# Impedance Matching (1). Maximum Power Transfer Choose an RL in order to maximize power delivered to RL.

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Impedance Matching (1)

Maximum Power Transfer Choose an RL in order to maximize power delivered to RL.

Numerical Example V TH =1 V R TH =50 Ω

Conclusion! Maximum power is delivered to the load resistor when R L is equal to R TH.

Max Power Transfer for Complex Source Impedance At resonant frequency, the series impedance of the inductor and capacitor is zero.

Summary R L >R S R S >R L

L Network Different L netowrk Difference bewteen highpass and low pass Examine butterworth filter from the point of view of matching….

Resistance Transformation (See derivation in the handout) R P must be larger than R S

Matlab Calculation

Simulation Results

High Pass Match Note: There is not a DC path to ZL. RS must be larger than RL! See derivation! QS=sqrt(RS/RL-1) QS=1/(ωR L C) QS=RS/(ωL)

Matlab Calculation

Dealing With Complex Load Absorption Approach Resonance Technique

Match Via Absorption Approach Ignore stray component Match the load resistance to the source resistance with an L-match Subtract the stray component from the L-match value

Absorption Example

Calculation Neglecting Stray Components

Account for Stray Components This technique will not work if the stray components is much larger than L match components. E.g. if 2pF is replaced by 6 pF, then this technique will not work.

Resonant Approach Resonate any stray reactance with an equal and opposite reactance at the frequency of interest!

Example Resonate the 40 pF with a parallel L.

Parallel Resonant Network

Determine the Matching Network

Resonant Approach Example

Series to Parallel Conversion for RC Circuits

Series to Parallel Conversion for RL Circuits

Intuition If the Q is sufficiently large, L S ≈L P and C S ≈C P. R P is Q 2 times R S.

Summary R L >R S R S >R L

Smith Chart Derivation

Smith Chart Derivation (2)

Smith Chart Construction (The center line represents an axis where X=0.) (+) (-)

z L =1±j

Adding a Series Capacitance to an Impedance

Use Smith Chart Matching

SmithChartMatch

Smith Chart Utility 1. Select Smith Chart Match 2.Click on Tools, then select Smith chart utility 3. Select first option

Change the Load Impedance to 75 Ohms

Negative Capacitance! Negative capacitance

Add a Series Inductor (1) (2) Double click on the smith chart to drop the component

Comparison with Matlab Vs. ADS ADSMatlab Shunt Cap1.511 pF1.5 pF Series L5.72 nH5.627 nH

Adding an Inductor in Series Insertion of a series inductor to an impedance moves the impedance upward, causing a rotation clockwise along a constant circle of resistance

Series Inductance Neg L High LLow L fixed frequency Insertion of a series inductor to an impedance moves the impedance upward, causing a rotation clockwise along a constant circle of resistance

Adding a Capacitor in Series Insertion of a series capacitor to an impedance move impedance downward, causes a rotation counter clockwise along a constant circle of resistance

Series Capacitance High C Low L Neg C fixed frequency Insertion of a series capacitor to an impedance move impedance downward, causes a rotation counter clockwise along a constant circle of resistance

Method 1

Method 2 1. Find the Z. 2. Rotate Smith Chart 180 degrees

Smith Chart Construction (The center line represents an axis where X=0.) (+) (-) Conductance circle Inductive susceptance Rotate the impedance chart by 180 degrees Capacitive susceptance

Adding a Shunt Capacitance Insertion of a shunt capacitor causes a rotation clockwise along a constant circle of admittance

Adding a Shunt Capacitance High C Low C Neg C fixed frequency Insertion of a shunt capacitor causes a rotation clockwise along a constant circle of admittance

Adding a Shunt Inductance Insertion of a shunt inductor causes a rotation counter clockwise along a constant circle of admittance

Shunt Inductance Neg Ind High L Low L fixed frequency Insertion of a shunt inductor causes a rotation counter clockwise along a constant circle of admittance

Next Class Pi Network T Network Smith Chart Genesis

The Pi Network The virtual resistance must be less than RS and RL.

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