Introduction. A microwave circuit is an interconnection of components whose size is comparable with the wavelength at the operation frequency

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1 Introduction A microwave circuit is an interconnection of components whose size is comparable with the wavelength at the operation frequency Type of Components: Interconnection: it is not an ideal connection (zero resistance) as in the case of lumped-element networks Pseudo-lumped components (they present, in a first approximation, the same frequency behavior of the lumped components, having however a finite size) Distributed components (transmission line sections or stubs arbitrarily terminated) In microwave circuits the junction between two (or more) component do not correspond to an ideal node (as in lumped circuits). The junction must be suitably represented (typically by means of its scattering parameters) in order to take into account the phenomena which always arise at a discontinuity in a transmission line.

2 Discontinuity between two transmission lines L 1 =10 cm L 2 =10 cm Z c Z c No good! Coax 1: R 1 =5 cm, r 1 =2.17 cm Coax 2: R 2 =3 cm, r 1 =1.3 cm Z c Matrix Z c Z c R R = = = Ω ln 60 ln 50 r1 r2 Discontinuity Correct model. Why?

3 Higher modes excitation Incident TEM wave Transmitted TEM wave TEM wave Transverse E, H fields Amplitude of higher modes excited At the discontinuity the E field cannot be transverse Higher modes are then generated, allowing the fulfillment of the continuity conditions. These modes however cannot propagate (they are below cutoff) The energy of the em field associated to the higher modes is confined closely to the discontinuity, and it affects the incident wave like a lumped reactance connected at the junction of the two coaxial lines

4 cattering Matrix of the discontinuity The scattering parameters of the following configuration can be evaluated by means of an em simulator (numerical solution of the em fields). It has (f 0 =1 GHz, Zc=50 Ω): = = , = = Port 1 L 1 =10 cm L 2 =10 cm 10 cm 10 cm Port 1 50 Ω Discontinuity 50 Ω Port 2 Port 2 Microwave network To get the parameters of discontinuity we need to move the reference sections (ports 1 and 2) by 10 cm inwards: 2π f φ = β 10 cm = 10 cm = rad ( 120 ) ν = = exp j2φ = , = = exp j2φ = ( ) ( )

5 Frequency dependance -10 Return Loss vs. Frequency DB( (1,1) ) Coaxtep Frequency (MHz) The effects of the discontinuities are generally frequencydependent

6 List of some discontinuities and components in the library of MWOffice

7 Example: microstrip implementation of a double stub matching network T junction 1 2 Z c, L 0 Z c, L 1 Z c, L 0 Z 1, L 1 Z 2, L 2 Open Ideal scheme

8 Discontinuities inclusion PORT P=1 Z=50 Ohm ID=TL3 W=2.2 mm L=10 mm MTEE$ ID=TL4 ID=TL6 W=0.622 mm L=79.68 mm MTEE$ ID=TL5 ID=TL1 W=2.2 mm L=5 mm PORT P=2 Z=50 Ohm ID=TL7 W=1.33 mm L=32.92 mm ID=TL8 W=1.22 mm L=8 mm MOPEN$ ID=TL2 MOPEN$ ID=TL9 MUB Er= 2.55 H= 0.8 mm Rho= T=.035 mm 1 Tand= 0

9 Comparison 0-10 Ideal Doppio tub 1 2 Z IMPED ID=Z1 R=17 Ohm X=66 Ohm Microtrip Frequency (GHz)

10 Computing the models of the discontinuities Equivalent circuits (lumped and/or distributed elements, numerically evaluated) Analytical formulas (simplest cases, low accuracy) Electromagnetic analysis (by means of em simulators, very time consuming) Different models for the same discontinuity are often available (high accuracy=long computation time)

11 Microstrip discontinuities: junctions tep (2-port) Tee (3-port) 1 2 Cross (4-port) ID=TL1 W=1 mm MTEPX$ ID=M1 Offset=0 mm 1 2 ID=TL3 ID=TL1 W=1 mm MTEE$ ID=TL ID=TL3 ID=TL1 W=1 mm MCRO$ ID=TL2 1 2 ID=TL5 W=2.5 mm L=5 mm 3 ID=TL3 ID=TL4 L=5 mm 4 ID=TL4 L=5 mm

