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High Speed Communication Circuits and SystemsLecture 3S-Parameters and Impedance TransformersMichael H. PerrottFebruary 9, 2004Copyright 2004 by Michael H. PerrottAll rights reserved.M.H. Perrott1

What Happens When the Wave Hits a Boundary? Reflections can occurIncident WaveZLExxHyyReflected WavezZLHyM.H. PerrottEx2

What Happens When the Wave Hits a Boundary? At boundary- Orientation of H-field flips with respect to E-field Current reverses direction with respect to voltageIncident WaveIZLExxyIReflected WavezIZLExHyM.H. PerrottHyI3

What Happens At The Load Location? Voltage and currents at load are ratioed according to theload impedanceIncident WaveIiVoltage at LoadZLViCurrent at LoadxyIiReflected WavezIrRatio at LoadZLVrM.H. PerrottIr4

Relate to Characteristic Impedance From previous slide Voltage and current ratio in transmission line set by itcharacteristic impedance Substituting:M.H. Perrott5

Define Reflection Coefficient Definition:- No reflection if L 0 Relation to load and characteristic impedances Alternate expression- No reflection if ZLM.H. Perrott Zo6

Parameterization of High Speed Circuits/Passives Circuits or passive structures are often connected totransmission lines at high frequencies- How do you describe their behavior?Linear NetworkTransmission Line 1M.H. PerrottTransmission Line 27

Calculate Response to Input Voltage Sources Assume source impedances match their respectivetransmission linesSame valueby definitionSame valueby definitionLinear NetworkZ1Z1Z2Z2Vin1Vin2Transmission Line 1M.H. PerrottTransmission Line 28

Calculate Response to Input Voltage Sources Sources create incident waves on their respectivetransmission lineCircuit/passive network causes- Reflections on same transmission line- Feedthrough to other transmission lineVi1Linear NetworkVi2Z1Z2Z1Vin1M.H. PerrottZ2Vr1Vr2Vin29

Calculate Response to Input Voltage Sources Reflections on same transmission line areparameterized by L- Note that is generally different on each side of thecircuit/passive networkLΓL1Vi1ΓL2Linear NetworkVi2Z1Z2Z1Vin1Z2Vr1Vr2Vin2How do we parameterize feedthrough tothe other transmission line?M.H. Perrott10

S-Parameters – Definition Model circuit/passive network using 2-port techniques- Similar idea to Thevenin/Norton modelingΓL1Vi1ΓL2Linear NetworkVi2Z1Z2Z1Vin1 Z2Vr1Vr2Vin2Defining equations:M.H. Perrott11

S-Parameters – Calculation/MeasurementΓL1Vi1ΓL2Linear NetworkVi2Z1Z2Z1Vin1M.H. PerrottZ2Vr1Vr2Vin212

Note: Alternate Form for S21 and S12ΓL1Vi1ΓL2Linear NetworkVi2Z1Z2Z1Vin1M.H. PerrottZ2Vr1Vr2Vin213

Block Diagram of S-Parameter 2-Port ModelS-Parameter Two-Port ModelVi1Vi2Z2S21Z1Z1Z1S12Z2Z2S11S22Vr1 Vr2Key issue – two-port is parameterized with respect tothe left and right side load impedances (Z1 and Z2)- Need to recalculate S , S , etc. if Z or Z changes- Typical assumption is that Z Z 50 Ohms112111M.H. Perrott2214

Macro-modeling for Distributed, Linear NetworksZ1ZsVslength d1LinearCircuits &Passives(1)Z2length d2LinearCircuits &Passives(2) Key parameters for a transmission line Key parameters for circuits/passivesZ3length d3ZL Vout- Characteristic impedance (only impacts S-parametercalculations)- Delay (function of length and )- Loss (ignore for now)- S-parametersWe would like an overall macro-model for simulationM.H. Perrott15

Macro-modeling for Distributed, Linear NetworksZ1ZsVslength d1d1velocity LC d1delay1 LinearCircuits &Passives(1)Z2length d2delay2 με d2LinearCircuits &Passives(2)Z3length d3delay3 ZL Voutμε d3με d1Model transmission line as a delay element- If lossy, could also add an attenuation factor (which is afunction of its length) Model circuits/passives with S-parameter 2-portsModel source and load with custom blocksM.H. Perrott16

Macro-modeling for Distributed, Linear NetworksZ1ZsVslength d1LinearCircuits &Passives(1)d1velocity LC d1 length d2delay2 delay1 με d2Z1delay1Z1 Zslength d3delay3 ZL Voutμε d3ZL Z2 ZR Z3Vi1Vr2delay2Zs 2-port(2)S-param.2-port(1)Zs-Z1M.H. PerrottZ3με d1ZL Z1 ZR Z2VsLinearCircuits &Passives(2)Z2ZL-Z3ZL Z3delay317

