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ECE 451 Advanced Microwave Measurements Transmission Lines

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ECE 451 Advanced Microwave Measurements Transmission Lines ECE 451 Advanced Microwave Measurements Transmission Lines Jose E. Schutt-Aine Electrical & Computer Engineering University of Illinois [email protected] ECE 451 –Jose Schutt‐Aine 1 Maxwell’s Equations B E Faraday’s Law of Induction t D HJ Ampère’s Law t D Gauss’ Law for electric field B 0 Gauss’ Law for magnetic field Constitutive Relations BH D E ECE 451 –Jose Schutt‐Aine 2 Why Transmission Line? z Wavelength : propagation velocity = frequency ECE 451 –Jose Schutt‐Aine 3 Why Transmission Line? In Free Space At 10 KHz : = 30 km At 10 GHz : = 3 cm Transmission line behavior is prevalent when the structural dimensions of the circuits are comparable to the wavelength. ECE 451 –Jose Schutt‐Aine 4 Justification for Transmission Line Let d be the largest dimension of a circuit circuit z If d << , a lumped model for the circuit can be used ECE 451 –Jose Schutt‐Aine 5 Justification for Transmission Line circuit z If d ≈ , or d > then use transmission line model ECE 451 –Jose Schutt‐Aine 6 Telegraphers’ Equations L I + V C - z L: Inductance per unit length. C: Capacitance per unit length. VI V L Assume jLI zt time-harmonic z dependence IV I C jt jCV zt VI,~ e z ECE 451 –Jose Schutt‐Aine 7 TL Solutions L I + V C - z 2 2V VV I 2 LCV jLjLjCV 2 zz z z z 2 2 I II V 2CLI jCjLjCI 2 zz z z z ECE 451 –Jose Schutt‐Aine 8 TL Solutions (Frequency Domain) z Zo forward wave backward wave Forward Wave Backward Wave LC jz jz Vz() Ve Ve L VVjz jz Z Iz() e e o ZZ C oo Forward Wave Backward Wave ECE 451 –Jose Schutt‐Aine 9 TL Solutions 1 Propagation constant LC Propagation velocity v LC v L Wavelength Characteristic impedance Z f o C Forward Wave Backward Wave Vzt(,) V cos t z V cos t z VV Izt(,) cos t z cos t z ZZ oo Forward Wave Backward Wave ECE 451 –Jose Schutt‐Aine 10 Reflection Coefficient At z=0, we have V(0)=ZRI(0) But from the TL equations: VVV(0) ZR VV VV VV I(0) Zo ZooZ Which gives VV R ZR Zo where R is the load reflection coefficient ZR Zo ECE 451 –Jose Schutt‐Aine 11 Reflection Coefficient -If ZR = Zo, R=0, no reflection, the line is matched -If ZR = 0, short circuit at the load, R=-1 -If ZR inf, open circuit at the load, R=+1 V and I can be written in terms of R jz jz jz 2 jz Vz() V eR e Vz() Ve 1 R e V Ve jz jz jz 2 jz Iz() eR e Iz() 1 R e Zo Zo ECE 451 –Jose Schutt‐Aine 12 Generalized Impedance jz jz Vz() eeR Zz() Zo jz jz Iz() eR e Impedance ZRo jZtan l Zl Zo transformation Z jZtan l oR equation ECE 451 –Jose Schutt‐Aine 13 Generalized Impedance - Short circuit ZR=0, line appears inductive for 0 < l < /2 Z()ljZl o tan - Open circuit ZR inf, line appears capacitive for 0 < l < /2 Z Zl() o j tan l -If l = /4, the line is a quarter-wave transformer Z 2 Zl()o ZR ECE 451 –Jose Schutt‐Aine 14 Generalized Reflection Coefficient Backward traveling wave at z