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Unformatted text preview: EE 4002 RF Circuit Design
Chapter 4 Sparameters a1 b1 Scattering Parameters are defined:
Port 1 a2 b2 S 11 S 21 S12 S 22 Port 2 Waves present at port n: an = 2 Pinc = V + 2 Z0 V = Z0 + bn = 2 Prefl = V  2 Z0 V = Z0 V + = a Z0 Vn = Z 0 (an + bn )
an = V  = b Z0 V = V + +V  Total voltage at a port. 1 ( Vn + Z 0 I n ) 2 Z0 bn = 1 ( Vn  Z 0 I n ) 2 Z0 V + V  I = I+ + I = +  V = V +V Z0 1 +  P = P + P = Pinc + Pref = Re { VI *} 2
Forward wave Reverse wave + 2 V 2 V V 1 1 + V  Pin = Re + V ) V = Re  ( 2 Z0 Z 0 2 Z 0 +  * 1V 1V P=  2 Z0 2 Z0 + 2  2 a b Pin =  2 2 2 2 The sparameters are power wave descriptors that define the relationship between incident and reflected power waves. a1 b1
Port 1 a2 S 11 S 21 S12 S 22 Port 2 b2 S11 = b1 a1 =
a2 = 0 port 1 refelcted wave port 1 incident wave b2 S 21 = a1
S 22 = b2 a2 =
a1 = 0 a2 = 0 port 2 transmitted wave = port 1 incident wave port 2 refelcted wave port 2 incident wave b1 S12 = a2 a1 = 0 port 1 transmitted wave = port 2 incident wave a1 b1
Port 1 a2 S 11 S 21 S12 S 22 Port 2 b2 In order to compute the Sparameter matrix we need to ensure that: a1 = 0 and a2 = 0
What does this mean? That there is no power wave returning to the DUT This is only possible when the TL connected to port 2 is Terminated into its characteristic impedance. b1 S11 = a1 a2 = 0 b2 S 21 = a1 a2 = 0 Implications of the Sparameters
Port 1 S 11 S 21 S12 S 22 Port 2 V1 b1 in = + = V1 a1 = S11 ,
a2 = 0 1 + S11 and VSWR = 1  S11 Reflection coefficient and VSWR can now be computed directly from Sparameters a1 b1
Port 1 a2 S 11 S 21 S12 S 22 + 2 1 Port 2 b2
2 1 2 2 P = Pinc + Pref = a1  b1 1 2
+ 2 1 ( ) 1V = 2 Z0 ( 1 )
in a1 2 = 1  S11 2 2 ( ) If S11 = 0, we have the maximum power available from the source:
2 V a1 1 Pinc = = , Pref = 0 2 Z0 2 Similarly for the analysis at port 2: 1 2 2 P2 = Pinc + Pref = a2  b2 2 ( ) 1V = 2 Z0 + 2 2 ( 1 )
2 out a2 = 2 2 ( 1 S )
2 22 RL = 10 log in = 20 log S11
V1+ 2 Return loss can be calculated directly From the Sparameters V2 b2 S 21 = a1 =
a2 = 0 V2 V1+ Z0 V2 Z 0 V2 = + = Z 0 V1 (V1 + Z 0 I1 ) 2 Z 0 Voltage leaving port 2 vs. voltage entering port 1 is the voltage gain from source to load, also known as the forward voltage gain. We can then write the gain in terms of the generator voltage, substituting: VG1 = V1 + Z 0 I1 2V S 21 = VG1  2 We can square the voltage gain to find the forward power gain. G0 = S 21 2 V = VG1 2  2 2 We can switch the connection of the source and load to obtain the remaining Sparameters:  + V1 V2 S 22 = b2 a2 = out =
a1 = 0 Z out  Z 0 Z out + Z 0 b1 S 21 = a2 =
a1 = 0 V1 V2+ Z0 V1 Z 0 V1 2V1 = + = = Z 0 V2 (V2 + Z 0 I 2 ) 2 Z 0 VG 2 Now we observe the reverse voltage gain in terms of generator voltage. Also, the reverse power gain: G0 = S12 2 Example: Find the Sparameters and resistors needed to create a 3 dB attenuator using a Tnetwork. The attenuator will be used in a system with characteristic line impedance of 50 A 3 dB attenuator should be matched to the line impedance and have a voltage gain of 3 dB or 0.707 Matching the line implies: impedance is 50 S11 = S 22 = 0 this will occur if the input and output Z in Z out Z in Z out R3 ( R2 + 50) Z in = R1 + = 50 R3 + R2 + 50 Z out R3 ( R1 + 50) = R2 + = 50 R3 + R1 + 50 R1 = R2 Tparameters: the Chain Scattering Matrix
We can rewrite the power wave expressions in terms of input and output ports: a T11 1 = b1 T21 T12 b2 a T
22 2 b S 1 11 = b2 S 21 S12 a1 a S
22 2 1 11 a T = A A b T 1 21 A A T12 b2 A A T22 a 2 A A 1B 11B a T = B B b T 1 21 T12B b2B B B T22 a 2 We observe that: 2A 1B b a = A B a b 2 1 T12B b2B B B T22 a 2 The chain scattering matrix plays a similar role as the ABCDmatrix. It is useful for cascading systems. 1A 11A T12A T11B a T = B A A A b T 1 21 T22 T21 If we have a component or system with known Sparameters that we wish to connect in cascade with each other, we can each system's Sparameters to Tparameters, find the overall Tmatrix. a T11 1 = b1 T21 T12 b2 T22 a2 b S 1 11 = b2 S 21 S12 a1 S22 a2 a1 = b2T11 + a2T12 b1 = b2T21 + a1T22
a1 T11 = b2 b1 = a1S11 + a2 S12 b2 = a1S 21 + a2 S 22
b2 S 21 = a1 a2 = 0 a2 = 0 1 T11 = S 21
Inverse of forward voltage gain Continuing the conversion from T to S S 22 T12 =  S 21 S11 T21 = S21  ( S11S22  S12 S 21 ) S T22 = = S21 S 21 If we know the Tmatrix of a cascaded system, we can then convert back to Sparameters and calculate: Insertion loss VSWR Voltage Gain Power Gain T21 S11 = T11
1 S 21 = T11 T S12 = T11
T12 S 22 =  T11 Matrix Conversions If we then convert Sparameters to Zparameters, we can then convert to any parameter set. The conversion from S to Z can be found on Page 177 and 178 Signal Flow Chart Modeling
Complicated RF systems can now be studied using Sparameters similarly to concepts used in controls systems. 1. Nodes used to identify network signals launched or received, sources, loads, or interconnections 2. Branches needed to connect paths between sources and loads Terminated TL segment with incident and reflected Sparamter description Conventional Form Signal flow form ...
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This note was uploaded on 01/31/2012 for the course EE 4002 taught by Professor Scalzo during the Fall '06 term at LSU.
 Fall '06
 Scalzo

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