Note for CppSim SimulationsZL Z2 ZR Z3ZL Z1 ZR Z2Z1Vsdelay1Z1 ZsVi1Vr2delay2Zs port(1)Zs-Z1Vi1delay2Vr1Vi2VoutZL-Z3ZL Z3delay3 CppSim does block-by-block computation Prevent artificial delays by- Feedback introduces artificial delays in simulation- Ordering blocks according to input-to-output signal flow- Creating an additional signal in CppSim modules topass previous sample values- Note: both are already done for you in Homework #1M.H. Perrott18

S-Parameter Calculations – Example 1TransmissionLine JunctionVi1Z1Vi2Z2Vr1Vr2Derive S-Parameter 2-Port Set Vi2 0M.H. Perrott Set Vi1 019

S-Parameter Calculations – Example 2Vi1Transmission LineJunction with CapacitorZ1Vr1 Vi2Z2CVr2Derive S-Parameter 2-PortSame as before:But now:M.H. Perrott20

S-Parameter Calculations – Example 3Vi1Vi2Z1Vr1Z2L1C1C2Vr2Derive S-Parameter 2-Port The S-parameter calculations are now more involved This is a homework problem- Network now has more than one nodeM.H. Perrott21

Impedance TransformersM.H. Perrott22

Matching for Voltage versus Power Transfer Consider the voltage divider networkIRSVsRL For maximum voltage transfer For maximum power transferVoutWhich one do we want?M.H. Perrott23

Note: Maximum Power Transfer DerivationRSVsIRLVout Formulation Take the derivative and set it to zeroM.H. Perrott24

Voltage Versus Power For most communication circuits, voltage (or current)is the key signal for detection- Phase information is important- Power is ambiguous with respect to phase information Example: VoltagePowerFor high speed circuits with transmission lines,achieving maximum power transfer is important- Maximum power transfer coincides with having zeroreflections (i.e., L 0)Can we ever win on both issues?M.H. Perrott25

Broadband Impedance Transformers Consider placing an ideal transformer between sourceIinIoutand loadRSVsRinRLVinVout1:N Transformer basics (passive, zero loss) Transformer input impedanceM.H. Perrott26

What Value of N Maximizes Voltage Transfer? Derive formula for Vout versus Vin for given N value Take the derivative and set it to zeroM.H. Perrott27

What is the Input Impedance for Max Voltage Transfer? We know from basic transformer theory that inputimpedance into transformer is We just learned that, to maximize voltage transfer, wemust set the transformer turns ratio to Put them togetherSo, N should be set for max power transfer into transformerto achieve the maximum voltage transfer at the load!M.H. Perrott28

Benefit of Impedance Matching with Transformers Transformers allow maximum voltage and powertransfer relationship to coincideRSVsRinIinIoutRLVinVout1:N Turns ratio for max power/voltage transfer Resulting voltage gain (can exceed one!)M.H. Perrott29

The Catch It’s hard to realize a transformer with goodperformance over a wide frequency range- Magnetic materials have limited frequency response- Inductors have self-resonant frequencies, losses, andmediocre coupling to other inductors without magneticmaterial For wireless applications, we only need transformersthat operate over a small frequency range- Can we take advantage of this?M.H. Perrott30

Consider Resonant Circuits (Chap. 4 of Lee’s Book)Series Resonant CircuitCsZin Parallel Resonant CircuitLsRsZinCpLpRpKey insight: resonance allows Zin to be purely realdespite the presence of reactive elementsM.H. Perrott31

Comparison of Series and Parallel RL CircuitsSeries RL CircuitParallel RL CircuitLsZin RsZinLpRpEquate real and imaginary parts of the left and rightexpressions (so that Zin is the same for both)- Also equate Q valuesM.H. Perrott32

Comparison of Series and Parallel RC CircuitsSeries RC CircuitParallel RC CircuitCsZin RsZinCpRpEquate real and imaginary parts of the left and rightexpressions (so that Zin is the same for both)- Also equate Q valuesM.H. Perrott33

A Narrowband Transformer: The L Match Assume Q 1Series to ParallelTransformationLsZinCRsZin So, at resonance Transformer steps up impedance!M.H. PerrottCLpRp34

Alternate Implementation of L Match Assume Q 1Parallel to SeriesTransformationLZinCpRpLZin So, at resonance Transformer steps down impedance!M.H. PerrottCsRs35

The Match Combines two L sectionsL1 L2 LL1L2LZinC1C2RZinC1Steps UpImpedance C2RSteps DownImpedanceProvides an extra degree of freedom for choosingcomponent values for a desired transformation ratioM.H. Perrott36

The T Match Also combines two L sectionsL1ZinC1 C2 CL2CL1RZinL2C1Steps DownImpedance C2RSteps UpImpedanceAgain, benefit is in providing an extra degree offreedom in choosing component valuesM.H. Perrott37

Tapped Capacitor as a TransformerC1ZinLC2RL To first order: Useful in VCO designSee Chapter 4 of Tom Lee’s book for analysisM.H. Perrott38

S-parameters We would like an overall macro-model for simulation 15. M.H. Perrott Macro-modeling for Distributed, Linear Networks Z1 Z3 Zs V s ZL Linear Circuits & Passives (1) Z2 Linear Circuits & Passives length d1 length d2 (2) length d3 delay1 velocity d1 LCd1 μεd1