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# RF Circuit Design - [Ch3-2] Power Waves and Power-Gain Expressions

Power Waves and Power-Gain Expressions

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### RF Circuit Design - [Ch3-2] Power Waves and Power-Gain Expressions

1. 1. Chapter 3-2 Power Waves and Power-Gain Expressions Chien-Jung Li Department of Electronics Engineering National Taipei University of Technology
2. 2. Department of Electronic Engineering, NTUT Maximum Power Transfer LZ   sE sZ V I   source impedance load impedance Phasor s s L E I Z Z   • The average power dissipated in the load     2 2 2 2 2 2 1 1 1 2 2 2 s s L L rms L L L s L s L s L E E R P I R I R R Z Z R R X X             • The maximum power dissipated in the load when s LX X s LR R s LZ Z  • Maximum power transfer theorem and that is (conjugate matched) • Can we link up the “conjugate matched impedances” and “reflection coefficients” ? 2/31
3. 3. Department of Electronic Engineering, NTUT Power Waves • In this section we discuss the analysis of lumped circuits in terms of a new set of waves, called power waves. LZ   sE sZ V I   source impedance load impedance  Since there is no transmission line, and therefore the characteristic impedances is not defined. oZ d l LZ 0  0d  IN d     0 L o L o Z Z Z Z has no meaning. No transmission line in between Can we define the reflection coefficient w/o transmission lines?  s sV E Z I 3/31
4. 4. Department of Electronic Engineering, NTUT Normalized Impedances (I) Reference: [1] K. Kurokawa, “Power waves and the scattering matrix.” IEEE Trans. Microwave Theory and techniques, vol. 13, pp.194-202, Mar. 1965. LZ   sE sZ V I   s s sZ R jX  L L LZ R jX  • Normalize the impedances with respect to Rs 1 s s s s s X z r jx j R     L L L L L s s R X z r jx j R R     4/31
5. 5. Department of Electronic Engineering, NTUT Normalized Impedances (II) 1 1 s s z rz z z r        U jV  Γ-plane U V 1 1 z z     0  1   1  • Recall the Smith Chart (Γ-plane) 1 s s s s s X z r jx j R     L L L L L s s R X z r jx j R R                 L s L s L L s ss L s L s s L s L s L L s s L s L s r j x x r r jx r jxz r z z Z Z z r r j x x r r jx r jx z z Z Z                          z should contains the resistance and reactance of the load (rL and xL), and the reactance of the source (xs) • When , the reflection coefficient (maximum power delivering to the load) L sZ Z  0  5/31
6. 6. Department of Electronic Engineering, NTUT Power-waves Representation of One-port Network (I)    1 2 p s s a V Z I R     1 2 p s s b V Z I R   Res sR Z • Reflected power wave is equal to zero when the load impedances is conjugately matched to the source impedance, i.e., . pb  L sZ Z where LZ   sE sZ pa pb V I   s p s L s p s L s s V Zb V Z I Z ZI Va V Z I Z ZZ I            p pa b  • Normalized power waves pL s L L p bZ Z Z Z a       and 6/31
7. 7. Department of Electronic Engineering, NTUT Available Power From Source      1 2 2 s p s s s s s E a E Z I Z I R R  2 2 4 s p s E a R    1 2 p s s a V Z I R  s sV E Z I• For and    2 2 2 , 1 2 8 s AVS p p rms s E P a a R is the power available from the source. • Maximum power is delivered to the load when  L sZ Z       2 21 1 Re Re 2 2 s L L L s L E P I Z Z Z Z PL attains its maximum value when , and is given by  L sZ Z ,maxL AVSP P   2 ,max 1 8 s L AVS s E P P R 7/31
