# Invariance Property of theμ Stability Test with respect to the Characteristic Impedance

Authors: Amr A. Ibrahim; Ali M. Darwish; H. Alfred Hung
Journal: International Journal of Microwave Science and Technology
 (2012)

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## Abstract

Stability of electrical amplifiers is of crucial importance. Among the popular stability tests is theμ-test which has many advantages over other tests like the K-Δ test. However, the value of μ parameter is dependent on the input/output terminal characteristic impedance Zo used and this raises the concern that the predictions of the test are dependent on the choice of Zo. This paper proves that the conclusions of the μ-test regarding stability/instability remain invariant with Zo. This proof is necessary for gaining confidence in the results of the μ-test and should benefit circuit designers. Similar proofs should be extended to all other stability tests for additional insights into their validity under different circuit termination.


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## Body

## 1. Introduction

The stability of a microwave amplifier, or equivalently its resistance to oscillations, is one of the main concerns when designing active microwave circuits. This classical problem of stability of linear two-port network has been intensively investigated in the technical literature over the last fifty years [1–7]. Several sets of conditions were proposed to check stability. All these sets of conditions, which were derived by steady state analysis, have been shown to be equivalent [1–6]. These tests determine whether it is possible to find a set of passive terminations that will cause the terminated two-port network to have unstable poles given that the network parameters do not have right-hand poles (Rollett’s proviso) [3]. If there are no such terminations, the two-port network is unconditionally stable. Among these tests, the most popular are the K-Δ test [3] and the μ-test [5]. These two tests are widely used in many CAD programs and textbooks discussing the design of both amplifiers as well as oscillators [8, 9]. The K-Δ relies on two conditions to assess the stability of microwave amplifiers. On the other hand, the μ-test relies only on one parameter to assess the stability of the amplifier circuit. Usually, the μ-test is calculated from the S-parameters of the circuits under test. However, in circuits composed of lumped components, the choice of Zo is ambiguous and hence the μ-test value may change based on the value of Zo. In this paper, it is proven that the μ-test result does not depend on the choice Zo, as long as Zo takes real positive values. Meaning that if the circuit is stable for a certain Zo then it will be also stable for any other Zo (i.e., μ value will always be greater than one), and vice versa. Moreover, it is shown that the worst case condition (μ value  ~ 1) will occur, when Zo tends to zero or infinity. One of the benefits of this analysis is that the μ-test can also be applied to circuits composed of lumped components with arbitrary Zo.Some stability tests that vary withZo can yield misleading results. For example, a common oscillation test is to evaluate the round trip gain |ΓinΓS|. The test requires that |ΓinΓS| > 1 for oscillations [9], where Γin  is the input reflection coefficient of the oscillator circuit and ΓS is the source reflection coefficient as shown in Figure 1. Suppose we have a negative resistance oscillator with input impedance of −25 + j10 Ohm and source impedance of 50 − j10 Ohm. Then |ΓinΓS| may be plotted, as a function of Zo as in Figure 2. It is clear that |ΓinΓS|> 1 for small values of Zo, and |ΓinΓS| < 1 otherwise. The variance of |ΓinΓS| with Zo makes the test’s conclusions questionable. Thus, the invariance of a stability test with Zo is crucial and cannot be assumed without a proof. This paper addresses this concern for the popular μ-test.Figure 1
The amplifier circuit.Figure 2
The round trip gain|ΓinΓS| as a function of Zo used in the evaluation of Γin and ΓS.


