02/10/05 Matched reciprocal lossless.doc 1/4 Jim Stiles The Univ. of Kansas Dept. of EECS Matched, Lossless, Reciprocal Devices Often, we describe a device or network as matched, lossless, or reciprocal. Q: What do these three terms mean?? A: Let’s explain each of them one at a time! Matched A matched device is another way of saying that the input impedance at each port is equal to Z0 when all other ports are terminated in matched loads. As a result, the reflection coefficient of each port is zero—no signal will be come out of a port if a signal is incident on that port (and only that port). In other words, we want: 0 for all mmmmVSV m− +== a result that occurs when: 0 for all mmSm= We find therefore that a matched device will exhibit a scattering matrix where all diagonal elements are zero.02/10/05 Matched reciprocal lossless.doc 2/4 Jim Stiles The Univ. of Kansas Dept. of EECS Therefore: 00.10.20.1 0 0.30.2 0.3 0jj⎡⎤⎢⎥⎢⎥=⎢⎥⎢⎥⎣⎦S is an example of a scattering matrix for a matched, three port device. Lossless For a lossless device, all of the power that delivered to each device port must eventually finds its way out! In other words, power is not absorbed by the network—no power to be converted to heat! Consider, for example, a four-port device. Say a signal is incident on port 1, and that all other ports are terminated. The power incident on port 1 is therefore: 21102VPZ++= while the power leaving the device at each port is: 22112110022mmmmVSVPSPZZ−−− +== =02/10/05 Matched reciprocal lossless.doc 3/4 Jim Stiles The Univ. of Kansas Dept. of EECS The total power leaving the device is therefore: ()1234222211 21 31411111222211 21 31411outPPPPPSP SP SP SPSSSSP−−−−+++++=+++=+++= +++ Note therefore that if the device is lossless, the output power will be equal to the input power, i.e., 1outPP+=. This is true only if: 222211 21 31411SSSS+++= If the device is lossless, this will likewise be true for each of the other ports: 222212 22 3242222213 23 3343222214 24 34 44111SSSSSSSSSSSS+++=+++=+++= We can state in general then: 211 for all NmnmSn==∑ In fact, it can be shown that a lossless device will have a unitary scattering matrix, i.e.: H=SS I02/10/05 Matched reciprocal lossless.doc 4/4 Jim Stiles The Univ. of Kansas Dept. of EECS where H indicates conjugate transpose and I is the identity matrix. The columns of a unitary matrix form an orthonormal set—that is, the magnitude of each column is 1 (as shown above) and dissimilar column vector are mutually orthogonal. In other words, the inner product (i.e., dot product) of dissimilar vectors is zero: 11 11 2210 for all Nij ij i j NiNjnSS SS SS S S i j∗∗ ∗ ∗==+ ++ = ≠∑" Reciprocal Reciprocity results when we build a passive (i.e., unpowered) device with simple materials. For a reciprocal network, we find that the elements of the scattering matrix are related as: mn nmSS= For example, a reciprocal device will have 21 12SS= or 32 23SS=. We can write reciprocity in matrix form as: T=SS where T indicates (non-conjugate)
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