Impedance Transformation. Topics Quality Factor Series to parallel conversion Low-pass RC High-pass...

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Impedance Transformation

Transcript of Impedance Transformation. Topics Quality Factor Series to parallel conversion Low-pass RC High-pass...

Page 1: Impedance Transformation. Topics Quality Factor Series to parallel conversion Low-pass RC High-pass RL Bandpass Loaded Q Impedance Transformation Coupled.

Impedance Transformation

Page 2: Impedance Transformation. Topics Quality Factor Series to parallel conversion Low-pass RC High-pass RL Bandpass Loaded Q Impedance Transformation Coupled.

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

Page 3: Impedance Transformation. Topics Quality Factor Series to parallel conversion Low-pass RC High-pass RL Bandpass Loaded Q Impedance Transformation Coupled.

Quality Factor

Page 4: Impedance Transformation. Topics Quality Factor Series to parallel conversion Low-pass RC High-pass RL Bandpass Loaded Q Impedance Transformation Coupled.

Quality Factor

Q is dimensionless

Page 5: Impedance Transformation. Topics Quality Factor Series to parallel conversion Low-pass RC High-pass RL Bandpass Loaded Q Impedance Transformation Coupled.

Quality factor of an inductor

(Imax)

= =ω=→ =

Q=(ωL)/R

Please note that Qis also equal to Q=Im(Z)/Re(Z)

Page 6: Impedance Transformation. Topics Quality Factor Series to parallel conversion Low-pass RC High-pass RL Bandpass Loaded Q Impedance Transformation Coupled.

Quality factor of Parallel RL circuit

Q=Im(Z)/Re(Z)

Z==

Q=ωL(Rp)2/(ω2L2Rp)=Rp/ωL

Page 7: Impedance Transformation. Topics Quality Factor Series to parallel conversion Low-pass RC High-pass RL Bandpass Loaded Q Impedance Transformation Coupled.

Quality factor of a Capacitor

= =ω=→ =

Q=ωCR

Please note that Qis also equal to Q=Im(Z)/Re(Z)

Z is the impedanceof parallel RC

Page 8: Impedance Transformation. Topics Quality Factor Series to parallel conversion Low-pass RC High-pass RL Bandpass Loaded Q Impedance Transformation Coupled.

Quality factor of a Capacitor in Series with a Resistor

Q=1/(ωCRS)

Please note that Qis also equal to Q=Im(Z)/Re(Z)

Z is the impedanceof series RC

Page 9: Impedance Transformation. Topics Quality Factor Series to parallel conversion Low-pass RC High-pass RL Bandpass Loaded Q Impedance Transformation Coupled.

Low-Pass RC Filter

Page 10: Impedance Transformation. Topics Quality Factor Series to parallel conversion Low-pass RC High-pass RL Bandpass Loaded Q Impedance Transformation Coupled.

High-Pass Filter

lpf= pf

𝐿=𝑅2𝐶

Page 11: Impedance Transformation. Topics Quality Factor Series to parallel conversion Low-pass RC High-pass RL Bandpass Loaded Q Impedance Transformation Coupled.

LPF+HPF

lpf= pf

Page 12: Impedance Transformation. Topics Quality Factor Series to parallel conversion Low-pass RC High-pass RL Bandpass Loaded Q Impedance Transformation Coupled.

LPF+HPF (Magnified)

Page 13: Impedance Transformation. Topics Quality Factor Series to parallel conversion Low-pass RC High-pass RL Bandpass Loaded Q Impedance Transformation Coupled.

Resistor Removed

Page 14: Impedance Transformation. Topics Quality Factor Series to parallel conversion Low-pass RC High-pass RL Bandpass Loaded Q Impedance Transformation Coupled.

Design Intuition

Page 15: Impedance Transformation. Topics Quality Factor Series to parallel conversion Low-pass RC High-pass RL Bandpass Loaded Q Impedance Transformation Coupled.

