Mu and Mu_prime in S parameter analysis

Dear Abhinav2020,

Abhinav2020 said:

know what they are but here it is written for both mu and mu_prime that them being greater than 1 is the sole necessary as well as sufficient condition for unconditional stability of the 2-port network. How can this be independently true for both mu and mu_prime? 

I think the key is in both the definitions provided in the Troubleshooting Article and the last two statements. Specifically, the definition of mu is noted as:

"This measurement gives the distance from the center of the Smith chart to the nearest output (load) stability circle."

while that of mu_prime is:

"This measurement gives the distance from the center of the Smith chart to the nearest unstable-input (source) stability circle."

and the last two statements are:

"The mu test not only is a test for unconditional stability, but the magnitude of mu is a measure of the stability. In other words, if one two port has a larger mu, it is more stable. 

The advantage of the mu test is that only a single parameter needs to be evaluated. There are no auxiliary conditions like the Kf, B1f test (based on Rollett’s and Kurakawa’s derivations)."

Hence, they both measure two different distances and BOTH provide an identical indication of unconditional stability. However, the magnitude of parameter mu may be used to compare the relative stability of this 2-port to a different 2-port with its value of mu. However, the same cannot be said for mu_prime. The relative magnitude of mu_prime of two different 2-port networks does not indicate anything concern their relative stabilities.

 Does that make any sense to you? I hope I was not too confusing with my comment...let me know if I should clarify!

Shawn

Dear Abhinav2020,

Abhinav2020 said:

know what they are but here it is written for both mu and mu_prime that them being greater than 1 is the sole necessary as well as sufficient condition for unconditional stability of the 2-port network. How can this be independently true for both mu and mu_prime? 

I think the key is in both the definitions provided in the Troubleshooting Article and the last two statements. Specifically, the definition of mu is noted as:

"This measurement gives the distance from the center of the Smith chart to the nearest output (load) stability circle."

while that of mu_prime is:

"This measurement gives the distance from the center of the Smith chart to the nearest unstable-input (source) stability circle."

and the last two statements are:

"The mu test not only is a test for unconditional stability, but the magnitude of mu is a measure of the stability. In other words, if one two port has a larger mu, it is more stable. 

The advantage of the mu test is that only a single parameter needs to be evaluated. There are no auxiliary conditions like the Kf, B1f test (based on Rollett’s and Kurakawa’s derivations)."

Hence, they both measure two different distances and BOTH provide an identical indication of unconditional stability. However, the magnitude of parameter mu may be used to compare the relative stability of this 2-port to a different 2-port with its value of mu. However, the same cannot be said for mu_prime. The relative magnitude of mu_prime of two different 2-port networks does not indicate anything concern their relative stabilities.

 Does that make any sense to you? I hope I was not too confusing with my comment...let me know if I should clarify!

Shawn

I don't think what you are saying is true because even in the case of mu_prime, as it is written "Having mu_prime > 1 is the single necessary and sufficient condition for unconditional stability of the 2-port network". So mu and mu_prime are identical in this aspect. When you say "they both measure two different distances and BOTH provide an identical indication of unconditional stability. However, the magnitude of parameter mu may be used to compare the relative stability of this 2-port to a different 2-port with its value of mu", it means that for a 2-Port network N1 having mu = mu1 and another 2-Port network N2 having mu=mu2, if mu1>mu2, N1 is more stable than N2. But the same thing can be said about mu_prime

I suggest that you take a look at the original article where these parameters are derived: https://doi.org/10.1109/22.179894 or https://scholar.google.com/scholar?cluster=7864836647632565806