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Published byEmanuel Collman Modified about 1 year ago

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Impedance Transformation

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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

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Quality Factor

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Q is dimensionless

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

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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

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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

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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

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Low-Pass RC Filter

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High-Pass Filter

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LPF+HPF

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LPF+HPF (Magnified)

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Resistor Removed

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Design Intuition

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Circuit Quality Factor Q=3.162/( )=0.99

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Mathematical Analysis

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Transfer Function of a Bandpass Filter Resonant frequency

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Cutoff Frequency

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Bandwidth Calculation

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

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

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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.

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Summary Q=0.99 Q=4.79 Q=0.49

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

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

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

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Implement the Inductor

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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

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

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Series to Parallel Conversion

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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.

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Derivation Q S =Q P

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RPRP Q S =1/(ωC S R S )

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Cp Q S =1/(ωC S R S )

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Summary

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Series to Parallel Conversion for RL Circuits

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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

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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

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

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Derivation

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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

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

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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

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Include the Effect of Parasitic Resistor

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