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Impedance Transformation

Impedance Transformation. Topics. Quality Factor Series to parallel conversion Low-pass RC High-pass RL Bandpass Loaded Q Impedance Transformation Coupled Resonant Circuit Recent implementation, if time permits. Quality Factor. Quality Factor. Q is dimensionless.

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Impedance Transformation

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  1. Impedance Transformation

  2. Topics • Quality Factor • Series to parallel conversion • Low-pass RC • High-pass RL • Bandpass • Loaded Q • Impedance Transformation • Coupled Resonant Circuit • Recent implementation, if time permits

  3. Quality Factor

  4. Quality Factor Q is dimensionless

  5. Quality factor of an inductor (Imax) ω=→= == Please note that Q is also equal to Q=Im(Z)/Re(Z) Q=(ωL)/R

  6. Quality factor of Parallel RL circuit Q=Im(Z)/Re(Z) Z== Q=ωL(Rp)2/(ω2L2Rp)=Rp/ωL

  7. Quality factor of a Capacitor ω=→= == Z is the impedance of parallel RC Please note that Q is also equal to Q=Im(Z)/Re(Z) Q=ωCR

  8. Quality factor of a Capacitor in Series with a Resistor Z is the impedance of series RC Please note that Q is also equal to Q=Im(Z)/Re(Z) Q=1/(ωCRS)

  9. Low-Pass RC Filter

  10. High-Pass Filter lpf=pf

  11. LPF+HPF lpf=pf

  12. LPF+HPF (Magnified)

  13. Resistor Removed

  14. Design Intuition

  15. Circuit Quality Factor Q=3.162/(5.129-1.95)=0.99

  16. Mathematical Analysis

  17. Transfer Function of a Bandpass Filter Resonant frequency

  18. Cutoff Frequency

  19. Bandwidth Calculation

  20. Equivalent Circuit Approach At resonant frequency, XP=1/(ωoCp)

  21. Effect of the Source Resistance Q=3.162/(0.664)=4.76

  22. Effect of the Load Resistor 6 dB drop at resonance due to the resistive divider. Q=3.162/(7.762-1.318)=0.49 The loading will reduce the circuit Q.

  23. Summary Q=0.99 Q=4.79 Q=0.49

  24. Design Constraints • Specs • Resonant Frequency: 2.4 GHz • RS=50 Ohms • RL=Infinity • List Q, C & L

  25. Values • Specs: • Resonant Frequency: 2.4 GHz • RS=50 Ohms • RL=Infinity

  26. Design Example Q=2.4/(2.523-2.286)=10.12 BW=237 MHz

  27. Implement the Inductor

  28. http://www-smirc.stanford.edu/spiralCalc.html

  29. Resistance of Inductor • R=Rsh(L/W) • Rsh is the sheet resistance • Rsh is 22 mOhms per square for W=6um. • If the outer diameter is 135 um, the length is approximately 135um x4=540 um. • R=22 mOhms x (540/6)=1.98 Ohms Q=(ωL)/R=(2π2.4G0.336 nH)/1.98 Ω=2.56

  30. Include Resistor In the Tank Circuitry Q=2.427/(3.076-1.888)=2.04 Inclusion of parasitic resistance reduces the circuit Q from 10.

  31. Series to Parallel Conversion

  32. Series to Parallel Conversion We have an open at DC! We have resistor RP at DC! It is NOT POSSIBLE to make these two circuits Identical at all frequencies, but we can make these to exhibit approximate behavior at certain frequencies.

  33. Derivation QS=QP

  34. RP QS=1/(ωCSRS)

  35. Cp QS=1/(ωCSRS)

  36. Summary

  37. Series to Parallel Conversion for RL Circuits

  38. Resistance of Inductor • R=Rsh(L/W) • Rsh is the sheet resistance • Rsh is 22 mOhms per square for W=6um. • If the outer diameter is 135 um, the length is approximately 135um x4=540 um. • R=22 mOhms x (540/6)=1.98 Ohms Q=(ωL)/R=(2π2.4G0.336 nH)/1.98 Ω=2.56 Rp=RS(1+QSQS)=1.98 Ohms(1+2.56x2.56)=14.96 Ohms Lp=LS(1+1/(QSQS))=331.5 pH(1+1/2.56/2.56)=382.08 nH

  39. Insertion Loss Due to Inductor Resistance At resonant frequency, voltage divider ratio is 14.96Ω/(14.96Ω+50Ω)=0.2303 Convert to loss in dB, 20log10(0.23)=-12.75 dB

  40. Use Tapped-C Circuit to Fool the Tank into Thinking It Has High RS

  41. Derivation

  42. Previous Design Values • Specs: • Resonant Frequency: 2.4 GHz • RS=50 Ohms • RL=Infinity

  43. Design Problem • Knowns & Unknowns • Knowns: • RS=50 Ohms • CT=13.26 pF • Unknowns: • C1/C2 • R’S

  44. Calculations • CT=C1/(1+C1/C2) • C1=CT(1+C1/C2)

  45. Include the Effect of Parasitic Resistor

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