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

Q is dimensionless

Quality factor of an inductor (I max ) Q=(ωL)/R Please note that Q is also equal to Q=Im(Z)/Re(Z)

Quality factor of Parallel RL circuit Q=Im(Z)/Re(Z) Q=ωL(R p ) 2 /(ω 2 L 2 R p )=R p /ωL

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

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

Low-Pass RC Filter

High-Pass Filter

LPF+HPF

LPF+HPF (Magnified)

Resistor Removed

Design Intuition

Circuit Quality Factor Q=3.162/( )=0.99

Mathematical Analysis

Transfer Function of a Bandpass Filter Resonant frequency

Cutoff Frequency

Bandwidth Calculation

Equivalent Circuit Approach At resonant frequency, XP=1/(ω o C p )

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

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

Summary Q=0.99 Q=4.79 Q=0.49

Design Constraints Specs – Resonant Frequency: 2.4 GHz – R S =50 Ohms – R L =Infinity List Q, C & L

Values QCL pF6.63 nH pF3.315 nH pF331.5 pH Specs: Resonant Frequency: 2.4 GHz R S =50 Ohms R L =Infinity

Design Example Q=2.4/( )=10.12 BW=237 MHz

Implement the Inductor

Resistance of Inductor R=R sh (L/W) – R sh 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

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

Series to Parallel Conversion

We have an open at DC! We have resistor R P 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.

Derivation Q S =Q P

RPRP Q S =1/(ωC S R S )

Cp Q S =1/(ωC S R S )

Summary

Series to Parallel Conversion for RL Circuits

Resistance of Inductor R=R sh (L/W) – R sh 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 R p =R S (1+Q S Q S )=1.98 Ohms(1+2.56x2.56)=14.96 Ohms L p =L S (1+1/(Q S Q S ))=331.5 pH(1+1/2.56/2.56)= nH

Insertion Loss Due to Inductor Resistance At resonant frequency, voltage divider ratio is 14.96Ω/(14.96 Ω+50 Ω)= Convert to loss in dB, 20log 10 (0.23)= dB

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

Derivation

Previous Design Values QCL pF6.63 nH pF3.315 nH pF331.5 pH Specs: Resonant Frequency: 2.4 GHz R S =50 Ohms R L =Infinity

Design Problem Knowns & Unknowns Knowns: R S =50 Ohms CT=13.26 pF Unknowns: C 1 /C 2 R’ S

Calculations C T =C 1 /(1+C 1 /C 2 ) C 1 =C T (1+C 1 /C 2 ) C 1 /C 2 R’ S C1C1 C2C Ω26.52 pF 2450Ω39.78 pF19.89 pF 3800Ω53.04 pF17.68 pF

Include the Effect of Parasitic Resistor