8. 8. Department of Electronic Engineering, NTUT Impedance Mismatch         2 2 *1 1 1 1 1 Re 2 2 8 8 2 L p p s s s s s s P a b V Z I V Z I V Z I V Z I V I R R             2 2 21 1 1 2 2 2 L p p AVS pP a b P b      21 2 p AVS Lb P P Power dissipated in the load = Available power from source – Reflected power • When the impedances are mismatched, the power delivering to the load is Reflected power = Available power from source – Power dissipated in the load 8/31
9. 9. Department of Electronic Engineering, NTUT Generalized Scattering Parameter (I) 1 11 1 12 2p p p p pb S a S a  2 21 1 22 2p p p p pb S a S a    1 1 1 1 1 1 2 pa V Z I R   2 2 2 2 2 1 2 pa V Z I R    1 1 1 1 1 1 2 pb V Z I R Two-port Network [Sp] 2pa 2pb 1pa 1pb Port 1 Port 2  1E 1Z 2I1I  1V   2V    2E 2Z  1 2 2 2 2 1 2 pb V Z I R    • Considering a two-port network, the generalized scattering matrix [Sp] is found with respect to a reference impedance Re{Z1} at port 1 and to Re{Z2} at port 2. If Z1 = Z2 = Zo, [Sp] = [S]. 9/31
10. 10. Department of Electronic Engineering, NTUT Generalized Scattering Parameter (II) 2 1 11 1 0p p p p a b S a   Two-port Network [Sp] 2 0pa  2pb 1pa 1pb Port 1 Port 2  1E 1Z 2I1I  1V   2V  2Z 1 11 1 12 2p p p p pb S a S a  2 21 1 22 2p p p p pb S a S a  1 1 11 1 1 T p T Z Z S Z Z      2 2 2 1 1 11 1 1 1 2 2 IN p p AVS pP a b P S    1TZ • Can we find the power by using [S] but not [Sp] ? Sure! We will talk about this later. 10/31
11. 11. Department of Electronic Engineering, NTUT Example • Calculate the power waves and the power delivered to the load in the circuit. 100 50LZ j     10 0sE   100 50sZ j   V I       100 50 10 5.59 26.57 100 50 100 50 L s L s Z j V E Z Z j j               10 0.05 A 100 50 100 50 s L s E I Z Z j j            1 1 10 0.5 2 2 2 100 p s s s s s s a V Z I E Z I Z I R R                1 1 1 1 10 0.05 100 50 0.05 100 50 0 2 2 2 100 p s s s s s b V Z I E Z I Z I j j R R                2 21 1 0.125 W 2 2 L p pP a b   (Try ) 1 Re 2 LP VI  11/31
12. 12. Department of Electronic Engineering, NTUT Example (I) • Calculate the generalized parameter Sp11 and Sp21 at 1 GHz in the lossless, reciprocal, two-port network. Then calculate Sp22 and Sp12. 2 10Z   1.59 nHL    1E 1 50 50Z j     1V   2V 10LZ j  1TZ 1I 2I 1 1 1 1 1 1 0.167 0T T Z V E E Z Z      1 1 1 1 1 0.0118 45 T E I E Z Z       2 1 0.118 45V E    2 1 0.0118 45I E       1 1 1 1 1 1 2 pa V Z I R   2 2 2 2 2 1 2 pa V Z I R    1 1 1 1 1 1 2 pb V Z I R    2 2 2 2 2 1 2 pb V Z I R 1 0.071 0pa   1 0.061 78.69pb   2 0pa  2 0.037 45pb     For Sp11 and Sp21 12/31
13. 13. Department of Electronic Engineering, NTUT Example (II)    2 1 1 1 11 1 1 10 10 10 50 50 0.85 78.69 10 10 50 50 p p T p p Ta b j jZ Z S a Z Z j j               2 2 21 1 0 0.037 45 0.525 45 0.071 0 p p p p a b S a          2 10Z   1.59 nHL    2E 1 50 50Z j     1V   2V 10LZ j  2TZ 1I 2I  For Sp22 and Sp12 1 2 0.833 0V E   1 2 0.0118 45I E     2 2 0.92 5.19V E    2 2 0.0118 45I E    1 0pa  1 0.083 45pb    2 0.158 0pa   2 0.134 11.32pb   1 2 2 2 22 2 2 20 0.85 11.3 p p T p p Ta b Z Z S a Z Z         1 1 12 2 0 0.083 45 0.525 45 0.158 0 p p p p a b S a          13/31
14. 14. Department of Electronic Engineering, NTUT Power-Gain Expressions (I) Transistor [S] 2a 2b 1a 1b Port 1 Port 2   sE sZ out LZ in s L s o s s o Z Z Z Z     L o L L o Z Z Z Z     1 11 1 12 2b S a S a  2 21 1 22 2b S a S a  • Consider a microwave amplifier with the source and load reflection coefficients and measured in a Zo system:s L • For the transistor, the input and output traveling waves measured in a Zo system (this is very practical) : 14/31
15. 15. Department of Electronic Engineering, NTUT Power-Gain Expressions (II)   sE sZ s LZ L Transistor [S] The reflection coefficients and S-parameters are separately measured in a Zo (usually 50 Ω) system Transistor [S] 2a 2b 1a 1b   sE sZ out LZ in s L After connecting them all together The goal is to find the input and output power relations. 1b 1a 2a 2b 15/31
