Invariance Property of the 𝜇 Stability Test with respect to the Characteristic Impedance

Abstract

Stability of electrical amplifiers is of crucial importance. Among the popular stability tests is the μ-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 𝑍_𝑜 used and this raises the concern that the predictions of the test are dependent on the choice of 𝑍_𝑜. This paper proves that the conclusions of the μ-test regarding stability/instability remain invariant with 𝑍_𝑜. 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.

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 𝑍_𝑜 is ambiguous and hence the μ-test value may change based on the value of 𝑍_𝑜. In this paper, it is proven that the μ-test result does not depend on the choice 𝑍_𝑜, as long as 𝑍_𝑜 takes real positive values. Meaning that if the circuit is stable for a certain 𝑍_𝑜 then it will be also stable for any other 𝑍_𝑜 (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 𝑍_𝑜 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 𝑍_𝑜.

Some stability tests that vary with 𝑍_𝑜 can yield misleading results. For example, a common oscillation test is to evaluate the round trip gain |Γ_inΓ_𝑆|. The test requires that |Γ_inΓ_𝑆| > 1 for oscillations [9], where Γ_in is the input reflection coefficient of the oscillator circuit and Γ_𝑆 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 |Γ_inΓ_𝑆| may be plotted, as a function of 𝑍_𝑜 as in Figure 2. It is clear that |Γ_inΓ_𝑆|> 1 for small values of 𝑍_𝑜, and |Γ_inΓ_𝑆| < 1 otherwise. The variance of |Γ_inΓ_𝑆| with 𝑍_𝑜 makes the test’s conclusions questionable. Thus, the invariance of a stability test with 𝑍_𝑜 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 the μ-test does not depend on the choice of 𝑍_𝑜 of terminal ports. It is convenient to start the proof from the impedance domain (𝑍) since it does not depend on the choice of 𝑍_𝑜. For an amplifier circuit to be stable (according to the μ-test), the condition||Γ_in||<1(1) must be satisfied for any passive load termination (𝑍_𝐿) [5]. The μ-test defines a parameter, μ, such that𝜇=1||Γ_max||,(2) where |Γ_max| is the maximum reflection coefficient (for any possible load). If |Γ_max| < 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 by𝑍_in=𝑍₁₁−𝑍¹²𝑍₂₁𝑍_𝐿+𝑍₂₂=𝑍¹¹𝑍_𝐿+Δ_𝑍𝑍_𝐿+𝑍₂₂,(3) where Δ_𝑍 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 𝑍_𝐿 domain (Re(𝑍_𝐿)=0) is mapped into a circle in the 𝑍_in domain as shown in Figure 3 with centre 𝐶_in and radius 𝑅_in given by𝐶_in=2𝑍¹¹Re𝑍₂₂−𝑍₁₂𝑍₂₁𝑅_in2Re𝑍₂₂,=||||𝑍¹²𝑍₂₁2Re𝑍₂₂||||.(4) For the amplifier to be absolutely stable, the mapped circle in the 𝑍_in domain must lie completely in the right half plane orRe𝐶_in>𝑅_in.(5) Furthermore, the greater the inequality Re(𝐶_in)>𝑅_in, the higher the stability. In the previous equation, it was assumed that the region (Re(𝑍_𝐿)>0) is mapped to the interior of the circle in the 𝑍_in 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 𝑍_in domain, one can map this circle to the Γ_in domain using the bilinear transformation𝑍_in=𝑍^𝑜Γ_in+𝑍_𝑜−Γ_in+1.