How To Draw Stability Circles In Smith Chart
Parent Category: 2022 HFE
By Own Rehman
Introduction:
Stability circles are a tool, used to examine and clarify the stability of an amplifier (in the case under discussion) using a graphical technique, with the assist of a Smith Chart. (Note: A free, demo version of smith chart software is bachelor on the web from Fritz Dellsperger).
This monograph presents the stability circle tool for engineers. It is understood that many CAD programs tin can generate these, but it is always useful to understand the stability circle on an intuitive level as a good applied science exercise.
We limit the stability circle tool description to a two port device.
Fundamentals
Reflection coefficients:
To recap some bones quantities we note that the reflection coefficient is given by:
Г = Vr/Vt(1)
Hither Vt is the transmitted signal and Vr is the reflected point from the load. Г is a circuitous number and is described by its magnitude and phase angle.
In the simplest cases where the imaginary part of the reflection coefficient is aught,
Г = -1 (maximum negative reflection, when the line is short circuited at its end.),
Г = 0 (no reflection from the load and the two port is perfectly matched to the load),
Г = +1 (maximum positive reflection; line is open circuit at the load).
VSWR
The voltage standing wave ratio is given in terms of the reflection coefficient as:
VSWR =Vmax/Vmin = (1+ρ)/(1-ρ)(ii)
where ρ is the magnitude of the reflection coefficient.
Reflection coefficient in terms of impedances.
If the source and load impedances are known and then the reflection coeffcient tin can exist written equally:
Г = (ZL-ZS)/(ZL+ZS)(3)
where ZL is the impedance towards the load and ZS is the impedance towards the source.
Another quantity that is useful in assay is the Return Loss which is simply the magnitude of the reflection coefficient in dB.
RL(dB) = -20Log(Г)(four)
For more on this and related topics please read the book: "Practical Impedance Matching" by Ain Rehman, published by Amazon.
Basics of stability circles
In this monograph, we will focus on ii ports and stability circles relating to two ports. Normally this is the most useful analysis for amplifiers and related circuits.
To stabilize a virtual or actual ii port, we need to investigate what types of termination tin can event in instability or oscillation.
If a two port has been found to be potentially unstable then:
one.0 There are some source terminations that crusade the output reflection coefficient to become greater than one.
two.0 There are also some output terminations that crusade the input reflection coeffcient to become greater than 1.
Lets call the terminations that cause reflection coefficients to become one, type a terminations and terminations that keep the reflection coefficients under i, type b terminations. A tertiary blazon of termination is also defined. Type c terminations are deadline terminations.
Thus, to stabilize an amplifier, for example, nosotros first need to find out what the type a terminations are, and then use a circuit to change the behavior of the amplifier from unstable to stable. (Techniques to stablize an amplifier are discussed in a companion article to be published).
We must as well realize, that when a ii port is connected in such a mode that it has external load and source impedances, its input and output reflection coefficients modify from s11 and s22 ( in terms of s parameters) to TIN and TOUT.
Interested readers may download the calculators available in the Bespeak Processing Group Inc website (world wide web.signalpro.biz) under the "Complementary" menu. These calculators let the user to apace summate Can and TOUT given the s parameters of the 2 port and profoundly simplify the calculations.
An active two port is defined by the following reflection coefficients:
ГS is the source reflection coeffcient,
ГL is the load reflection coefficient,
ГIN is the ii port input reflection coeffcient,
ГOUT is the 2 port output reflection coefficient.
To recap, TOUT is given past:
TOUT = s22+(s12*s21*ГS)/(1-s11* ГS)(5)
Setting the magnitude of TOUT to one and solving for TIN gives an equation for a circle. This circle is called the source stability circumvolve.
The center of the source stability circumvolve is at:
CS =[ (s11-S22*) *Δ]*/[abs(s11)2 –abs(Δ)2 ](vi)
and the radius of the circumvolve is at:
RS = abs(s21*s12)/[(abs(s11)2)-abs(Δ)2 ](vii)
Δ = s11*s22 – s21*s12(7.1)
Similarly the load stability circle is found past setting the magnitude of Tin to 1. Values of TOUT that satisfy this equation requite the load stabilty circle.
Estimation of the stability circles
Interpreting stability circles can be a difficult job commonly, simply the post-obit technique can exist used to simplify the interpretation with the given weather condition:
A. If the magnitude of TOUT > 1 oscillations may take place at the output.
B. If the magnitude of TOUT<1 so the output port will exist stable.
C. If the magnitude of TOUT =1, then it is a deadline case.
So we demand to run into if the region inside the stability circle represents a type a or blazon b termination. (Type a is unstable and blazon b is stable).
Cull 50 Ohm for the source termination. (The rationale for this is, that we assume 50Ω is the impedance used for extracting s- parameters originally, specifically s22. Nosotros demand to make certain this is the case though.)
The above pick is to see if the magnitude of |s22|>one when the source impedance is 50 Ohms. If the magnitude of |s22|<1 then obviously the 50 Ohm impedance is a type b termination, i.e a favorable termination for stable ouput.
Conversely if |s22|>ane and then 50 Ohm will be classified as a blazon a, or unfavorable termination.
Please refer to figure 1 that tin can clarify these remarks.
Effigy 1 • Circumference of the stability circumvolve represents borderline terminations.
Note that the circumference of the stability circle represents borderline terminations.
With a bit of practise one can get proficient at identifying the stable and unstable regions.
In the to a higher place, the source stability circumvolve was used for the assay. Information technology should be obvious that in a similar manner the load stabilty circle can be used and analyzed.
Summary:
Stability circles are a tool to analyze the stability of an amplifier or related circuits using a graphical technique. The idea is to find source and load terminations that volition cause oscillation or not. The unfavorable terminations are chosen blazon a and the favorable terminations are called blazon b. Deadline terminations are called type c. Using stability circles ane can analyze a excursion like an amplifier for stable operation.
About the Author
Ain Rehman is founder and chief engineer at Signal Processing Group.
Source: http://www.highfrequencyelectronics.com/index.php?option=com_content&view=article&id=2473:understanding-stability-circles&catid=215&Itemid=189
Posted by: murphycattat.blogspot.com
0 Response to "How To Draw Stability Circles In Smith Chart"
Post a Comment