Direct connection to separated tuners

Here we analyze the approach used in [Reference Köther and Berben2] and [Reference Liu and Tsironis3], where tuners are connected directly to the DUT. For this, we assume that the connection network ${\bi {S}^{\prime}}_c$ is a direct connection from port 1 to port 3 and port 2 to port 4. The connection network is perfectly matched, reciprocal, lossless, and has zero length. The results of these simplifications are shown in (2).

(2)$$\eqalign{{{S}^{\prime}}_{11} = \,& {{S}^{\prime}}_{22} = {{S}^{\prime}}_{33} = {{S}^{\prime}}_{44} = 0 \cr {{S}^{\prime}}_{12} = \,& {{S}^{\prime}}_{21} = {{S}^{\prime}}_{34} = {{S}^{\prime}}_{43} = 0 \cr {{S}^{\prime}}_{14} = \,& {{S}^{\prime}}_{41} = {{S}^{\prime}}_{23} = {{S}^{\prime}}_{32} = 0 \cr {{S}^{\prime}}_{13} = \,& {{S}^{\prime}}_{31} = {{S}^{\prime}}_{24} = {{S}^{\prime}}_{42} = 1.} $$

Using S of the concatenated network and generalized mixed-mode pseudo scattering S-parameters, we get the structure depicted in Fig. 2. Applying (2) we arrive at the result for ${\bi {S}^{\prime}}_{mix}$ shown in (3) and (4) [Reference Dobrowolski11].

(3)$${\bi {S}^{\prime}}_{mix} = \displaystyle{1 \over 2}\cdot \left[{\matrix{ {S_{\rm \Delta } + S_{\rm \Sigma }} & {S_{\rm \Delta }-S_{\rm \Sigma }} \cr {S_{\rm \Delta }-S_{\rm \Sigma }} & {S_{\rm \Delta } + S_{\rm \Sigma }} \cr } } \right] = \left[{\matrix{ {{{S}^{\prime}}_{dd}} & {{{S}^{\prime}}_{dc}} \cr {{{S}^{\prime}}_{cd}} & {{{S}^{\prime}}_{cc}} \cr } } \right]\comma \;$$

(4)$${\rm {\Gamma }^{\prime}}_{diff} = {\rm \;}{\rm {\Gamma }^{\prime}}_{com} = {S}^{\prime}_{dd} = {S}^{\prime}_{cc} = \displaystyle{{S_{\rm \Delta } + S_{\rm \Sigma }} \over 2}.$$

Fig. 2. Block diagram of a simplified load network for a differential device using mixed-mode port representation.

The matrix entries ${S}^{\prime}_{dd}$ and ${S}^{\prime}_{cc}$ of ${\bi {S}^{\prime}}_{mix}$ correspond to the differential and common mode reflection coefficients the device sees. ${S}^{\prime}_{dc}$ and ${S}^{\prime}_{cd}$ show the mode conversion that is introduced by the load. Because ${S}^{\prime}_{dd}$ and ${S}^{\prime}_{cc}$ are equal, the common and differential mode loads are always the same. Additionally, mode conversion is introduced by the load if S _Δ and S _Σ are not equal.

Connection via a 180° hybrid

In this sub-section, we assumed that the connection network ${\bi {S}^{\prime \prime \prime}}_c$ is an ideal 180° hybrid as defined in (5). The results from (6) to (8) show that differential and common mode reflection coefficients are independent of each other, and reduce to the reflection coefficients of the tuners. Mode conversion is zero for an ideal 180° hybrid.

(5)$$\eqalign{{{S}^{\prime \prime}}_{11} = \,& {{S}^{\prime \prime}}_{22} = {{S}^{\prime \prime}}_{33} = {{S}^{\prime \prime}}_{44} = 0 \cr {{S}^{\prime \prime}}_{12} = \,& {{S}^{\prime \prime}}_{21} = {{S}^{\prime \prime}}_{34} = {{S}^{\prime \prime}}_{43} = 0 \cr {{S}^{\prime \prime}}_{13} = \,& {{S}^{\prime \prime}}_{31} = {{S}^{\prime \prime}}_{14} = {{S}^{\prime \prime}}_{41} = {{S}^{\prime \prime}}_{24} = {{S}^{\prime \prime}}_{42} = \displaystyle{1 \over {\sqrt 2 }} \cr {{S}^{\prime \prime}}_{23} = \,& {{S}^{\prime \prime}}_{32} = {-}\displaystyle{1 \over {\sqrt 2 }}\comma \;} $$