Circuit Quality Factor

Q=3.162/(5.129-1.95)=0.99

Page 16: Impedance Transformation. Topics Quality Factor Series to parallel conversion Low-pass RC High-pass RL Bandpass Loaded Q Impedance Transformation Coupled.

Mathematical Analysis

Page 17: Impedance Transformation. Topics Quality Factor Series to parallel conversion Low-pass RC High-pass RL Bandpass Loaded Q Impedance Transformation Coupled.

Transfer Function of a Bandpass Filter

Resonant frequency

Page 18: Impedance Transformation. Topics Quality Factor Series to parallel conversion Low-pass RC High-pass RL Bandpass Loaded Q Impedance Transformation Coupled.

Cutoff Frequency

Page 19: Impedance Transformation. Topics Quality Factor Series to parallel conversion Low-pass RC High-pass RL Bandpass Loaded Q Impedance Transformation Coupled.

Bandwidth Calculation

𝑄=ω𝑜𝑅𝐶

Page 20: Impedance Transformation. Topics Quality Factor Series to parallel conversion Low-pass RC High-pass RL Bandpass Loaded Q Impedance Transformation Coupled.

Equivalent Circuit Approach

At resonant frequency, XP=1/(ωoCp)

Page 21: Impedance Transformation. Topics Quality Factor Series to parallel conversion Low-pass RC High-pass RL Bandpass Loaded Q Impedance Transformation Coupled.

Effect of the Source Resistance

Q=3.162/(0.664)=4.76

Page 22: Impedance Transformation. Topics Quality Factor Series to parallel conversion Low-pass RC High-pass RL Bandpass Loaded Q Impedance Transformation Coupled.

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.

Page 23: Impedance Transformation. Topics Quality Factor Series to parallel conversion Low-pass RC High-pass RL Bandpass Loaded Q Impedance Transformation Coupled.

Summary

Q=0.99

Q=4.79

Q=0.49

𝑄=ω𝑜𝑅𝐶

Page 24: Impedance Transformation. Topics Quality Factor Series to parallel conversion Low-pass RC High-pass RL Bandpass Loaded Q Impedance Transformation Coupled.

Design Constraints

• Specs– Resonant Frequency: 2.4 GHz– RS=50 Ohms

– RL=Infinity

• List Q, C & L

𝑄=ω𝑜𝑅𝐶

Page 25: Impedance Transformation. Topics Quality Factor Series to parallel conversion Low-pass RC High-pass RL Bandpass Loaded Q Impedance Transformation Coupled.

ValuesQ C L

0.5 0.663 pF

6.63 nH

1 1.326 pF

3.315 nH

10 13.26 pF

331.5 pH

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

Page 26: Impedance Transformation. Topics Quality Factor Series to parallel conversion Low-pass RC High-pass RL Bandpass Loaded Q Impedance Transformation Coupled.

Design Example

Q=2.4/(2.523-2.286)=10.12

BW=237 MHz

Page 27: Impedance Transformation. Topics Quality Factor Series to parallel conversion Low-pass RC High-pass RL Bandpass Loaded Q Impedance Transformation Coupled.

Implement the Inductor

Page 28: Impedance Transformation. Topics Quality Factor Series to parallel conversion Low-pass RC High-pass RL Bandpass Loaded Q Impedance Transformation Coupled.

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

Page 29: Impedance Transformation. Topics Quality Factor Series to parallel conversion Low-pass RC High-pass RL Bandpass Loaded Q Impedance Transformation Coupled.

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

Page 30: Impedance Transformation. Topics Quality Factor Series to parallel conversion Low-pass RC High-pass RL Bandpass Loaded Q Impedance Transformation Coupled.

Include Resistor In the Tank Circuitry

Q=2.427/(3.076-1.888)=2.04

Inclusion of parasitic resistancereduces the circuit Q from 10.

Page 31: Impedance Transformation. Topics Quality Factor Series to parallel conversion Low-pass RC High-pass RL Bandpass Loaded Q Impedance Transformation Coupled.