12 Bend (2-port) ID=TL1 MBENDA ID=TL2 ANG=90 Deg MCURVE ID=TL2 ID=TL1 ANG=45 Deg R=2 mm ID=TL1 MUBEND$ ID=TL4 =2 mm M=0.5 ID=TL3 ID=TL3 ID=TL3

13 Terminations (1-port) Open end Via hole Radial tub ID=TL1 MOPENX$ ID=MO1 ID=TL1 VIA1P ID=V1 D=1.5 mm H=1 mm T=0.05 mm RHO=1 ID=TL1 MRTUB2W ID=TL2 Ro=7 mm Theta=50 Deg

14 Pseudo-lumped components eries inductor L s l s Z c Y L = 1 1 jωl jωl s 1 1 jωl jωl s s s Y = 1 1 jz tan l jz l ( β ) sin ( β ) c s c s 1 1 jz sin l jz l ( β ) tan ( β ) c s c s ( ) ( ) Y Y βl 0 tan βl sin βl βl L s s s s ωl Z βl s c s L s Zl v c s

15 hunt capacitor: l s C p Z c Z C = 1 1 jωc jωc p 1 1 jωc jωc p p p Z = 1 1 jy tan l jy l ( β ) sin ( β ) c s c s 1 1 jy sin l jy l ( β ) tan ( β ) c s c s ( ) ( ) Z Z βl 0 tan βl sin βl βl L s s s s ωc Y βl p c s C p Yl v c s

16 Other components Interdigital capacitor piral Inductors 1 2 ID=TL1 MICAP$ ID=MI1 W=1 mm =1 mm G=1 mm L=10 mm N=4 WP=1 mm ID=TL2 ID=TL1 W=1 mm L=2 mm EPB=1 TDB=0 TB=0.001 mm RhoB=1 ID=TL2 W=1 mm L=2 mm

17 Two-port circuits matrix for reciprocal, lossless networks: * = U = = φ + φ 2φ =± π Only three real parameters are needed for defining (for instance 11, φ 11, φ 22 )

18 How to get a transformer at microwaves? The transformer is a 2-port widely used in many circuital applications The ideal component is reciprocal and lossless. At microwave frequencies it is however very difficult to be realized. In those application where the goal is the impedance (admittance) scaling, the component employed is the impedance (admittance) inverter: n:1 K Z L Z L Zin = n 2 Z L Ideal Transformer Z = in K 2 Z L Ideal Impedance inverter

19 parameters of the impedance inverter Z 0 K Z 0 ( ) ( 2 K Z ) K Z Z K Z = = 22 = Z K + Z ( ) j2 K = = ± Z K + Z0 π φ = φ = π φ12 = φ21 = ± 2 12 has been obtained by imposing the lossless conditions: 2 2 K Z 0 12 = 1 11 = K + Z0 π π φ11 + φ22 2 φ12 =± π φ12 = φ11 ± =± 2 2 2

20 Practical implementation of inverters Note: the parameter K of the ideal inverter is independent on frequency. The physical implementation however can only approximate this condition in a limited frequency band -jk -jk jj Z c =K jk -jj -jj λ 0 /4 J=1/K φ φ φ jb φ Z c X Z c Y c Y c 1 1 2X φ = tan, X = 2 1 K 2 Zc ( KZ) 2 Y 1 ( ) c JYc c 1 1 2B φ = tan, B = J 2 Note: for X (B) positive, φ is <0 and K/Z c (J/Y c ) is <1.

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22 Equivalent model for a 2-port lossless circuit φ 1 φ 2 Z c K Z c Choice 1 ( K <Z c ): K = Z c π π φ =, φ = Choice 2 ( K >Z c ): K = Z c φ =, φ = π K = 12 + φ 1 + φ 2 (must be 0 or π) 2