6.776 High Speed Communication Circuits Lecture 4 S-Parameters and Impedance Transformers
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6.776 High Speed Communication Circuits Lecture 4 S-Parameters and Impedance Transformers Massachusetts Institute of Technology February 10, 2005 Copyright © 2005 by Hae-Seung Lee and Michael H. Perrott What Happens When the Wave Hits a Boundary? Reflections can occur Incident Wave ZL Ex x Hy y Reflected Wave z ZL Hy H.-S. Lee & M.H. Perrott Ex MIT OCW 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 voltage Incident Wave I ZL Ex x y I Reflected Wave z I ZL Ex Hy H.-S. Lee & M.H. Perrott Hy I MIT OCW What Happens At The Load Location? Voltage and currents at load are ratioed according to the load impedance Incident Wave Ii Voltage at Load ZL Vi Current at Load x y Ii Reflected Wave z Ir Ratio at Load ZL Vr H.-S. Lee & M.H. Perrott Ir MIT OCW Relate to Characteristic Impedance From previous slide Voltage and current ratio in transmission line set by it characteristic impedance Substituting: H.-S. Lee & M.H. Perrott MIT OCW Define Reflection Coefficient Definition: - No reflection if Γ L =0 Relation to load and characteristic impedances Alternate expression - No reflection if Z L H.-S. Lee & M.H. Perrott = Zo MIT OCW Parameterization of High Speed Circuits/Passives Circuits or passive structures are often connected to transmission lines at high frequencies - How do you describe their behavior? Linear Network Transmission Line 1 H.-S. Lee & M.H. Perrott Transmission Line 2 MIT OCW Calculate Response to Input Voltage Sources Assume source impedances match their respective transmission lines Same value by definition Same value by definition Linear Network Z1 Z1 Z2 Z2 Vin1 Vin2 Transmission Line 1 H.-S. Lee & M.H. Perrott Transmission Line 2 MIT OCW Calculate Response to Input Voltage Sources Sources create incident waves on their respective transmission line Circuit/passive network causes - Reflections on same transmission line - Feedthrough to other transmission line Vi1 Linear Network Vi2 Z1 Z2 Z1 Vin1 H.-S. Lee & M.H. Perrott Z2 Vr1 Vr2 Vin2 MIT OCW Calculate Response to Input Voltage Sources Reflections on same transmission line are parameterized by ΓL - Note that Γ is generally different on each side of the circuit/passive network L ΓL1 Vi1 ΓL2 Linear Network Vi2 Z1 Z2 Z1 Vin1 Z2 Vr1 Vr2 Vin2 How do we parameterize feedthrough to the other transmission line? H.-S. Lee & M.H. Perrott MIT OCW S-Parameters – Definition Model circuit/passive network using 2-port techniques - Similar idea to Thevenin/Norton modeling ΓL1 Vi1 ΓL2 Linear Network Vi2 Z1 Z2 Z1 Vin1 Z2 Vr1 Vr2 Vin2 Defining equations: H.-S. Lee & M.H. Perrott MIT OCW S-Parameters – Calculation/Measurement ΓL1 Vi1 ΓL2 Linear Network Vi2 Z1 Z2 Z1 Vin1 H.-S. Lee & M.H. Perrott Z2 Vr1 Vr2 Vin2 MIT OCW Note: Alternate Form for S21 and S12 ΓL1 Vi1 ΓL2 Linear Network Vi2 Z1 Z2 Z1 Vin1 H.-S. Lee & M.H. Perrott Z2 Vr1 Vr2 Vin2 MIT OCW Block Diagram of S-Parameter 2-Port Model S-Parameter Two-Port Model Vi1 Vi2 Z2 S21 Z1 Z1 Z1 S12 Z2 Z2 S11 S22 Vr1 Vr2 Key issue – two-port is parameterized with respect to the 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 Ohms 11 21 1 1 H.-S. Lee & M.H. Perrott 2 2 MIT OCW S-Parameter Calculations – Example 1 Transmission Line Junction Vi1 Z1 Vi2 Z2 Vr1 Vr2 Derive S-Parameter 2-Port Set Vi2 = 0 H.-S. Lee & M.H. Perrott Set Vi1 = 0 MIT OCW S-Parameter Calculations – Example 2 Vi1 Transmission Line Junction with Capacitor Z1 Vr1 Vi2 Z2 C Vr2 Derive S-Parameter 2-Port Same as before: But now: H.-S. Lee & M.H. Perrott MIT OCW S-Parameter Calculations – Example 3 Vi1 Vi2 Z1 Vr1 Z2 L1 C1 C2 Vr2 Derive S-Parameter 2-Port The S-parameter calculations are now more involved This is a homework problem - Network now has more than one node H.-S. Lee & M.H. Perrott MIT OCW Impedance Transformers H.-S. Lee & M.H. Perrott MIT OCW Matching for Voltage versus Power Transfer Consider the voltage divider network Given the Thevenin equivalent source with Vs and Rs, how do we deliver maximum voltage or power to the load? I RS For maximum voltage transfer Vs RL Vout For maximum power transfer Which one do we want? H.