3D Smith chart constant quality factor semi-circles contours for positive and negative resistance circuits

The article proves first that the constant quality factor (Q) contours for passive circuits, while represented on a 2D Smith chart, form circle arcs on a coaxal circle family. Furthermore, these circle arcs represent semi-circles families in the north hemisphere while represented on a 3D Smith chart. On the contrary we show that the constant Q contours for active circuits with negative resistance form complementary circle arcs on the same family of coaxal circles in the exterior of the 2D Smith chart. Also, we find out that these constant Q contours represent complementary semi-circles in the south hemisphere while represented on the 3D Smith chart for negative resistance circuits. The constant Q - computer aided design (CAD) implementation of the Q semi-circles on the 3D Smith chart is then successfully used to evaluate the quality factor variations of newly fabricated Vanadium dioxide inductors first, directly from their reflection coefficient, as the temperature is increased from room temperature to 50 degrees Celsius ({\deg}C). Thus, a direct multi-parameter frequency dependent analysis is proposed including Q, inductance and reflection coefficient for inductors. Then, quality factor direct analysis is used for two tunnel diode small signal equivalent circuits analysis, allowing for the first time the Q and input impedance direct analysis on Smith chart representation of a circuit, including negative resistance

In this work we first prove that the constant Q curves represent circle arcs mapped on coaxal circle families [18] while providing to the best of our knowledge for the first time their equations: i.e. centre-radius-Q dependency.
Then, we prove that these circle arcs represent simple semicircles on the North hemisphere for all passive circuits while analysed on the 3D Smith chart computer aided design (CAD) tool [19][20][21]. After displaying the constant Q-arcs for negative resistance circuits, we determine that these are semi-circles in the south hemisphere. In order to prove the utility of our CAD implementation, we show that, while grounding the second port of newly fabricated Vanadium dioxide two port inductors [22], one may get the Q-frequency dependency directly from the S 11reflection coefficient parameter, thus avoiding classical 2D Qfrequency plots previously employed by us [22][23] or other authors [24][25], in these types of evaluations. The proposed visualization on the 3D Smith chart proves its effectiveness for passive circuits, especially when quality factors do not exceed big values [22][23][24][25], being particularly useful for Vanadium dioxide temperature variations studies on Q-where Q degrades as temperature increases, but potentially applicable in all inductors frequency dependent factor evaluations. Here we test the temperature dependence of Q for fabricated inductors with VO 2 by sweeping it from 25°C to 50°C-directly from the vector network analyser with the newly developed technique. Further we show the utility of the new CAD implementation for negative resistance circuits and we analyse the quality factor of various tunnel diodes, when negative resistance occurs, and the Smith chart cannot be used anymore.
On the other hand, the reflection coefficient of one port network (where R 1 is the port resistance, usually 50 Ω) can be defined as: where r and x denote the normalized resistance, respectively reactance: and and denote the real and the imaginary part of the reflection coefficient. Based on (1) and (4) one can easily obtain Q as (5).
If n≠ 0 is constant, we observe that the expression (5) represents radial lines in the normalized impedance (z plane). Fig. 1 (a) shows that for |n|=3 we obtain two lines as r and x are swept from -∞ to+∞. Fig. 1 (b) shows the family of the radial lines obtained for various values of n.

Proof:
We know from [18][19] that any transformation of form (6) represents an inversive transformation mapping always generalized circles (circles or infinite lines) into generalized circles.
By using the bipolar coordinates ( [27])), we obtain that the centre (C) and radius (rad) of this family of circles are: For n>0 we obtain simply circles arcs inside of Smith Chart as observed too, only in [11].
For n=0, we have a circle of infinity radius, i.e. the  r axis. We consider that the distance between two constant Q-circles is given by the distance between their intersections with  i axis (7), then the distance between two consecutives circles is given in Table I. Fig. 2 showing the Q=n>0 circles. Computing the distance between two consecutive circles, for negative resistance circuits (with r ≤ 0 implying n ≤ 0) we obtain the results in Table II. Displaying now the contours for n ≤ 0, we get the circle arcs in Fig.3. At the limit, for n=0, we have a circle of infinite radius, i.e. the r axis. Finally, Fig. 4 displays all the coaxal circles together for -∞≤n≤∞