(6) For a circle having a centre (𝐶_in) and radius (𝑅_in) in the 𝑍_in domain, the mapped circle in the Γ_in domain will have a center (𝐶_Γ) and radius (𝑅_Γ) given by𝐶_Γ=||𝐶ⁱⁿ||²−𝑅²_in−𝑍^2𝑜+𝑗2𝑍_𝑜Im𝐶_in𝑅_Γ||𝐶_in||²−𝑅²_in+𝑍^2𝑜+2𝑍_𝑜Re𝐶_in,=2𝑅ⁱⁿ𝑍_𝑜||𝐶_in||²−𝑅²_in+𝑍^2𝑜+2𝑍_𝑜Re𝐶_in.(7) Since the interior of the circle in the 𝑍_in domain is mapped to the interior of the circle in the Γ_in domain, (1) can be rewritten as||𝐶_Γ||+𝑅_Γ<1.(8) To check this condition, define𝐶_in=𝑎+𝑗𝑏.(9) From (5), one can write𝑎²=𝑅²_in+𝛿²,(10) where 𝛿²>0 is a parameter that can indicate higher stability. Substituting with (10) in (7), one can write||𝐶_Γ||+𝑅_Γ=𝑏²+𝛿²+𝑍^2𝑜²−2𝛿𝑍_𝑜²+2𝑅_in𝑍_𝑜𝑏²+𝛿²+𝑍^2𝑜+2𝑎𝑍_𝑜.(11) In this case, the μ-test is simply the inverse of the previous expression. Thus, based on (2),𝜇=1||𝐶_Γ||+𝑅_Γ=𝑏²+𝛿²+𝑍^2𝑜+2𝑎𝑍_𝑜𝑏²+𝛿²+𝑍^2𝑜²−2𝛿𝑍_𝑜²+2𝑅_in𝑍_𝑜.(12) The value of the previous expression is always greater than 1 given that (5) is satisfied irrespective of the value of 𝑍_𝑜. 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 𝑍_in 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 𝑍_𝑜 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 the 𝑍-domain at certain frequency (𝑓_𝑜) as𝑍_amp=⎡⎢⎢⎣−3165−−𝑗𝑗25485210.5−−𝑗𝑗248⎤⎥⎥⎦,(13) which corresponds to the following 𝑆-parameters (in a 50-Ohms system):𝑆_amp=⎡⎢⎢⎣0.1∠−10.5^∘0.05∠−46.5^∘7∠−68.5^∘0.15∠−112.7^∘⎤⎥⎥⎦.(14) For this amplifier, the μ-test, given by (15), is calculated for different values of 𝑍_𝑜 and the result is shown in Figure 4(a),𝜇=1−||𝑆¹¹||²|||𝑆₂₂−Δ_𝑆𝑆₁₁|||+||𝑆₁₂𝑆₂₁||,(15) where Δ_𝑆 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 𝑍_𝑜 tends to zero or infinity. The same conclusion can be obtained by observing the mapped circles of the region (Re(𝑍_𝐿)>0) in the Γ_in domain as a function of 𝑍_𝑜 (Figure 5(a)), where all the mapped circles are located inside the unity circle (|Γ_in|=1). Now consider a second case where the amplifier 𝑍-parameters are changed to𝑍_amp=⎡⎢⎢⎣−1115..86−−𝑗𝑗33173.3−09.8.47−−𝑗𝑗380.4.6⎤⎥⎥⎦(16) which corresponds to the following 𝑆-parameters (in a 50-Ohms system):𝑆_amp=⎡⎢⎢⎣0.1∠−11.3^∘0.15∠−45.9^∘7∠−68.8^∘0.15∠−114.7^∘⎤⎥⎥⎦.(17) The μ-test as a function of 𝑍_𝑜 is shown in Figure 4(b). As the figure indicates, the value of μ is less than 1, irrespective of 𝑍_𝑜. Also, Figure 5(b) shows that the mapped circles of the region (Re(𝑍_𝐿)>0) in the Γ_in domain intersect with the unity circle at all values of 𝑍_𝑜. The previous two examples illustrate the application of the theory.

(a)

(b)

Figure 4 

μ-test as a function of 𝑍𝑜 for (a) unconditionally stable amplifier and (b) conditionally stable amplifier.

(a)

(b)

Figure 5 

Mapped circles in the Γin domain for (a) unconditionally stable amplifier and (b) conditionally stable amplifier.

4. Conclusion

One of the most popular tests for stability is the μ-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 𝑍_𝑜 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.