Parent Category: 2021 HFE

*By Ain Rehman*

**Introduction:**

Stability circles are a tool, used to examine and analyze the stability of an amplifier (in the case under discussion) using a graphical technique, with the help of a Smith Chart. (Note: A free, demo version of smith chart software is available on the web from Fritz Dellsperger).

This monograph presents the stability circle tool for engineers. It is understood that many CAD programs can generate these, but it is always useful to understand the stability circle on an intuitive level as a good engineering practice.

**We limit the stability circle tool description to a two port device.**

*Fundamentals*

Reflection coefficients:

To recap some basic quantities we note that the reflection coefficient is given by:

Г = Vr/Vt(1)

Here Vt is the transmitted signal and Vr is the reflected signal from the load. Г is a complex number and is described by its magnitude and phase angle.

In the simplest cases where the imaginary part of the reflection coefficient is zero,

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)(2)

where ρ is the magnitude of the reflection coefficient.

**Reflection coefficient in terms of impedances.**

If the source and load impedances are known then the reflection coeffcient can be written as:

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 analysis is the Return Loss which is simply the magnitude of the reflection coefficient in dB.

RL(dB) = -20Log(Г)(4)

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 two ports and stability circles relating to two ports. Usually this is the most useful analysis for amplifiers and related circuits.

To stabilize a virtual or actual two port, we need to investigate what types of termination can result in instability or oscillation.

If a two port has been found to be potentially unstable then:

1.0 There are some source terminations that cause the output reflection coefficient to become greater than 1.

2.0 There are also some output terminations that cause the input reflection coeffcient to become greater than 1.

Lets call the terminations that cause reflection coefficients to become 1, type a terminations and terminations that keep the reflection coefficients under 1, type b terminations. A third type of termination is also defined. Type c terminations are borderline terminations.

Thus, to stabilize an amplifier, for example, we 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 also realize, that when a two port is connected in such a way that it has external load and source impedances, its input and output reflection coefficients change from s11 and s22 ( in terms of s parameters) to TIN and TOUT.

Interested readers may download the calculators available in the Signal Processing Group Inc website (www.signalpro.biz) under the “Complementary” menu. These calculators allow the user to quickly calculate TIN and TOUT given the s parameters of the two port and greatly 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 two port input reflection coeffcient,

ГOUT is the two port output reflection coefficient.

To recap, TOUT is given by:

TOUT = s22+(s12*s21*ГS)/(1-s11* ГS)(5)

Setting the magnitude of TOUT to 1 and solving for TIN gives an equation for a circle. This circle is called the source stability circle.

The center of the source stability circle is at:

CS =[ (s11-S22*) *Δ]*/[abs(s11)^{2} –abs(Δ)^{2} ](6)

and the radius of the circle is at:

RS = abs(s21*s12)/[(abs(s11)^{2})-abs(Δ)^{2} ](7)

Δ = s11*s22 – s21*s12(7.1)

Similarly the load stability circle is found by setting the magnitude of TIN to 1. Values of TOUT that satisfy this equation give the load stabilty circle.

**Interpretation of the stability circles**

Interpreting stability circles can be a difficult chore normally, but the following technique can be used to simplify the interpretation with the given conditions:

A. If the magnitude of TOUT > 1 oscillations may take place at the output.

B. If the magnitude of TOUT<1 then the output port will be stable.

C. If the magnitude of TOUT =1, then it is a borderline case.

Then we need to see if the region inside the stability circle represents a type a or type b termination. (Type a is unstable and type b is stable).

Choose 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. We need to make sure this is the case though.)

The above choice is to see if the magnitude of |s22|>1 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|>1 then 50 Ohm will be classified as a type a, or unfavorable termination.

Please refer to figure 1 that can clarify these remarks.

Figure 1 • Circumference of the stability circle represents borderline terminations.

Note that the circumference of the stability circle represents borderline terminations.

With a bit of practice one can become adept at identifying the stable and unstable regions.

In the above, the source stability circle was used for the analysis. It should be obvious that in a similar way 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 will cause oscillation or not. The unfavorable terminations are called type a and the favorable terminations are called type b. Borderline terminations are called type c. Using stability circles one can analyze a circuit like an amplifier for stable operation.

**About the Author**

Ain Rehman is founder and chief engineer at Signal Processing Group.

### May 2023

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