Lecture 4. Maximum Transfer of Power. The Purpose of Matching. Lecture 4 RF Amplifier Design. Johan Wernehag Electrical and Information Technology

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1 Johan Wernehag, EIT Lecture 4 RF Amplifier Design Johan Wernehag Electrical and Information Technology Design of Matching Networks Various Purposes of Matching Voltage-, Current- and Power Matching Design by Lumped Circuit Elements L, Pi and T Networks Design by Using the Smith Chart Design by Line Structures Transformation by a Transmission Line Quarter-Wave Transformer Line Section with Optimised Length and Z 0 Stubs Lecture 4 Passive Components Lumped Components Resistors Capacitors Inductors Transformers Substrate and Conductor Materials Transmission Lines Coaxial Line Microstrip Stripline Discontinuities Bends, corners Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 3 The Purpose of Matching Maximum Transfer of Power Relative load signal level Minimum noise power Maximum output power Maximum transfer of power In most cases bandwidth and filtering are important parameters too Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 4 Current matching Voltage matching Power matching Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 5

2 Complex Conjugate Matching Z or G Representation Impedance Maximum transfer of power when Reflection coefficient (Z 0 is resistive) resonance Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 6 Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 7 Available Power from Source, P AVS The Need of a Matching Network The maximum amount of power that can be delivered from the source is defined as: NOTE! The unit of V S is here V RMS! If Z L and/or Z S are fixed a matching network is needed to ensure proper matching Active power Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 8 Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 9

3 Network Design by Lumped Circuits L Network, Four Different Variations L network low-pass high-pass fixed circuit Q Low-pass, R L > R S Low-pass, R L < R S Pi and T network low-pass high-pass desired circuit Q High-pass, R L > R S High-pass, R L < R S - Series component in serie with the smallest impedance - Shunt component in with the largest impedance Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 10 Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 11 Designing Reactive Circuit Elements in the Smith Chart Experiment: Design a matching network by using the Smith chart and the VNA step 3 step 1 Add susceptance (connect an inductor in parallel) until you end up in Y = Yo = 1/50 mho. 1/wC= Ω 1 = Ω = 42 step 2 Subtract reactance (connect a capacitor in series) until you reaches the conductance circle g = 1/50 mho. 1/wC= Ω 55 Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 12 Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 13

4 Eight possible L-type Networks Matching Networks by Line Structures Transformation by a serial transmission line line section with optimised length and Z 0 quarter-wave transformer matching by multiple sections Stubs short-circuited and open stubs symmetrical stubs Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 14 Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 15 Impedance Transformation by a Single Serial Line The Quarter-Wave Transformer For a line at the length l/4 is if Z L is resistive Z in G in Z L G L may be used for matching between arbitrary resistive source and load impedances. a) lossless line b) lossy line Useful to remember! Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 16 Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 17

5 Quarter-Wave Transformer at Multiple Sections Quarter-Wave Transformer at Multiple Sections (cont.) Transformation in several and minor impedance steps may provide a larger bandwidth If the characteristic impedances are distributed according to binomial coefficients maximum-flatness is achieved. Ex. three sections: Binomial coefficients Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 18 Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 19 Impedance Transformation by Stubs and Serial Lines Length and Termination of the Stubs Short-circuited or open line sections may be used as reactive shunt elements. Combined with a serial line matching can be achieved between arbitrary loci in the Smith chart. Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 20 Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 21

6 Designing the Length of the Stub Symmetrical Stubs Single stub Symmetrical stubs Note: 2 The length of the symmetrical stubs is designed to individually provide the half value of the requested susceptance. Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 22 Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 23 high-frequency model Resistors Frequency characteristics - example Resistor Ideal resistor Resistive f SRF = Self Resonance Frequency Self resonance Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 24 Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 25

7 high-frequency model Capacitors Frequency characteristics - example Capacitor capacitive inductive R ESR = equivalent series resistance Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 26 Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 27 high-frequency model Inductors Frequency characteristics - example Inductor inductive capacitive R P = equivalent parallel resistance Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 28 Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 29

8 Inductor Transformer The Q-factor frequency dependence Q The losses increases rapidly versus the frequency SRF (self resonance frequency) Windings on a ferrite rod or toroid core Skin effect Parasitic capacitance ω These materials provides a high permeability that unfortunately decreases at higher frequencies Usable at best up to a few GHz C parasitic doesn t affect the Q if the inductor is used in a resonant circuit! Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 30 Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 31 Loss in Substrate Materials Properties of some Substrate Materials For dielectric materials often the loss tangent used to specify the losses: for material with low loss where d is the loss angle The loss tangent is also related to the quality factor or Q-factor of the material: Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 32 Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 33

9 Properties of some Metals Skin Effect The magnetic flux inside the conductor will effectively push the current to a narrow region close to the surface Current density Skin depth: = the distance were the current density has decreased by a factor e A large circumference is more essential than a large cross section area of the conductor! Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 34 Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 35 Example of Skin Depth Dimension of the Coaxial Cable d i [m] ε = 1 (air) ε D d iron 40 - f [Hz] aluminium gold copper silver ,0 1,2 1,4 1,6 1,8 2 2, Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 36 Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 37

10 Microstrip Design of Microstrip, Diagram 1 Determine the proper W for a specified Z 0 and h Cross section of a microstrip structure W Z 0 h H E å r ground plane 1 Complex geometry and non-uniform fields makes a complicated flux image. e r in the substrate is therefore not usable for calculation of for example electrical length in the line structure. Instead the effective permittivity, e eff, based on the dimensions of the microstrip structure, may be used W/h e r Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 38 Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 39 Design of Microstrip, Diagram 2 Design of Microstrip, Summary e eff Determine the effective permittivity e eff Specified Z 0, substrate height h and permittivity ε r : Z Select the width W in diagram 1 e r Select the effective permittivity ε eff in diagram 2 e eff W/h e r 3. Calculate the effective wavelength e r W/h - 2 W/h Specified electrical length 4. Calculate the physical length Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 40 Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 41

11 Stripline Discontinuities W The design of circuits containing transmission lines must take into account the impact of junctions and transitions between different line widths. ground plane circuit diagram design substrate e r conductor h Z 0,1 Z 0,1 l 1 ground plane l 1 l 2 l 3 l 3 Z 0,2 l 2 How to model such a geometry? Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 42 Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 43 T-junction, equivalent model Discontinuities (cont.) Discontinuities (cont.) Symmetrical step, an abrupt transition between two different line widths W 1 W 2 Transmission line, open-circuit termination conductor C f Methods for manual calculation of the circuit elements in the equivalent model may be found in handbooks. ground plane e r er Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 44 Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 45

12 Discontinuities (cont.) Some other cases are corners and bends serial gap (to realize a serial capacitance) undesired coupling between nearby structures Crosstalk CAD tools are necessary for analysis and design of more complex circuits ADS and others In worst case a complete structure may be simulated by the finite element method and Maxwell s equations which is both time and memory consuming Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 46

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