16. 16. Department of Electronic Engineering, NTUT Input Reflection Coefficient 1 1 in b a   2 2La b  2 21 1 22 2Lb S a S b   21 1 2 221 L S a b S    Transistor [S] 2a 2b 1a 1b   sE sZ out LZ in s L • After connecting the circuits together, the first step is to find the new input coefficient , which is the result coming from and .in  S L     1 11 1 12 2b S a S a  2 21 1 22 2b S a S a   1 12 21 11 1 221 L in L b S S S a S        12 21 1 11 1 12 2 11 1 1 221 L L L S S b S a S b S a a S         a1 is your input, so the goal here is to find the reflected wave b1 1 11 1 12 2b S a S a  a1 is your input, to find b1 = you need to find a2 to find a2 = you need to find b2  the relationship between b2 and a1 16/31
17. 17. Department of Electronic Engineering, NTUT Output Reflection Coefficient 2 2 0s out E b a    1 1sa b  1 11 1 12 2sb S b S a   12 2 1 111 s S a b S    12 21 2 21 1 22 2 2 22 2 111 s s s S S b S b S a a S a S         12 212 22 2 110 1 s s out sE S Sb S a S        Transistor [S] 2a 2b 1a 1b   sE sZ out LZ in s L • After connecting the circuits together, the second step is to find the new output coefficient , which is the result coming from and .out  S s   1 11 1 12 2b S a S a  2 21 1 22 2b S a S a      The same procedure as finding is applied.in 17/31
18. 18. Department of Electronic Engineering, NTUT The Available Power and Input Power (I)   sE sZ s 1a 1b • After finding out the input/output refection coefficients, let’s now deal with the power. in Since we have got , we can discard the circuits connected after the source right here. in 1 1s sV E I Z    1V 1I   1 1 1 1 1 1 1s s s s o V V V V V E I I Z E Z Z                    1 1 1 1 1 1 1s s s s s o o o V V V V V E Z V E Z Z V Z Z Z                   1 1 o s o s o s s o Z Z Z V E V Z Z Z Z            • Use the normalized power waves 1 1 1 1 s o s o s s o s s oo o o E Z Z ZV V a a b Z Z Z ZZ Z Z               where , , ands o s o s E Z a Z Z   1 1 o V b Z   s o s s o Z Z Z Z     18/31
19. 19. Department of Electronic Engineering, NTUT The Available Power and Input Power (II) 1 1inb a  1 1 1s s s s ina a b a a       1 1 s s in a a       2 2 2 2 2 2 1 1 1 2 11 1 1 1 1 2 2 2 2 1 in in in s s in P a b a a            • The available power from source 2 2 2 2 2 2 2 22 2 1 11 1 1 1 2 2 2 11 1 in s s s AVS in s s s ss s P P a a a                   2 2 2 2 2 2 1 111 2 1 1 s in in in s AVS AVS s s in s in P a P P M               • Ms is known as the source mismatch factor (or mismatch loss).   sE sZ s 1a 1b in   1V 1I Pin 19/31
20. 20. Department of Electronic Engineering, NTUT The Available Power and Output Power (II) LZ L out Since we have got , the circuits looking into the output port (with source) can be simplified as a Thevenin’s equivalent circuit. out   thE outZ 2a 2b   LV LI LZ L out  2 2 2 2 2 2 2 1 1 1 1 2 2 2 L LP b a b     • The power delivered to the load ZL 2 2 2 11 2 1 L L th out L P b       • The available power from the network 2 2 1 1 2 1L out AVN L th out P P b         2 2 2 1 1 1 L out L AVN AVN L out L P P P M          • ML is known as the load mismatch factor (or mismatch loss). 20/31
21. 21. Department of Electronic Engineering, NTUT Definition of the Power Gains Transistor [S]  sE sZ LZ PAVNPAVS PLPin Ms interface interface ML • The power gain L p in P G P  • The transducer power gain L T p s AVS P G G M P   • The available power gain AVN T A AVS L P G G P M   p TG G A TG G • When the Input and output are matched: p T AG G G  From the amplifier input to load From the source to load 21/31