## 2. Mathematical Treatment

In this section, it is proven that theμ-test does not depend on the choice of Zo of terminal ports. It is convenient to start the proof from the impedance domain (Z) since it does not depend on the choice of Zo. For an amplifier circuit to be stable (according to the μ-test), the condition(1)|Γin|<1
must be satisfied for any passive load termination (ZL) [5]. The μ-test defines a parameter, μ, such that(2)μ=1|Γmax⁡|,
where |Γmax|  is the maximum reflection coefficient (for any possible load). If |Γmax| < 1, then the network is stable, μ is greater than 1, and no negative impedances can be produced, and vice versa.The input impedance of the amplifier circuit is given by(3)Zin=Z11-Z12Z21ZL+Z22=Z11ZL+ΔZZL+Z22,
where ΔZ is the determinant of the impedance matrix.Equation (3) has the form of a bilinear transformation. The bilinear transformation is a complex mapping technique and has the property of mapping circles in one domain into circles in another domain with lines as the limiting case. Under this transformation, the imaginary axis in the ZL domain (Re(ZL)=0) is mapped into a circle in the Zin domain as shown in Figure 3 with centre Cin and radius Rin given by(4)Cin=2Z11Re(Z22)-Z12Z212Re(Z22),Rin=|Z12Z212Re(Z22)|.
For the amplifier to be absolutely stable, the mapped circle in the Zin domain must lie completely in the right half plane or(5)Re(Cin)>Rin.
Furthermore, the greater the inequality Re(Cin)>Rin, the higher the stability. In the previous equation, it was assumed that the region (Re(ZL)>0) is mapped to the interior of the circle in the Zin domain which will happen given that the condition (Re(Z22)>0) is satisfied.Figure 3
Mapping from the (ZL) impedance plane into Zin plane followed by mapping into Γin plane.Now assuming that (5) is satisfied and the amplifier is absolutely stable in the Zin domain, one can map this circle to the Γin domain using the bilinear transformation(6)Zin=ZoΓin+Zo-Γin+1.
For a circle having a centre (Cin) and radius (Rin) in the Zin domain, the mapped circle in the Γin domain will have a center (CΓ) and radius (RΓ) given by(7)CΓ=|Cin|2-Rin2-Zo2+j2ZoIm⁡(Cin)|Cin|2-Rin2+Zo2+2ZoRe(Cin),RΓ=2RinZo|Cin|2-Rin2+Zo2+2ZoRe(Cin).
Since the interior of the circle in the Zin domain is mapped to the interior of the circle in the Γin domain, (1) can be rewritten as(8)|CΓ|+RΓ<1.
To check this condition, define(9)Cin=a+jb.
From (5), one can write(10)a2=Rin2+δ2,
where δ2>0 is a parameter that can indicate higher stability. Substituting with (10) in (7), one can write(11)|CΓ|+RΓ=(b2+δ2+Zo2)2-(2δZo)2+2RinZo(b2+δ2+Zo2)+2aZo.
In this case, the μ-test is simply the inverse of the previous expression. Thus, based on (2),(12)μ=1|CΓ|+RΓ=(b2+δ2+Zo2)+2aZo(b2+δ2+Zo2)2-(2δZo)2+2RinZo.
The value of the previous expression is always greater than 1 given that (5) is satisfied irrespective of the value of  Zo. That is because the first term in the numerator is greater than the first term in the denominator, and the same is true for the second term in the numerator and denominator. Also, at the stability edge in the Zin domain where δ→0, the value of μ will approach 1. It should be noted that the value of μ will also approach 1 in the limiting case as Zo approaches zero or infinity. The previous steps of the proof are summarized in Figure 3.


## 3. Verification of Theory Using Two Examples

Two examples are given to validate the previous results. The first example is for an unconditionally stable amplifier and the second one is for a conditionally stable amplifier. First, consider the parameters of an amplifier circuit in theZ-domain at certain frequency (fo) as(13)Zamp=[36-j241.5-j4-15-j58520-j28],
which corresponds to the following S-parameters (in a 50-Ohms system):(14)Samp=[0.1∠-10.5°0.05∠-46.5°7∠-68.5°0.15∠-112.7°].
For this amplifier, the μ-test, given by (15), is calculated for different values of Zo and the result is shown in Figure 4(a),(15)μ=1-|S11|2|S22-ΔSS11¯|+|S12S21|,
where ΔS is determinant of the S-parameters matrix, and the upper bar indicates conjugation. It is clear from Figure 4(a) that the value of μ is always greater than 1 and it approaches 1 as Zo tends to zero or infinity. The same conclusion can be obtained by observing the mapped circles of the region (Re(ZL)>0) in the Γin domain as a function of Zo (Figure 5(a)), where all the mapped circles are located inside the unity circle (|Γin|=1). Now consider a second case where the amplifier Z-parameters are changed to(16)Zamp=[1.8-j310.87-j8.4-115.6-j373.3-9.4-j30.6]
which corresponds to the following S-parameters (in a 50-Ohms system):(17)Samp=[0.1∠-11.3°0.15∠-45.9°7∠-68.8°0.15∠-114.7°].
The μ-test as a function of Zo is shown in Figure 4(b). As the figure indicates, the value of μ is less than 1, irrespective of Zo. Also, Figure 5(b) shows that the mapped circles of the region (Re(ZL)>0) in the Γin domain intersect with the unity circle at all values of Zo. The previous two examples illustrate the application of the theory.μ-test as a function of Zo for (a) unconditionally stable amplifier and (b) conditionally stable amplifier.
(a)(b)Mapped circles in theΓin domain for (a) unconditionally stable amplifier and (b) conditionally stable amplifier.
(a)(b)


## 4. Conclusion

One of the most popular tests for stability is theμ-test due to its simplicity (one parameter, instead of two) and its ability to indicate relative stability. This paper proves that the conclusions reached through the μ-test are invariant with characteristic impedance Zo choices. This proof is necessary for using the μ-test with confidence. A stability test which does not meet this condition may lead to inaccurate conclusions. Two examples illustrating the application of the theory were provided.



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*Source: 865625-2012-05-08.xml*