1. N-port Network
Port reference
Line Impedance
Port Voltage & Current

2. Impedance Matrix
Ratio of i-th port Open Circuit Voltage to the j-th port Current with all ports except j-th open-circuiting

3. Admittance Matrix
Ratio of i-th port Short Circuit Current to the j-th port Voltage with all ports except j-th short-circuiting

4. *Lorentz Reciprocity Theorem

5. 1) S encloses no sources.
2) S bounds perfect conductor.
3) S is a sphere at infinity
Same as 2)

6. Reciprocal Network:

7. Lossless Network:
The real power delivered to the network

8. Scattering Parameter
Circuit variables:
Reference plane
Port 2
Port 1

9. Why Use S-Parameters? relatively easy to obtain at high frequencies
measure voltage traveling waves with a vector network analyzer
don't need shorts/opens which can cause active devices to oscillate or self-destruct
relate to familiar measurements (gain, loss, reflection coefficient ...)
can cascade S-parameters of multiple devices to predict system performance
can compute H, Y, or Z parameters from S-parameters if desired
can easily import and use S-parameter files in our electronic-simulation tools
At high frequencies, it is very hard to measure total voltage and current at the device ports. One cannot simply connect a voltmeter or current probe and get accurate measurements due to the impedance of the probes themselves and the difficulty of placing the probes at the desired positions. In addition, active devices may oscillate or self-destruct with the connection of shorts and opens. Clearly, some other way of characterizing high-frequency networks is needed that doesn't have these drawbacks.
For these reasons, scattering or S-parameters were developed. S-parameters have many advantages over the previously mentioned H, Y or Z-parameters. They relate to familiar measurements such as gain, loss, and reflection coefficient. They are relatively easy to measure, and don't require the connection of undesirable loads to the device under test. The measured S-parameters of multiple devices can be cascaded to predict overall system performance. If desired, H, Y, or Z-parameters can be derived from S-parameters if desired. And very important for RF design, S-parameters are easily imported and used with electronic-simulation tools. S-parameters are the shared language between simulation and measurement.
An N-port device has N2 S-parameters. So, a two-port device has four S-parameters. The numbering convention for S-parameters is that the first number following the "S" is the port where energy emerges, and the second number is the port where energy enters. So, S21 is a measure of power coming out port two as a result of applying an RF stimulus to port one. When the numbers are the same (e.g. S11), it indicates a reflection measurement.

10. Scattering Parameter
Definition of Circuit variables(with traveling waves)

11. Scattering Parameter Lossless network: S-matrix is unitary matrix
Total power incident toward network (Pin ) must be equal to total power reflected from network (Pout )

12. Scattering Parameter
Reciprocal network: S-matrix is symmetric matrix

13. Scattering Parameter Shifting property Port 1 n-port network Port n
A shift in reference plane:the phase of Sij is shifted by electrical length’s of shift in terminal planes i and j.

14. Measuring S-Parameters(2 –port)11 =
Reflected
Incident b 1 a 2 21
Transmitted 22 12 Z
Load
DUT
Forward
Reverse
Port matching condition: S-parameters are measured under the condition
that TL’s connected each ports are terminated with matched load
S11 and S21 are determined by measuring the magnitude and phase of the incident, reflected and transmitted signals when the output is terminated in a perfect Zo load. This condition guarantees that a2 is zero. S11 is equivalent to the input complex reflection coefficient or impedance of the DUT, and S21 is the forward complex transmission coefficient. Likewise, by placing the source at port 2 and terminating port 1 in a perfect load (making a1 zero), S22 and S12 measurements can be made. S22 is equivalent to the output complex reflection coefficient or output impedance of the DUT, and S12 is the reverse complex transmission coefficient.
The accuracy of S-parameter measurements depends greatly on how good a termination we apply to the port not being stimulated. Anything other than a perfect load will result in a1 or a2 not being zero (which violates the definition for S-parameters). When the DUT is connected to the test ports of a network analyzer and we don't account for imperfect test port match, we have not done a very good job satisfying the condition of a perfect termination. For this reason, two-port error correction, which corrects for source and load match, is very important for accurate S-parameter measurements.

15. Transmission (ABCD) Matrix

18. S Z Y
ABCD

19. Equivalent
PI-Network

20. Scattering Transfer Parameter (T-parameter of two port network)

21. Various 2-port network connection

22. Signal Flow Graph
1. Signal flow graph can be used to represent and analyze the transmission and reflection
of waves in microwave network
2. State variables are those used for S-parameter
3. Using flow graph technique we can derive expressions such as power gains, and voltage
gains of complete microwave network with ease

24. Power Gain Equations

25. Power Gain Transducer Power Gain Matched power gain
Operating Power Gain

26. Power Gain Available Power Gain Unilateral Power Gain
Maximum unilateral power gain

27. Power Gain (unilateral case: S12=0)
Unilateral figure of merit: U

28. Impedance Matching (for two port network with S-parameter)
Complexity, Bandwidth, Implementation, Adjustability
1. without reflection
2. with maximum power transmission:
loss and noise may be inserted due to multiple reflection
3. Low noise, gain, power, input and/or output VSWR, bandwidth..
Matching
Network

29. Matching Network(lumped elements)
The design of lossless matching network is accomplished by moving
along constant-resistance and constant conductance circles
L-network A B

30. Case of , using network A

31. Case of , using network A
Method #1
Method #2

32. Case of , using network B

33. Case of , using network B
Method #1
Method #2

34. Transmission lines open and short

35. Transmission lines open and short
Frequency response
1. Open

36. 2. short

37. Single Stub Tuning
Shunt Stub Tuning
Series Stub Tuning

38. Shunt Stub Tuning: Analytic Solution

39. Shunt Stub Tuning

41. Double Stub Tuning

42. Double Stub Tuning

43. Double Stub Tuning

45. Quarter-wavelength transformer
Impedance chart

48. Theory of Small Reflection
Single section transformer

50. Multi-section transformer

51. Binomial Transformer-design

52. Binomial Transformer-characteristics

54. Chebyshev Transformer-design
Chebyshev polynomials

56. Chebyshev Transformer-design

57. Chebyshev Transformer-design
With N=3

60. Tapered Lines

61. Exponential Taper

62. Triangular Taper