(6)$${\bi {S}^{\prime \prime}}_{mix} = \left[{\matrix{ {S_{\rm \Delta }} & 0 \cr 0 & {S_{\rm \Sigma }} \cr } } \right] = \left[{\matrix{ {{{S}^{\prime \prime}}_{dd}} & {{{S}^{\prime \prime}}_{dc}} \cr {{{S}^{\prime \prime}}_{& cd}} & {{{S}^{\prime \prime}}_{cc}} \cr } } \right]\comma \;$$

(7)$${\rm {\Gamma }^{\prime \prime}}_{diff} = {S}^{\prime \prime}_{dd} = S_{\rm \Delta }\comma \;$$

(8)$${\rm {\Gamma }^{\prime \prime}}_{com} = {S}^{\prime \prime}_{cc} = S_{\rm \Sigma }.$$

Direct connection to connected tuners

A third approach of mixed-mode load-pull is to connect the outputs of the tuners to each other while the inputs are connected directly to the differential output of the DUT. This approach was presented for the first time in [Reference Veit, Gadringer and Leitgeb1]. The resulting load matrix ${\bi {S}^{\prime}}_L$ provides non-zero transmission coefficients S _T as highlighted in (9). Performing the same calculations as before using the assumptions from (2) and (9) results in the mixed-mode network ${\bi {S}^{\prime \prime \prime}}_{mix}$ summarized in (10) to (12). Compared to the results in section “Direct connection to separated tuners”, now different differential and common mode reflection coefficients can be realized. Yet, they cannot be selected independently from each other. The degree in which they are depending on each other can be tuned by the transmission coefficient S _T. Hence, achieving a specific combination of differential and common mode loads is more complicated compared to the load-pull approach as discussed in section “Connection via a 180° hybrid”, but no 180° hybrid is needed.

(9)$${\bi {S}^{\prime}}_L = \left[{\matrix{ {S_{\rm \Delta }} & {S_T} \cr {S_T} & {S_{\rm \Sigma }} \cr } } \right]\comma \;$$

(10)$${\bi {S}^{\prime \prime \prime}}_{mix} = \displaystyle{1 \over 2}\left[{\matrix{ {S_{\rm \Delta } + S_{\rm \Sigma }-2\cdot S_T} \hfill & {S_{\rm \Delta }-S_{\rm \Sigma }} \hfill \cr {S_{\rm \Delta }-S_{\rm \Sigma }} \hfill & {S_{\rm \Delta } + S_{\rm \Sigma } + 2\cdot S_T} \hfill \cr } } \right]\comma \;$$

(11)$${\rm {\Gamma }^{\prime \prime \prime}}_{diff} = \displaystyle{{S_{\rm \Delta } + S_{\rm \Sigma }} \over 2}-S_T\comma \;$$

(12)$${\rm {\Gamma }^{\prime \prime \prime}}_{com} = \displaystyle{{S_{\rm \Delta } + S_{\rm \Sigma }} \over 2} + S_T.$$

Impact of insertion and return loss

The first parameters we discuss are insertion loss (IL) and return loss (RL). By the magenta colored arrows in Fig. 4, the impact of insertion loss is visualized. As shown in (7), for an ideal 180° hybrid, the differential and common mode impedances seen by the DUT (S _dd, S _cc) are equal to the impedances set at the corresponding tuners (S _Δ, S _Σ). Insertion loss reduces the magnitude of the set reflection coefficients S _Δ and S _Σ, thereby limiting the available tuning range. The light blue colored arrows in Fig. 4 demonstrate the influence of symmetric return loss at the ports facing the DUT. In this context, symmetric means that the return losses at port 1 and port 2 have the same magnitude and phase (S ₁₁ = S ₂₂). These return losses introduce additional reflections, which add to the reflections produced by the tuners, causing a shift of S _dd and S _cc which depends on the magnitude and phase of the return losses. As long as S ₁₁ and S ₂₂ are equal, no mode conversion occurs, and the impedance shift can be compensated by changing the values of S _Δ and S _Σ accordingly. The same behavior is caused by return losses at the ports facing the tuners. This again can be compensated by changing the tuner settings of each individual tuner.

