Scattering Parameters for Two-port Circuit

Presentation in Experiments and Practice of Electrical and Electronics Engineering for fourth-year technical college students.

Transcript

1. Scattering Parameters for Two-port Circuit Kazuaki OHYAMA Tokyo Metropolitan College of Industrial Technology Public Version
2. 2026/4/20 Introduction: Background 1 Wireless Power Transmission Wireless IoT https://newswitch.jp/p/31332 https://mitssolutions.asia/products/bea nair/sensors/wireless-iot-accelerometer- sensors/beandevice-wifi-wilow-ax-3d/
3. 2026/4/20 • Conductor wire – For kHz order • Coaxial cable – For MHz, GHz order Introduction: Transmission Devices 2 https://www.kawai-cable.co.jp/products/vct-sb.html https://eshop.aisan.co.jp/?p=13748
4. 2026/4/20 Distributed Line: Wavelength in a Circuit • Wavelength 𝜆 = ! " 3 <latexit sha1_base64="WxRC5H9J/NGzm7Fn2SzeVE0B7dU=">AAAB63icbVDLSgNBEOyNrxhfUY9eFoPgKeyKr2PQi8cI5gHJEmYnnWTIzOw6MyuEJb/gxYMiXv0hb/6Ns8keNLGgoajqprsrjDnTxvO+ncLK6tr6RnGztLW9s7tX3j9o6ihRFBs04pFqh0QjZxIbhhmO7VghESHHVji+zfzWEyrNIvlgJjEGggwlGzBKTCZ1JT72yhWv6s3gLhM/JxXIUe+Vv7r9iCYCpaGcaN3xvdgEKVGGUY7TUjfRGBM6JkPsWCqJQB2ks1un7olV+u4gUrakcWfq74mUCK0nIrSdgpiRXvQy8T+vk5jBdZAyGScGJZ0vGiTcNZGbPe72mUJq+MQSQhWzt7p0RBShxsZTsiH4iy8vk+ZZ1b+sXtyfV2o3eRxFOIJjOAUfrqAGd1CHBlAYwTO8wpsjnBfn3fmYtxacfOYQ/sD5/AEbTY5M</latexit> 6= >
5. 2026/4/20 Introduction: Purposes • The impedance matching at the distributed line • The definition of the scattering parameters for a two-port circuit 1. Solve the telegram equations 2. Connect two distributed lines as a two-port circuit 3. Suppose a special situation 4
6. 2026/4/20 Distributed Line: Coaxial Cable Model (1) 5 <latexit sha1_base64="ES8A0ODLLfNd3GnERolM0ETqeus=">AAACGHicbVDLSsNAFJ34rPEVdekmWAQXUhOR6rLoxoVgRfugSQiT6aQdOpOEmYlQQj/Djb/ixoUibrvzb5y0WWjbAxcO59zLvfcECSVCWtaPtrS8srq2XtrQN7e2d3aNvf2miFOOcAPFNObtAApMSYQbkkiK2wnHkAUUt4LBTe63njEXJI6e5DDBHoO9iIQEQakk3zjr+I/uqeMyKPucZe49wz048lxX7/h3CwzfKFsVawJzntgFKYMCdd8Yu90YpQxHElEohGNbifQyyCVBFI90NxU4gWgAe9hRNIIMCy+bPDYyj5XSNcOYq4qkOVH/TmSQCTFkgerM7xSzXi4u8pxUhldeRqIklThC00VhSk0Zm3lKZpdwjCQdKgIRJ+pWE/Uhh0iqLHUVgj378jxpnlfsaqX6cFGuXRdxlMAhOAInwAaXoAZuQR00AAIv4A18gE/tVXvXvrTvaeuSVswcgH/Qxr+RNKAj</latexit> ZS [⌦] ZL [⌦] : Internal impedance of 𝑣! 𝑡 <latexit sha1_base64="ES8A0ODLLfNd3GnERolM0ETqeus=">AAACGHicbVDLSsNAFJ34rPEVdekmWAQXUhOR6rLoxoVgRfugSQiT6aQdOpOEmYlQQj/Djb/ixoUibrvzb5y0WWjbAxcO59zLvfcECSVCWtaPtrS8srq2XtrQN7e2d3aNvf2miFOOcAPFNObtAApMSYQbkkiK2wnHkAUUt4LBTe63njEXJI6e5DDBHoO9iIQEQakk3zjr+I/uqeMyKPucZe49wz048lxX7/h3CwzfKFsVawJzntgFKYMCdd8Yu90YpQxHElEohGNbifQyyCVBFI90NxU4gWgAe9hRNIIMCy+bPDYyj5XSNcOYq4qkOVH/TmSQCTFkgerM7xSzXi4u8pxUhldeRqIklThC00VhSk0Zm3lKZpdwjCQdKgIRJ+pWE/Uhh0iqLHUVgj378jxpnlfsaqX6cFGuXRdxlMAhOAInwAaXoAZuQR00AAIv4A18gE/tVXvXvrTvaeuSVswcgH/Qxr+RNKAj</latexit> ZS [⌦] ZL [⌦] : Load impedance <latexit sha1_base64="PkJDcbKOaYFq9n50M5dHlgBLMIA=">AAAB+HicbVDLSsNAFL3xWeujUZduBovgQkoiUl0W3bisYB/QhDKZTtqhM0mYmYg19EvcuFDErZ/izr9x0mahrQcGDufcyz1zgoQzpR3n21pZXVvf2Cxtlbd3dvcq9v5BW8WpJLRFYh7LboAV5SyiLc00p91EUiwCTjvB+Cb3Ow9UKhZH93qSUF/gYcRCRrA2Ut+uPHpnPU9gPZIiE1O/b1edmjMDWiZuQapQoNm3v7xBTFJBI004VqrnOon2Myw1I5xOy16qaILJGA9pz9AIC6r8bBZ8ik6MMkBhLM2LNJqpvzcyLJSaiMBM5hHVopeL/3m9VIdXfsaiJNU0IvNDYcqRjlHeAhowSYnmE0MwkcxkRWSEJSbadFU2JbiLX14m7fOaW6/V7y6qjeuijhIcwTGcgguX0IBbaEILCKTwDK/wZj1ZL9a79TEfXbGKnUP4A+vzB/R3k04=</latexit> x [m] : Distance Coaxial Cable
