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1.1 Transmission Line Basic Concepts: Introduction to Narrow-Band Matching Networks

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1.1 Transmission Line Basic Concepts: Introduction to Narrow-Band Matching Networks European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems 1.1 Transmission line basic concepts: Introduction to narrow-band matching networks March 2010 Francesc Torres, Lluís Pradell, Jorge Miranda European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems Voltage and current in the transmission line V For any lossless transmission line: L IL V (z) V (z) V (z) V(z V+(z) Z ≠ Z 1 Z0 I(z) - L 0 I(z) V (z) V (z) V (z) Z0 -z z=0 where At z=0 Periodicity of V(z) and I(z): wavelength jz (z z) z n2 , z n2 V (z) V0 e V (0) V0 V0 VL 1 jz I(0) V V I 2 V (z) V0 e Z L z n n 0 Impedance: load Reflection coefficient: load V V V V V 1 V Z Z Z L Z Z L Z L L 0 L 0 0 0 L I L V V V LV 1 L V Z L Z0 1 European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems Reflection coefficient in the transmission line At any point z of the transmission line the impedance is computed as: V (z) 1 (z) Z(z) Z0 Z(z) Z0 (z) Z0 I(z) 1 (z) Z(z) Z0 At any point z of the transmission line the reflection coefficient is: V (z) V 2 jz 2 jz 2 jz (z) e (0)e Le V (z) V 2 jz Modulus (z) Le L Constant in z 180º 2z rad 2z deg Phase 2z 2 (z z) n2 , 2z n2 Linear n n Increasing with +z (towards load) z n Periodicity: half a wavelength 2 2 European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems The transmission line as impedance transformer () L 2 j (z) L 2 jz Z0, β ZL≠ Z0 |Γ(z) |=|ΓL| i () z z=0 Example: Compute the input impedance Zi of a circuit formed by a transmission line of length is λ/8 and loaded with ZL=0 (sc). 2 2 j j Z L Z0 2 j 8 2 L 1 i () Le e e j Z L Z0 1 i Zi Z0 jZ0 1 i 2 European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems Standing waves in the transmission line (i) V (z) V e jz 1 e jL e2 jz V 1 e jL e2 jz L L V V jz jL 2 jz jL 2 jz I(z) e 1 L e e 1 L e e Z0 Z0 At any point z where the term (1+ΓL) is real, voltage and current are real an their magnitude is either maximum or minimum: V V (z) V 1 max max L V L 2zmax 2 rad I I(z) 1 min min L Z0 z z max min 4 V V (z) V 1 min min L L 2zmin rad V I I(z) 1 max max L Z0 European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems Standing waves in the transmission line (ii) At any point z where the voltage is maximum the impedance is real an maximum, if the voltage is minimum the impedance is real and minimum: jz V (zmax ) V e 1 L 1 L Z(zmax ) Z0 Zmax I(zmax ) V jz 1 L e 1 L Vmax Imin (zmax ) Z0 jz Vmin Imax (zmin ) V (zmin ) V e 1 L 1 L Z(zmin ) Z0 Zmin I(zmin ) V jz 1 L e 1 L Z0 The voltage standing wave ratio (VSWR or S) is defined as Vmax 1 L S 1 0 L 1 S , L Vmin 1 L S 1 1 S Matched load: Z=Z0, Γ=0, SWR=1 3 European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems Standing waves in the transmission line (iii) Maximum and minimum impedances at in the transmission line: λ/4 λ/4 V (z) Vmax Vmin zmin zmax zmin z=0 Vmax 1 L Vmin 1 1 L Z0 Zmax Z0 Z0S Zmin Imin 1 L Imax Z0 1 L S VSWR is easy to measure and it is widely used to specify mismatch European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems Power in the transmission line (i) At any point z of the transmission line the net power is computed as: 1 * P(z) e V (z)·I * (z) e V e jz 1 (z) V e jz 1 * (z) Z0 2 V 2 2 1 (z) P 1 (z) e1 a1 a* 1 a 2 , a r jx Zo Since the modulus of the reflection coefficient is constant in z, the net transmitted power is constant at any z and equals the power