the wave icidet o, ad reflected from, some passive load. ike most equatios i egieerig, the variable ames ca chage, but the physics described by the mathematics will ot! Now, back to our discussio of S-parameters. We foud that if z P for all ports, the scatterig parameters could be directly writte i terms of wave amplitudes ad m. S = (whe z for all k ) m m k k ( ) Which we ca ow equivaletly state as: S m m = (whe all ports, except port, are termiated i matched loads ) Jim Stiles The Uiv. of Kasas Dept. of EECS

9 2/23/7 The Scatterig Matrix 723 9/13 Oe more importat ote otice that for the ports termiated i matched loads (i.e., those ports with o icidet wave), the voltage of the exitig wave is also the total voltage! β ( ) = z e e j z jβz m m m m = m e m e jβz jβz m m (for all termiated ports) Thus, the value of the exitig wave at each termiated port is likewise the value of the total voltage at those ports: ( ) = m m m m m (for all termiated ports) Ad so, we ca express some of the scatterig parameters equivaletly as: ( ) S = (for termiated port m, i.e., for m ) m m You might fid this result helpful if attemptig to determie scatterig parameters where m (e.g., S 21, S 43, S 13 ), as we ca ofte use traditioal circuit theory to easily determie the. total port voltage ( ) m Jim Stiles The Uiv. of Kasas Dept. of EECS

10 2/23/7 The Scatterig Matrix 723 1/13 However, we caot use the expressio above to determie the scatterig parameters whe m = (e.g., S 11, S 22, S 33 ). Thik about this! The scatterig parameters for these cases are: S = Therefore, port is a port where there actually is some icidet wave (port is ot termiated i a matched load!). Ad thus, the total voltage is ot simply the value of the exitig wave, as both a icidet wave ad exitig wave exists at port. = 2 ( ) = ( ) ( ) ( ) = ( ) 1 4-port microwave device Jim Stiles The Uiv. of Kasas Dept. of EECS

11 2/23/7 The Scatterig Matrix /13 Typically, it is much more difficult to determie/measure the scatterig parameters of the form S, as opposed to scatterig parameters of the form S m (where m ) where there is oly a exitig wave from port m! We ca use the scatterig matrix to determie the solutio for a more geeral circuit oe where the ports are ot termiated i matched loads! Q: I m ot uderstadig the importace scatterig parameters. How are they useful to us microwave egieers? A: Sice the device is liear, we ca apply superpositio. The output at ay port due to all the icidet waves is simply the coheret sum of the output at that port due to each wave! For example, the output wave at port 3 ca be determied by (assumig z P ): = S S S S More geerally, the output at port m of a N-port device is: N m m P = 1 ( ) = S z = Jim Stiles The Uiv. of Kasas Dept. of EECS

12 2/23/7 The Scatterig Matrix /13 This expressio ca be writte i matrix form as: = S Where is the vector: T 1, 2, 3,, N = ad is the vector: T 1, 2, 3,, N = Therefore S is the scatterig matrix: S S11 S1 = Sm1 S m The scatterig matrix is a N by N matrix that completely characterizes a liear, N-port device. Effectively, the scatterig matrix describes a multi-port device the way that describes a sigle-port device (e.g., a load)! Jim Stiles The Uiv. of Kasas Dept. of EECS

13 2/23/7 The Scatterig Matrix /13 But beware! The values of the scatterig matrix for a particular device or etwork, just like, are frequecy depedet! Thus, it may be more istructive to explicitly write: S ( ω) S11 ( ω) S1 ( ω) = Sm1 ( ω) Sm ( ω) Also realize that also just like the scatterig matrix is depedet o both the device/etwork ad the value of the trasmissio lies coected to it. Thus, a device coected to trasmissio lies with Z = 5Ω will have a completely differet scatterig matrix tha that same device coected to trasmissio lies with Z = 1Ω!!! Jim Stiles The Uiv. of Kasas Dept. of EECS