7. 2026/4/20 6 Coaxial Cable Distributed Line: Coaxial Cable Model (2)
8. 2026/4/20 Distributed Line: Unit Coaxial Cable Model 7 <latexit sha1_base64="v7kxYbKjCnnFH6BFZsftgVj2TkQ=">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</latexit> R [⌦/m], L [H/m] G [S/m], C [F/m]
9. 2026/4/20 Distributed Line: Telegram Equations (1) 8 <latexit sha1_base64="y9L0VvRxHfy34BDsqMyyZQwtPtc=">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</latexit> @v(x, t) @x = Ri(x, t) L @i(x, t) @t , @i(x, t) @x = Gv(x, t) C @v(x, t) @t .
10. 2026/4/20 Distributed Line: Telegram Equations (2) 9 Incident wave Reflected wave <latexit sha1_base64="Wx04HTQsNtITvt0QVKpAhpKvLzs=">AAACHnicbVDJSgNBEO1xjXGLevTSGBRFDTPidhGCXjwqmFHIRk+nkjTpnhm6a8Qw5Eu8+CtePCgieNK/sRODuD1oeP1eFVX1glgKg6777oyMjo1PTGamstMzs3PzuYVF30SJ5lDikYz0VcAMSBFCCQVKuIo1MBVIuAw6J33/8hq0EVF4gd0Yqoq1QtEUnKGV6rm9CrZBQzPSQP31my2KG2tHfm0Taul2pcWUYvSmt+nXtq3w9S/Uc3m34A5A/xJvSPJkiLN67rXSiHiiIEQumTFlz42xmjKNgkvoZSuJgZjxDmtB2dKQKTDVdHBej65apUHtivaFSAfq946UKWO6KrCVimHb/Pb64n9eOcHmYTUVYZwghPxzUDORFCPaz4o2hAaOsmsJ41rYXSlvM8042kSzNgTv98l/ib9T8PYL++e7+eLxMI4MWSYrZJ145IAUySk5IyXCyS25J4/kyblzHpxn5+WzdMQZ9iyRH3DePgCbrqD0</latexit> ) V (x, t) = V +e x + V e x. Propagation constant <latexit sha1_base64="rMSUXxeH5TXsUZxTE8CiAJUm05k=">AAACEnicbVDLSgMxFM3UV62vUZdugkXoINQZkepGKHahCxdV7AM7w5BJ0zY28zDJCGXoN7jxV9y4UMStK3f+jWk7iFYPXDg5515y7/EiRoU0zU8tMzM7N7+QXcwtLa+srunrG3URxhyTGg5ZyJseEoTRgNQklYw0I06Q7zHS8PqVkd+4I1zQMLiSg4g4PuoGtEMxkkpydePaNeExtMUtl0nhcvfGDn3SRfDc2Cucfr8qhjF09bxZNMeAf4mVkjxIUXX1D7sd4tgngcQMCdGyzEg6CeKSYkaGOTsWJEK4j7qkpWiAfCKcZHzSEO4opQ07IVcVSDhWf04kyBdi4Huq00eyJ6a9kfif14pl58hJaBDFkgR48lEnZlCGcJQPbFNOsGQDRRDmVO0KcQ9xhKVKMadCsKZP/kvq+0WrVCxdHOTLJ2kcWbAFtkEBWOAQlMEZqIIawOAePIJn8KI9aE/aq/Y2ac1o6cwm+AXt/Qv4HZsq</latexit> Z0 = p (R + j!L)/(G + j!C)) Characteristic impedance <latexit sha1_base64="DZjWAi5IjOLOG9ohr0X4z6vnKYk=">AAACFHicbVDLSgMxFM3UV62vqks3wSK0FMqMSHUjFLvQhYsqVgudUu6kaRubzIxJRihDP8KNv+LGhSJuXbjzb0wfiLYeuHByzr3k3uOFnClt219WYm5+YXEpuZxaWV1b30hvbl2rIJKEVknAA1nzQFHOfFrVTHNaCyUF4XF64/XKQ//mnkrFAv9K90PaENDxWZsR0EZqpvNuB4QAfIxddSd1nL3M37qBoB3A57ns6c+jnMsNmumMXbBHwLPEmZAMmqDSTH+6rYBEgvqacFCq7tihbsQgNSOcDlJupGgIpAcdWjfUB0FVIx4dNcB7RmnhdiBN+RqP1N8TMQil+sIznQJ0V017Q/E/rx7p9lEjZn4YaeqT8UftiGMd4GFCuMUkJZr3DQEimdkVky5IINrkmDIhONMnz5Lr/YJTLBQvDjKlk0kcSbSDdlEWOegQldAZqqAqIugBPaEX9Go9Ws/Wm/U+bk1Yk5lt9AfWxzdiipyF</latexit> = p (R + j!L)(G + j!C))
11. 2026/4/20 Distributed Line: Reflection (1) 10 Distributed Line <latexit sha1_base64="39seqtHVTdFx7lrTEWxvGS8FkWc=">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</latexit> V (x, t) = V +e x + V e x, I(x, t) = 1 Z0 (V +e x V e x).