-S. Lee & M.H. Perrott MIT OCW Note: Maximum Power Transfer Derivation RS Vs I RL Vout Formulation: Rs is given, RL is variable Take the derivative and set it to zero H.-S. Lee & M.H. Perrott MIT OCW 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: - For high speed circuits with transmission lines, achieving maximum power transfer is important Maximum power transfer coincides with having zero reflections (i.e., ΓL = 0) - Can we ever win on both issues? H.-S. Lee & M.H. Perrott MIT OCW Broadband Impedance Transformers Consider placing an ideal transformer between source Iin Iout and load R S Vs Rin RL Vin Vout 1:N Transformer basics (passive, zero loss) Transformer input impedance H.-S. Lee & M.H. Perrott MIT OCW What Value of N Maximizes Voltage Transfer? Derive formula for Vout versus Vin for given N value Take the derivative and set it to zero H.-S. Lee & M.H. Perrott MIT OCW What is the Input Impedance for Max Voltage Transfer? We know from basic transformer theory that input impedance into transformer is We just learned that, to maximize voltage transfer, we must set the transformer turns ratio to Put them together So, N should be set for max power transfer into transformer to achieve the maximum voltage transfer at the load. This also ensures no reflection. H.-S. Lee & M.H. Perrott MIT OCW Benefit of Impedance Matching with Transformers Transformers allow maximum voltage and power transfer relationship to coincide RS Vs Rin Iin Iout RL Vin Vout 1:N Turns ratio for max power/voltage transfer Resulting voltage gain (can exceed one!) H.-S. Lee & M.H. Perrott MIT OCW Problems with True Transformers It’s difficult to realize a transformer with good performance over a wide frequency range - Magnetic materials have limited frequency response (both low and high frequency limits) - Inductors have self-resonant frequencies, losses, and mediocre coupling to other inductors without magnetic material For wireless applications, we only need transformers that operate over a small frequency range (except UWB) - Can we take advantage of this?: use ‘impedance transformer” instead of a true transformer H.-S. Lee & M.H. Perrott MIT OCW Consider Resonant Circuits (Chap. 3 (2nd ed.) or 4 (1st ed.) of Text) Series Resonant Circuit Cs Zin Parallel Resonant Circuit Ls Rs Zin Cp Lp Rp Key insight: at resonance Zin becomes purely real despite the presence of reactive elements H.-S. Lee & M.H. Perrott MIT OCW Equivalence of Series and Parallel RL Circuits Series RL Circuit Parallel RL Circuit Ls Zin Rs Zin Lp Rp Equate real and imaginary parts of the left and right expressions (so that Zin is the same for both) - Also equate Q values H.-S. Lee & M.H. Perrott MIT OCW Series-Parallel Equivalence Analysis Series RL Circuit Parallel RL Circuit Ls Zin Rs H.-S. Lee & M.H. Perrott Zin Lp Rp MIT OCW Equivalence of Series and Parallel RC Circuits Series RC Circuit Parallel RC Circuit Cs Zin Rs Zin Cp Rp Equate real and imaginary parts of the left and right expressions (so that Zin is the same for both) - Also equate Q values H.-S. Lee & M.H. Perrott MIT OCW A Narrowband Impedance Transformer: The L Match Series to Parallel Transformation Ls Zin C At resonance Transformer steps up impedance! H.-S. Lee & M.H. Perrott Rs Zin C Lp Rp MIT OCW Alternate Implementation of L Match Parallel to Series Transformation L Zin Cp Rp L Zin At resonance Transformer steps down impedance! H.-S. Lee & M.H. Perrott Cs Rs MIT OCW The π Match Combines two L sections L1 + L2 = L L1 L2 L Zin C1 C2 R Zin C1 Steps Up Impedance C2 R Steps Down Impedance Provides an extra degree of freedom for choosing component values for a desired transformation ratio H.-S. Lee & M.H. Perrott MIT OCW The T Match Also combines two L sections L1 Zin C1 + C2 = C L2 C L1 R Zin L2 C1 Steps Down Impedance C2 R Steps Up Impedance Again, benefit is in providing an extra degree of freedom in choosing component values H.-S. Lee & M.H. Perrott MIT OCW Tapped Capacitor as a Transformer C1 Zin L C2 RL To first order: Useful in VCO design See Chap. 3 (2nd ed.) or 4 (1st ed.) of Text H.-S. Lee & M.H. Perrott MIT OCW