C. Applications in reconfigurable inductors with Vanadium dioxide switches in on/off state
The quality factor of an inductor is most commonly defined as (11) -by grounding its second port [23][24][25][28][29][30]. It is usually evaluated until x changes sign and becomes negative. Respectively, we may notice that the imaginary part of the reflection coefficient of a one port network can be written as (12) and this becomes zero when x becomes 0. Based on this observation, the zeros of the imaginary part of S 11 are given by the zeros of x. Secondly, (11) is identical with (5) under r>0 and x>0 conditions, which are always fulfilled for a passive inductor before its self-resonant frequency (when it becomes capacitive).
Plotting the 1 port reflection coefficient on the Smith chart delivers the values of the quality factor directly, unlike the two port reflection coefficients representation on the Smith chart which is unrelated to it. Displaying the one-port reflection coefficient (grounding the second port), we can directly detect the quality factors and while using the 3D Smith chart implementation, visualize the extracted inductance and frequency dependency in a concomitant view. Considering our previous work [22], the inductance can be represented in the 3D space via a homothety over the 113 parameter using (13) where ( ) represents the normalized extracted inductance.
In [19], the proposed display could not allow for the simultaneous displaying of 3 ( ), Q and 113 ( ). Fig. 9 shows the extracted inductance and Q of the inductors reported in [22] and using VO 2 as switching element using a classical approach. The results are measured at 20 °C (off) and at 100 °C (on). Let us consider now the representation proposed in the previous section. Fig. 10 (a) displays the S 113D of the inductors in on/off states, with the second port grounded, for the same frequency range as in Fig.10. One may directly read the value of the quality factor in each point of them using the CAD implementation proposed, while plotting the convenient constant Q semi circles. From the chosen rendering it renders clear that these values are between 7-10 for a wide frequency band, while in the case of the off state inductors, these values are decreasing towards 0. For the on state it can be seen clearly that these values do not decrease below 4 for the frequency range displayed. Fig 10 (b) adds the frequency representation proposed in [22] over the previous render, offering their dynamical view as frequency is swept. Fig 10 (c) shows the extracted inductance displayed over them, using a centre projection for each point. The results picture the increased value of the inductance in the off state as in comparison to the one in the on state, while both showing frequency linearity.

D. Applications in reconfigurable inductors with Vanadium dioxide switches in temperature sweeping
The proposed CAD approach is particularly useful in the case of VO 2 inductors, since these need to be tested at various temperatures, thus a fast detection of a failure, directly from the measured S parameters, would allow for skipping of the testing for the same inductor at a different temperature. Usually on a wafer, one which has in the tenths-hundreds number denomination of inductors, the proposed procedure represents a fast tool for detecting failures in the desired expected Q. Fig. 11 summarizes the proposed procedure. The S parameters can be exported as Touchstone two-port files and imported directly into the 3D Smith chart application. The application is developed using the Java programming language and for the 3D rendering and interaction with the Riemann sphere the Open Graphics Library (OpenGL) Application Programming Interface (API) is employed. The user can interact with the 3D space in which the Riemann sphere is rendered to manipulate the view of the 3D space and to adjust the parameters of the displayed circuits, as necessary.

FIGURE 11. Proposed CAD Q evaluation, for a wafer full of inductors requiring Q>Qmin on a specific frequency band.
In order to test the temperature sensitivity of the Vanadium Dioxide reconfigurable inductors, let us examine a new inductor, based on the design methodology in [19], but with longer switch length (minimizing losses in off state and increasing them in the on state). The measurement setup is shown in Fig. 12 (a)-and it includes a thermo chuck, whose temperature is increased up to 50 °C . Fig. 12 (b) shows the layout of the inductor, the same as in [22], in this case however with a 2 µm switch length instead of 600 nm as in [22].

FIGURE 12. (a) Setup for heating including Vector network analyser and a thermo heater (b) Layout of the inductors [18].
Let us now verify the Q frequency dependency while analysing the inductors in the 4-8 GHz and then check the minimum value in this band. The extracted Q and inductance are displayed in Fig. 13 (a) and (b) on a 2D display. The values of the Q decrease slightly up to 6, while the values of the extracted inductance stay stable with temperature increase -Fig 13(b). Using the new proposed CAD methodology, we can see in Fig.  14 (a), the 113 ( ). It can be clearly seen how the 113 ( ) does not decrease below 6 for none of the analysed temperatures. Exploiting the frequency dependency display, the dynamics in Fig. 14 (b) can be observed. Fig 14 (c) shows the extracted inductance in 3D -displaying its extremely stable values as temperature increases up to 50 °C .