12. 2026/4/20 Distributed Line: Reflection (2) 11 Distributed Line <latexit sha1_base64="39seqtHVTdFx7lrTEWxvGS8FkWc=">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</latexit> V (x, t) = V +e x + V e x, I(x, t) = 1 Z0 (V +e x V e x).
13. 2026/4/20 Distributed Line: Reflection Coefficient 𝛤 (2) • 𝛤 ≡ #! #" • 𝛤 = 0 Ø 𝑉" = 0: No reflected wave Ø Impedance matched 12 Distributed Line
14. 2026/4/20 Distributed Line: Cross-point of Two Lines (1) 13 Distributed Line A Distributed Line B
15. 2026/4/20 Distributed Line: Cross-point of Two Lines (2) 14 Distributed Line A Distributed Line B
16. 2026/4/20 Two-port circuit from Distributed Line 15 Distributed Line A Distributed Line B
17. 2026/4/20 Scattering Parameters: Scattering Matrix {𝑆!"} 16 <latexit sha1_base64="b0+soynN/3WUZWtCmtw8JDYK1yA=">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</latexit> ✓ b1 b2 ◆ = ✓ S11 S12 S21 S22 ◆ ✓ a1 a2 ◆
18. 2026/4/20 Scattering Parameters: Case of Port Matched 17 <latexit sha1_base64="b0+soynN/3WUZWtCmtw8JDYK1yA=">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</latexit> ✓ b1 b2 ◆ = ✓ S11 S12 S21 S22 ◆ ✓ a1 a2 ◆ <latexit sha1_base64="iO9ARtNA4baLlv+TuCX31GtAor0=">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</latexit> ) S11 = b1 a1 a2=0 ) S21 = b2 a1 a2=0
19. 2026/4/20 • 𝛤 = #! #" • 𝑆$$ = ' %# &# &$'( Reflection Coefficient and Scattering Parameter 18 Distributed Line
20. 2026/4/20 • 𝛤 = #! #" • 𝑆$$ = ' %# &# &$'( Reflection Coefficient and Scattering Parameter 19 Distributed Line <latexit sha1_base64="J86CAl0qOR4xeGu7adgzFIM05gs=">AAACCHicbVBNS8NAEN34WetX1KMHF4vgqSQi6kUQPeixotVCE8pmO7WLu0nYnQgl5OjFv+LFgyJe/Qne/Dduaw5afTDweG+GmXlRKoVBz/t0JianpmdmK3PV+YXFpWV3ZfXKJJnm0OSJTHQrYgakiKGJAiW0Ug1MRRKuo9uToX99B9qIJL7EQQqhYjex6AnO0EoddyPAPmjoJRpoHgjMg1OmFCuKw4tO7vtFx615dW8E+pf4JamREo2O+xF0E54piJFLZkzb91IMc6ZRcAlFNcgMpIzfshtoWxozBSbMR48UdMsqXWqPsRUjHak/J3KmjBmoyHYqhn0z7g3F/7x2hr2DMBdxmiHE/HtRL5MUEzpMhXaFBo5yYAnjWthbKe8zzTja7Ko2BH/85b/kaqfu79X3zndrR8dlHBWyTjbJNvHJPjkiZ6RBmoSTe/JInsmL8+A8Oa/O23frhFPOrJFfcN6/ANY9md0=</latexit> ) = S11
21. 2026/4/20 Conclusion • Two analog filters were measured by 1. The time-domain response 2. The frequency-domain response • Impedance matching can be done by – The reflection coefficient 𝛤 – Scattering parameters • The reflection coefficient 𝛤 is equal to 𝑆$$ 20