Lesson 1: Introduction and Backgrounds on Microwave Circuits. Giuseppe Macchiarella Polytechnic of Milan, Italy Electronic and Information Department

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1 Lesson 1: Introduction and Backgrounds on Microwave Circuits Giuseppe Macchiarella Polytechnic of Milan, Italy Electronic and Information Department

2 A very general definition A microwave filter is a -port junction exhibiting a selective fruency behavior in the transmission from the input port to the output port Passband(s): Fruency band(s) which is transferred from input to output without attenuation (ideally) Stopband(s): Fruency band(s) where the transmission is blocked (signal is reflected at input port)

3 Definition of transmission Input port Z c Reference sections Z c Output port Z c 1 -port network Z c Physical Structure Equivalent Representation Both the real structure and the uivalent circuit are characterized through the Scattering Parameters (S matrix) defined with respect to Z c. S 11, S = Reflection at the ports S 1 =S 1 =Transmission (<1 for lossless junctions) Attenuation (db) = - log 1 ( S 1 ) > for lossless junctions

4 Basic Classification Filter are classified according to the number and location of passbands and stopbands. The most basic classification considers 4 filters classes: Low-pass, High-pass, Band-pass, Band-stop Attenuation Stopband Stopband Stopband Stopband Stopband Passband Passband Passband Passband Passband Low-pass High-pass Band-pass Band-stop Fruency

5 A (db) Specification of filter ruirements: the Attenuation Mask A s4 A s1 A s Passband A s3 Stopband Stopband A p f s1 f s f p1 f p1 f s3 f s4 Fr.

6 Attenuation and matching Ideally, a filter is a lossless network. In this case the attenuation in the passband is produced only by the not perfect match at input/output ports. In the real world the filter components introduce dissipation, which becomes the main contribution to passband attenuation. Passband Losses affect very few (in general) the port matching In addition to the attenuation mask (which refer to overall attenuation) must be then specified also the matching ruirements at the ports

7 Effect produced by losses on attenuation shape S11-S1 Magnitude (db) Q =3 db Fruency (MHz)

8 Ruirements on transmission phase To have a distortion-less two port, the phase of transmission parameter should be linear in the passband Often, the ruirement is given on the group delay, which is defined as the derivative of transmission phase with respect the radian fruency. The group delay should be ideally constant in the passband Phase linearity is generally assessed a posteriori, after the attenuation ruirements are satisfied. If the ruirements on phase linearity is not verified, a phase ualizer may be ruired. In alternative, complex transmission zeros can be introduced in the response for phase ualization

9 Approaches to the design of microwave filters Image parameters methods (old technique based on attenuation produced by basic blocks, which are suitably interconnected). Synthesis of networks composed by commensurate transmission line sections and stubs (suited for broadband filters; poses strong bounds on the configuration of the filter structure) Equivalent Synthesis method, based on the uivalence between the real (distributed components) structure and a lumped component filter network (suitable for narrow and moderate bandwidth passband filters. The most used technique today

10 Advantages of the Equivalent Synthesis Method Precise control of the curve representing the filter response The synthesis is developed in the lumped-element world, where well established, analytical and numerical procedures are available The results of the synthesis can be expressed by means of universal parameters which maintain the same meaning both for lumped-element circuit and distributed microwave networks A first-order dimensioning of the physical structure can be easily performed using these universal parameters

11 Recalls on Microwave Circuits for Filter Design

12 General physical structure of a microwave filter CAVITY 4 k 45 CAVITY 5 k 53 IN CAVITY 1 k 14 k 1 k 4 k 5 CAVITY k 3 CAVITY 3 OUT Basic components: - Cavities (resonating on a specific mode) - Coupling structures (represented as ) Not always cavities and couplings can be identifies separately

13 Basic uivalent circuit of the cavity R C L G C L Series or parallel resonator Characteristic parameters: - Mode Resonant fruency (f ) - Equivalent slope parameter (L, C ) - Loss parameter (R, G )

14 Basic uivalent circuit of the coupling Couplings are generally modeled with impedance or admittance inverters Z L Zin ZL J Y L Y in J Y L The inverter has two basic functions: - Operate as impedance transformer - Change the nature of the load The inverter is an ideal component which can be approximated by real elements in a limited fruency band

15 S parameters of the Impedance Inverter S Z Z Z 11 S11 Z Z Z Real (positive or negative) The impedance inverter is a symmetrical and reciprocal -port network. It behaves as an ideal / phase shifter. Imposing the lossless condition and the value it is not sufficient to identify univocally the network (the ± sign of 1 remains undetermined)

16 Equivalent circuit for the impedance inverter /4 -jx -jx jb Z c jx -jb -jb =Zc =X J=1/=B Z C jx Z C Y C jb Y C Z C C tan 1 1 X tan ZC X ZC Z 1 Z C J Y C tan 1 1 B tan YC B JYC Y 1 JY C C