delivered to the load: 2 2 P(z) P 1 (z) P 1 L P P PL Where the power associated to the “positive” (incident) and “negative” (reflected) waves is: 2 2 V V 2 P , P , P L P Z0 Z0 4 European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems Power in the transmission line (ii) + Example: P =1 Watt in transmission line of impedance Z0=50Ω j 2 1 1 Z 50 j100, , , P P 0.5W L L 1 j L 2 2 Return loss definition: P 2 RL , RL 20·log ( ), P (dBW ) P (dBW ) RL(dB) P L 10 L In this case RL 3 dB, P 0dBW 3dB 3dBW The standing wave ratio (SWR) in the transmission line 1 S L 5.8 1 L European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems Matching networks If we have a mismatched load + 2 P PL P 1 L P Z0 ZL≠ Z0 - • A fraction of the incident power P+ P ≠0 is not delivered to the load • A fraction of the incident power ΓL≠ 0 returns to the generator ZL≠ Z0 2 P P + L P Matching A matching network must be Z0 - Network •Simple (passive) P =0 •Lossless (L, C, Transformer, transmission line, waveguide,…) Γi= 0 Zi= Z0 If lossless: PL Pi P All power is delivered to the load 5 European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems Normalized impedances and admittances An impedance can be normalized to a reference impedance Z0 Z Y 1 Z L ; Z 0 P+ L Z L Y Y 0 L L Z0 ZL≠ 1 P-≠0 In this case, the reflection coefficient: ΓL≠ 0 ZL≠ 1 P+ Matching Z L Z0 Z L 1 1YL Z0=1 L P-=0 Network Z L Z0 Z L 1 1 YL Γi= 0 Zi= 1 Working with normalized impedances is equivalent to work with reference transmission lines of Z0=1 European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems Example: lossless narrowband matching network (i) Zx =jX P+ Z0=1 YB =jB ZL=4-j2 P-=0 YL=0.2+j0.1 Zi= 1 Z1 In this case, the reflection coefficient: 1 1 1 Z1 jX Z1 1 jX YL jB 0.2 j(0.1 B) By equalling the real parts By equalling the imaginary parts 0.2 B 0.3 (0.1 B) X 2 1 1 X 1 2 2 2 2 0.2 (0.1 B) B2 0.5 0.2 (0.1 B) X 2 3 6 European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems Example: lossless narrowband matching network (ii) Solution 1 Solution 2 Zx =-j2 Zx =j2 YB =j0.3 Z0=1 ZL=4-j2 Z0=1 YB =-j0.5 ZL=4-j2 Zi= 1 Zi= 1 Sometimes a shunt-series solution does not exist and a series-shunt network must be used: Z0=1 Z0=1 ZL Zi= 1 Zi= 1 European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems The Smith Chart Z(z) → Impedance at any point z of the transmission line ReZz 0, ImZz , P.H. Smith, in 1939, developed a chart to represent any impedance Z(z) as a function of its related reflection coefficient ρ(z). This graphic tool is based in the fact that |ρ|≤1 which allows to represent all impedances in a finite area. The Smith chart is currently used as a universal tool to represent impedances. P.H. SMITH 7 European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems Relation Z(z) - ρ(z) An impedance can be normalized in relation to a reference impedance Zo as Z 1 Z 1 j Zrjx ej Z 1 ri Z0 1 Mathematically, this correspond to a bilinear transformation which translates a circle in the impedance domain into a circle in the reflection coefficient domain. x i r≥0 ≤1 Z r r European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems Relation Z(z) - ρ(z) Now, if ρ=ρr+jρi is substituted in the expression of the normalized impedance, the equations that relate the loci r and x constants as a function of the components ρr and ρi are obtained. This is a set of circumferences in the complex domain : 22 2 r 2 1112 ri;1 ri 2 rr11 xx r 1 Constant resistance circle: CENTRE ,0 RADIUS r 1 r 1 1 1 Constant reactance circle: CENTRE 1, RADIUS x x i x r=const x i x=const.
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