### Matching circuit & Smith Chart

```Impedance Matching (1)
Maximum Power Transfer
Choose an RL in order to maximize power delivered to RL.
Numerical Example
• VTH=1 V
• RTH=50 Ω
Conclusion!
• Maximum power is delivered to the
load resistor when RL is equal to RTH.
Max Power Transfer for Complex
Source Impedance
At resonant frequency, the series impedance of
the inductor and capacitor is zero.
Summary
RL>RS
RS>RL
L Network
• Different L netowrk
• Difference bewteen highpass and low
pass
• Examine butterworth filter from the
point of view of matching….
Resistance Transformation
RP must be larger than RS
(See derivation in the handout)
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/(ωRLC)
QS=RS/(ωL)
Matlab Calculation
• 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, LS≈LP and CS
≈CP.
• RP is Q2 times RS.
Summary
RL>RS
RS>RL
Smith Chart Derivation
Smith Chart Derivation (2)
Smith Chart Construction
(+)
(-)
(The center line represents an axis where X=0.)
zL=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
to 75 Ohms
Negative Capacitance!
Negative capacitance
(1)
(2)
Double click on the smith chart to drop the component
Matlab
Shunt Cap
1.511 pF
1.5 pF
Series L
5.72 nH
5.627 nH
Insertion of a series inductor to an impedance moves the impedance upward,
causing a rotation clockwise along a constant circle of resistance
Series Inductance
Low L
High L
Neg 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
Insertion of a series capacitor to an impedance move impedance
downward, causes a rotation counter clockwise along a constant circle of
resistance
Series Capacitance
Neg C
High C
Low L
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
Inductive susceptance
Conductance
circle
(-)
(+)
Rotate the impedance chart by 180 degrees
(The center line represents an axis where X=0.)
Capacitive susceptance
Insertion of a shunt capacitor causes a rotation clockwise along a
Neg C
High C
Low C
fixed frequency
Insertion of a shunt capacitor causes a rotation clockwise along a
Insertion of a shunt inductor causes a rotation counter clockwise along a
Shunt Inductance
Low L
High L
Neg Ind
fixed frequency
Insertion of a shunt inductor causes a rotation counter clockwise along a
Next Class
•
•
•
•
Pi Network
T Network
Smith Chart
Genesis
The Pi Network
The virtual resistance must be less than RS and RL.
```