Vz() ()z b Forward traveling wave atz Vzf ( ) jz Ve V 22jz jz ()zee jz R Ve V Reflection coefficient ()le2 jl transformation equation R 1() z Z()zZ o Zz() Zo ()z 1() z Z()zZ o ECE 451 –Jose Schutt‐Aine 15 Voltage Standing Wave Ratio (VSWR) jz2 jz We follow the magnitude of the Vz() Ve 1 R e voltage along the TL jz 22 jz jz Vz() Ve 1RR e V 1 e Maximum and minimum magnitudes given by Vmax 1R Vmin 1R ECE 451 –Jose Schutt‐Aine 16 Voltage Standing Wave Ratio (VSWR) Define Voltage Standing Wave Ratio as: V 1 VSWR max R Vmin 1 R It is a measure of the interaction between forward and backward waves ECE 451 –Jose Schutt‐Aine 17 VSWR – Arbitrary Load Shows variation of amplitude along line ECE 451 –Jose Schutt‐Aine 18 VSWR –For Short Circuit Load Voltage minimum is reached at load ECE 451 –Jose Schutt‐Aine 19 VSWR –For Open Circuit Load Voltage maximum is reached at load ECE 451 –Jose Schutt‐Aine 20 VSWR –For Open Matched Load No variation in amplitude along line ECE 451 –Jose Schutt‐Aine 21 Application: Slotted‐Line Measurement ‐ Measure VSWR = Vmax/Vmin ‐ Measure location of first minimum ECE 451 –Jose Schutt‐Aine 22 Application: Slotted‐Line Measurement At minimum, (z ) pure real R 2 jd min Therefore, ()demin R R 2 jd min So, RR e VSWR 1 VSWR 1 2 jd min Since R then R e VSWR 1 VSWR 1 1 R and ZZRo 1 R ECE 451 –Jose Schutt‐Aine 23 Summary of TL Equations Voltage Current V jz2 jz Iz() ejz 1 e2 jz Vz() Ve 1 R e R Zo ZRo jZtan l Impedance Transformation Zl Zo ZoR jZtan l 2 jl Reflection Coefficient Transformation ()leR 1() z Reflection Coefficient –to Impedance Zz() Z o 1() z Z()zZ Impedance to Reflection Coefficient ()z o Z()zZ o ECE 451 –Jose Schutt‐Aine 24 Example 1 Ig Zg =(10+j10) + Z in Zo = 50 ZR=(30+j40) - Vg =100 V d=l d=0 Consider the transmission line system shown in the figure above. (a) Find the input impedance Zin. ZRo Z (30 j 40) 50 R 0.5 90 ZZR o (30 j 40) 50 4 j .0.725 2 jl (dl )R e 0.5 90 e 0.5 72 ECE 451 –Jose Schutt‐Aine 25 Example 1 –Cont’ 1()l 10.572 ZZ50 in o 1()l 10.572 Zjin 64.361 51.738 39.86 50.54 (b) Find the current drawn from the generator Vg 100 0 I g 1.552 39.11 A ZZgin(10 j 10)(39.86 j 50.54) I g 1.207jA 0.981 ECE 451 –Jose Schutt‐Aine 26 Example 1 –Cont’ c) Find the time‐average power delivered to the load 1 Vl( ) ZI 64.361 51.73 1.5562 39.11 100.159 12.62 in g 2 11* PVlI Real ( ) Real 100.159 12.62 1.5562 39.11 22g 1 PVlIWReal ( )* 48.26 2 g ECE 451 –Jose Schutt‐Aine 27 Example 1 –Cont’ R=50 + Zo = 50 L=1 H - V g C=100 pF d=l d=0 In the system shown above, a line of characteristic impedance 50 is terminated by a series R, L, C circuit having the values R = 50 , L = 1 H, and C = 100 pF. (d) Find the source frequency fo for which there are no standing waves on the line. ECE 451 –Jose Schutt‐Aine 28 Example 1 –Cont’ No standing wave on the line if ZR = R = Zo which occurs at the resonant frequency of the RLC circuit. Thus, 11108 fo Hz15.92 MHz 2 LC 21010 610 2 ECE 451 –Jose Schutt‐Aine 29.
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