17 Modeling of a microwave junction with a lumped uivalent circuit (1-port) Microwave Junction Z J =R J ()+jx J () Z J =R J ( )+jx J () Reference Section Goal: model of the junction in a defined fruency range B around f (B<<f ) The model is represented by an uivalent lumpedelement impedance Z = R + X obtained by imposing: Constant real part: R =R J ( ) Same value at of the imaginary part: X ( )=X J ( ) Same value of derivative at : X ( )/ = X J ( ) /

18 Equivalent impedance Z The most elementary network exhibiting Z is a series resonator: R = R J ( ) C L Z 1 Z R j L C Imposing the previous conditions: 1 X L XJ C L X 1 X L C J 1XJ X J 1 1 XJ X J, C

19 Equations for the Equivalent Admittance G = G J ( ) Y 1 L C Y G j C L Imposing the previous conditions: 1 B C B B J L 1 B C L J C 1BJ B J 1 1 BJ B J, L

20 Remark on the uivalent network The uivalence between the microwave junction and the lumped-element uivalent circuit is exact only at f The deviation between Z J (Y J ) and Z (Y ) becomes larger and larger with the increase of B 4 3 Im(Z(1,1)) Junction X J Im(Z(1,1)) Equivalent Circuit X f =1 MHz X(f )= Fruency (MHz)

21 Special case: Resonant Junction (Cavity) In case of resonant junctions (X J ( )=, B J ( )=), the previous uations become: L C 1 X J 1 C L 1 B J 1 L C These uations define an uivalent circuit for a large class of cavity resonators. The type of resonator depends on the resonant mode and on the reference section

22 Example: TEM cavity realized with short-circuited/ open-circuited transmission line L L= / C L L Z C short Zc L= /4 C L C Y 4 C L L= / C L C YC open Zc L= /4 C L L Z 4 C

23 Capacity-loaded coaxial resonator Tuning screw L< /4 z Reference section short Yc C s L Reference section B C Y cot L tot s C Resonance condition: cot B C Y L tot s C Equivalent capacitance: 1 Btot 1 L C YC cot L sin L 1 L L= YC YC cot L 1 C 1 sinl

24 Losses in the cavity: the unloaded Q Cavity Q Energy stored in the junction Power dissipated in the junction R = R p ( ) C L Z Q L R p Rp RJ R R J is the sum of two terms: Rp, due to thefinite conductivity of metallic walls and Re to to medium dissipation (dielectric losses). Is has then: Q QJ Q

25 Modeling of a cavity coupled to loads C L r p Z C Z C L f Q r Z L for p C ZC B3 db Q L r p Y C J J Y C g p C L C f Q g J Y L for p C J YC B3 db Q C g p

26 Parameters of loaded cavities The loaded Q (QL): determines The 3 db bandwidth (B 3dB =f /Q L ). For a given cavity (L or C ), a value for (J) can be evaluated for obtaining the desired B 3dB : Z L Y C, J Q Q C C L The transducer attenuation between input and output at the resonant fruency The matching at the input (output) port. For small losses, it can be shown that the bandwidth B for a given value of the input reflection coefficient is related to B 3dB as follows: B B 3dB L

27 Power transmitted to load (transducer attenuation) Input Power Cavity Output Power Dissipated Power Input Power = Available Power Z C C L r p Dissipated Power = Power on r p Z C Output Power = Power on Zc A db 1log 1 Q Q L 1

28 Coupling of two cavities: uivalent circuit L C C L Characteristic parameter: the coupling coefficient k L J C for shunt resonators k determines the coupling bandwidth of the two resonators (the larger is k the larger is the bandwidth)

29 Evaluation of k from resonances Even and odd resonances of two coupled resonators: C L C L C L C ±j Short-j Open+j f o f 1 L L Odd Resonance (-j) L C -j -j j C L f e f 1 L L Even Resonance (+j) Short/Open f f f k e o, e o f f f

30 Practical evaluation of k L C C L R R L R Input/output coupling MHz -3.3 db Odd Resonance 5. MHz -4 db Even Resonance Fruency (MHz) X =1, R=.1 f e =5. MHz f o =1975. f f f MHz k f e e f f o o.5

31 Coupled cavities: example 1 L L Cavity 1 Cavity / / Zc Zc Inductive Iris (X L ) Zc jx Zc X 1 Z C, 1 tan X

32 example 1 (cont.) L L Zc Zc jx L Zc Zc INVERTER 1 L = tan X L XL (for Z ) C 1 Z g C Equiv. reactance: L ZC Coupling coefficient: k X L g L ZC

33 Resonators with coupled lines: Interdigital IN L= /4 open Z m Z m open Zeven, Zodd OUT L Z 4 m 1 Zm Zeven Zodd 1 1 Zeven Zodd sin L k 4Zeven Z odd 4 m L Zeven Zodd

34 Resonators with coupled lines: Comb C S C S IN L= /8 short Yeven, Yodd short OUT C Ym 1, J Yodd Y even m k C Yeven Yodd J C S Y m C S Y m 1 Ym Yeven Yodd 1 1 J Yeven Yodd tan L Note: Y Y Z Z Y Y Z Z odd even even odd even odd even odd m

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