Series to Parallel Conversion

Page 32: Impedance Transformation. Topics Quality Factor Series to parallel conversion Low-pass RC High-pass RL Bandpass Loaded Q Impedance Transformation Coupled.

Series to Parallel Conversion

We have an open at DC!

We have resistor RP at DC!

It is NOT POSSIBLE to make these two circuitsIdentical at all frequencies, but we can makethese to exhibit approximate behavior at certain frequencies.

Page 33: Impedance Transformation. Topics Quality Factor Series to parallel conversion Low-pass RC High-pass RL Bandpass Loaded Q Impedance Transformation Coupled.

Derivation

QS=QP

Page 34: Impedance Transformation. Topics Quality Factor Series to parallel conversion Low-pass RC High-pass RL Bandpass Loaded Q Impedance Transformation Coupled.

RP

QS=1/(ωCSRS)

Page 35: Impedance Transformation. Topics Quality Factor Series to parallel conversion Low-pass RC High-pass RL Bandpass Loaded Q Impedance Transformation Coupled.

Cp

QS=1/(ωCSRS)

Page 36: Impedance Transformation. Topics Quality Factor Series to parallel conversion Low-pass RC High-pass RL Bandpass Loaded Q Impedance Transformation Coupled.

Summary

Page 37: Impedance Transformation. Topics Quality Factor Series to parallel conversion Low-pass RC High-pass RL Bandpass Loaded Q Impedance Transformation Coupled.

Series to Parallel Conversion for RL Circuits

Page 38: Impedance Transformation. Topics Quality Factor Series to parallel conversion Low-pass RC High-pass RL Bandpass Loaded Q Impedance Transformation Coupled.

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 OhmsLp=LS(1+1/(QSQS))=331.5 pH(1+1/2.56/2.56)=382.08 nH

Page 39: Impedance Transformation. Topics Quality Factor Series to parallel conversion Low-pass RC High-pass RL Bandpass Loaded Q Impedance Transformation Coupled.

Insertion Loss Due to Inductor Resistance

At resonant frequency, voltage divider ratio is14.96Ω/(14.96 Ω+50 Ω)=0.2303

Convert to loss in dB, 20log10(0.23)=-12.75 dB

Page 40: Impedance Transformation. Topics Quality Factor Series to parallel conversion Low-pass RC High-pass RL Bandpass Loaded Q Impedance Transformation Coupled.

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

Page 41: Impedance Transformation. Topics Quality Factor Series to parallel conversion Low-pass RC High-pass RL Bandpass Loaded Q Impedance Transformation Coupled.

Derivation

Page 42: Impedance Transformation. Topics Quality Factor Series to parallel conversion Low-pass RC High-pass RL Bandpass Loaded Q Impedance Transformation Coupled.

Previous Design ValuesQ C L

0.5 0.663 pF

6.63 nH

1 1.326 pF

3.315 nH

10 13.26 pF

331.5 pH

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

Page 43: Impedance Transformation. Topics Quality Factor Series to parallel conversion Low-pass RC High-pass RL Bandpass Loaded Q Impedance Transformation Coupled.

Design Problem

Knowns & UnknownsKnowns: • RS=50 Ohms• CT=13.26 pFUnknowns:• C1/C2

• R’S

Page 44: Impedance Transformation. Topics Quality Factor Series to parallel conversion Low-pass RC High-pass RL Bandpass Loaded Q Impedance Transformation Coupled.

Calculations

• CT=C1/(1+C1/C2)

• C1=CT(1+C1/C2)

C1/C2 R’S C1 C2

1 200 Ω 26.52 pF 26.52 pF

2 450Ω 39.78 pF 19.89 pF

3 800Ω 53.04 pF 17.68 pF

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Page 46: Impedance Transformation. Topics Quality Factor Series to parallel conversion Low-pass RC High-pass RL Bandpass Loaded Q Impedance Transformation Coupled.

Include the Effect of Parasitic Resistor