A. Frequency dependent example
Let us consider the circuit given in Fig. 15, which is the equivalent circuit of a resonant tunnelling diode [30]. These diodes can be used as local oscillators in microwave and millimetre wave frequencies. Their small signal equivalent circuit is presented in Figure 13. Assuming now the values given in [30] for the negative resistance: R=-120 Ω, shunt capacitance C=0.7pF, while the series resistance R S =3.5 Ω and the series inductance L=3.5 nH, let us analyse the frequency dependency of its input impedance from its Q in between 5 GHz and 11 GHz. The quality factor of a tunnelling diode can be negative [30], while keeping the same classical definition. The evolution of the quality factor from values of -2 towards infinity can be seen in Fig. 16 (a)-(c) on a 3D Smith chart rendered with the constant normalized conductance (g) and susceptance (b) circles. The values of the normalized input admittance can be checked in each moment along the g and b circles also. While maintaining a negative input resistance, the device behaves capacitive with up to 8.3 GHz when Q=0, then from 8.3 GHz the device behaves inductive. Its Q becomes infinite in absolute value at 11 GHz, when its input resistance starts changing sign. All these evolutions can be easily checked and computed with the 3D Smith chart without the need of further calculations.

B. Single frequency point analysis
Let us now analyse the tunnel diode small signal equivalent circuit given in Fig. 17   We can verify with ease the correctness of the approach by computing the input impedance mathematically. However, the 3D Smith chart implementations allowed us to follow step by step its development in different nodes, with no need of arithmetical manipulations, and, simply by changing the rendering, we were able to read the exact values stepwise. The appendix shows how the implementation can be used for a bandstop filter too in order to extract the equivalent circuit directly from the S 11 parameters intersections with constant Q circles.

IV. Conclusions
In this article we have proved, for the first time, that the constant Q contours (nodal quality factors) (1) form circle arcs on a family of coaxal circles on the Smith chart. We provided, for the first time, (to the best of our knowledge) by means of bipolar equations, their explicit equations in terms of radius, circles centre-Q value relationship, by solving their implicit equations. Further, we have proved that, while evaluated on the 3D Smith chart, the constant Q contours represent semicircles in the north hemisphere for positive resistance circuits, respectively semi-circles on the south hemisphere for negative resistance circuits, all cantered in 3D Smith chart centre. This simple, compact, and practical circle shaped property has enabled us to use these Q semi-circles directly, in the reflection coefficients plane, for both passive and active circuits, for the direct Q evaluations from measured S parameters. In the case of Vanadium Dioxide reconfigurable inductors temperature sensitivity analysis: the proposed methodology allowed us the multi-parameter extraction (inductance, Q, reflection coefficient) (Fig. 14) directly from the measured devices, simplifying the extraction procedures-and allowing us a fast evaluation of their performances directly from the measuring setup. In the case of negative resistance circuits, the proposed Q visualization extended the use of constant Q contours for circuits with negative resistance too, impossible on a 2D Smith chart, exemplified here on tunnel diodes small signal equivalent circuits (Figs. [15][16][17][18].

Appendix
Let us consider the parallel R, L, C (which can be the equivalent circuit of a bandstop filter) circuit present in Fig.  19. Supposing one would need to determine the values of the elements R, L, C in Fig. 19 that would fit a measured S 11 of a bandstop filter, whose equivalent circuit is completely determined by Fig. 19.: one can use the new frequency dependent Q implementation. By representing the S 11 with the second port grounded we get Fig. 20. The input admittance of the bandstop filter (Fig.19) can be computed with (14). At resonance the imaginary part is zero and (15) is fulfilled where ω 0 is the angular resonance frequency, while f 0 the resonance frequency. The input admittance at resonance 110 becomes (16) and Q defined in (13) becomes (17).
In Fig. 20 we can easily determine f 0 and R: we check when S 11 crosses the Q=0 circle and read the values for the frequency and for the normalized resistance. We get f 0 =35.6 GHz and r=0.8=R/50, thus R=400Ω. Now let us compute (11) in a general form for the circuit given in Fig. 19: = − Imposing now Q=1 we get: where ω 1 =2*ᴫ f 1 is the angular frequency for which Q=1 (and f 1 the frequency for which Q=1) Getting back in Fig. 20 and checking where S 11 crosses the Q=1 circle we get f 1 = 34.6 GHz. Now getting back to (15) and ( Solving using Mathematica [31] numerically (20) and (21) we get one of the solutions: C=0.2 pF and L=0.1 nH. This enabled us to extract the equivalent circuit without any fitting procedure.