22. 22. Department of Electronic Engineering, NTUT Power Gain     2 2 2 2 2 1 1 1 2 1 1 2 L L p in in bP G P a       21 1 2 221 L S a b S    2 2 212 2 22 11 1 1 L p in L G S S        • The Power Gain Gp where Transistor [S]  sE sZ LZ PAVNPAVS PLPin Ms interface interface ML 22/31
23. 23. Department of Electronic Engineering, NTUT Transducer Power Gain • The Transducer Power Gain GT L L in in T p p s AVS in AVS AVS P P P P G G G M P P P P     2 2 2 2 2 2 21 212 2 2 2 22 11 1 1 1 1 1 1 1 1 s L s L T s in L s out L G S S S S                       2 2 2 1 1 1 s in s s in M         where Transistor [S]  sE sZ LZ PAVNPAVS PLPin Ms interface interface ML 23/31
24. 24. Department of Electronic Engineering, NTUT Available Power Gain • The Available Power Gain GA AVN L AVN AVN T A T AVS AVS L L L P P P P G G G P P P P M     2 2 212 2 11 1 1 1 1 s A s out G S S        Transistor [S]  sE sZ LZ PAVNPAVS PLPin Ms interface interface ML   2 2 2 1 1 1 L out L out L M         where 24/31
25. 25. Department of Electronic Engineering, NTUT Two-port Network Matrices • Several ways that are commonly used to represent the two-port network: Impedance matrix : z-parameter Admittance matrix : y-parameter Hybrid matrix : h-parameter ABCD matrix : ABCD parameters Scattering matrix : S-parameter • These matrices describe the relationship between the input/output voltages and currents except the scattering matrix which describes the relationship between the input/output traveling waves (or power waves). 25/31
26. 26. Department of Electronic Engineering, NTUT Two-port Network Representation  z-parameter  y-parameter  h-parameter  ABCD parameters 1 11 12 1 2 21 22 2 v z z i v z z i                   1 11 1 12 2v z i z i  2 21 1 22 2v z i z i  1 11 12 1 2 21 22 2 i y y v i y y v                   1 11 12 1 2 21 22 2 v h h i i h h v                   1 2 1 2 v vA B i iC D                Two-port network   1v 1i 2i   2v Port 1 Port 2 26/31
27. 27. Department of Electronic Engineering, NTUT Conversion Between the Network Parameter • This table is provided at page 62 in the textbook. 27/31
28. 28. Department of Electronic Engineering, NTUT Series Connection • Series Connection: use z-parameter 1 11 1 11 11 12 12 2 22 2 21 21 22 22 a b a b a b a b a b a b v iv v z z z z v iv v z z z z                           28/31
29. 29. Department of Electronic Engineering, NTUT Shunt Connection • Shunt Connection: use y-parameter 1 11 1 11 11 12 12 2 22 2 21 21 22 22 a b a b a b a b a b a b i vi i y y y y i vi i y y y y                           29/31
30. 30. Department of Electronic Engineering, NTUT Cascade Circuits • Cascade Circuits : use ABCD parameters (chain) 1 1 2 2 1 1 2 2 a a ba a a a b b a a ba a a a b b v v v vA B A B A B i i i iC D C D C D                                         30/31
31. 31. Department of Electronic Engineering, NTUT Summary • The power delivered to the load can be calculated by using three methods: (1) Real power dissipated at load ( ) (2) Power waves (generalized [Sp], linked with reflections) (3) Traveling waves ([S], it’s practical and useful in amplifier design)  Re 2L L LP V I  • Available power from source (maximum average power the source can provide when matched) :    2 2 2 , 1 2 8 s AVS p p rms s E P a a R 2 2 21 1 1 2 2 2 L p p AVS pP a b P b    • When mismatch occurs: Power wave Power wave L p inP G P L T AVSP G P • Power gains (defined with traveling waves, circuitries are separately measured in a Zo system) : 31/31