Fig. 4. Impact of non-symmetric return loss and poor isolation between the ports facing the DUT. S _dd,IL and S _dd,IL+RL (dashed red line) are the differential mode impedances seen by the DUT including the impact of insertion loss (IL) and non-symmetric return loss (RL), respectively. Similarly, S _cc,IL and S _cc,IL+RL (blue square) is the common mode impedance seen by the DUT including the impact of IL and non-symmetric RL, respectively. $S_{dd\comma IL + RL + ISO_{DUT}}$ (solid red line) and $S_{cc\comma IL + RL + ISO_{DUT}}$ (blue circle) are the impedances seen by the DUT including the impact of IL, RL, and finite isolation between the DUT ports (ISO_DUT). S _dc,IL (green cross) and S _dc,IL+RL (green square) are the mode conversion seen by the DUT considering IL and non-symmetric RL. Note that mode conversion is plotted in the overlying polar plot as it cannot be mapped to an impedance value.

Figure 5 shows the impact of non-symmetric return loss (S ₁₁ = −S ₂₂), which means the return losses at port 1 and port 2 are 180° out of phase. This influence is depicted by the magenta colored arrow. A 180° phase difference of the return losses causes a differential mode signal to be reflected as a common mode signal and vice versa. This causes mode conversion but does not directly influence S _dd and S _cc. The general case where S ₁₁ and S ₂₂ have different magnitude and phase causes a mixture of mode conversion and shift of S _dd and S _cc. Different to the shift of S _dd and S _cc, this type of mode conversion cannot be influenced by the tuners. Therefore, we refer to this as load-independent mode conversion.

Fig. 5. Impact of poor isolation between the load ports 3 and 4 of the 180° hybrid. S _dd,IL, S _cc,IL, and S _dc,IL are the impedances and mode conversion seen by the DUT including the impact of insertion loss. $S_{dd\comma IL + ISO_{Load}}$, $S_{cc\comma IL + ISO_{Load}}$, and $S_{dc\comma IL + ISO_{Load}}$ are the impedances and mode conversion seen by the DUT considering insertion loss and isolation. As mentioned in the first paragraph of section “Impact of non-ideal 180° hybrids”, the setting for the common mode load tuner is kept constant. But in this case, the introduced mode conversion causes $S_{cc\comma IL + ISO_{Load}}$ to become a circle similar to the swept S _dd.

Impact of finite isolation

This sub section deals with the impact of finite isolation between the isolated ports of a 180° hybrid. At first, the impact of poor isolation between the ports facing the DUT is marked by the green arrows in Fig. 5. For S _cc, poor isolation causes the same shift as a symmetric return loss. This is because the incoming signals which make up the common mode signal are equal; therefore, a reciprocal transmission from port 1 to port 2 cannot be distinguished from a symmetric return loss with same phase and magnitude. The only parameter changing for S _dd is the additional 180° phase shift introduced because the incoming signals at port 1 and port 2 are 180° out of phase. This causes the green arrows in Fig. 5 to point in opposite directions. This shift of S _dd and S _cc can be compensated by the tuners in the same way as symmetric return loss. Beside this, there is also the impact of finite isolation between the ports facing the tuners. The influence of this is shown in Fig. 6. Because of this imperfection, reflections of one tuner carry over to the other tuner, causing a coupling between the two tuners. On the one hand, this causes mode conversion which depends on the tuner settings, later referred to as load-dependent mode conversion. On the other hand, it leads to differential and common mode impedances (S _dd, S _cc) which now depend on both tuners (S _Δ, S _Σ). In the example shown in Fig. 6, this causes S _cc to become a circle in the Smith chart if the phase of S _Δ is swept from 0 to 2π. In order to compensate for this behavior, one would need to tune both tuners jointly, thereby losing one of the key benefits of using a 180° hybrid. Please note that especially in the case of isolation, we chose unrealistically poor values to produce readable plots. As shown in Table 1, isolation values are much better than most other imperfections and can often be neglected.

Fig. 6. Comparison of simulated and calculated impact of magnitude and phase imbalance. S _dd and S _cc show the differential and common mode impedances seen by the DUT. The appendixes “sim.” or “calc.” indicate the simulated points using Monte Carlo simulation or the calculated area using the simplified formulas (13) and (14). S _dc shows the mode conversion due to magnitude and phase imbalance using Monte Carlo simulation only.

Impact of magnitude and phase imbalance

As last topic, we discuss the impact of magnitude and phase imbalance between the transmission parameters of a 180° hybrid. A hybrid has four transmission parameters and in general each of them may have a different magnitude and phase error relative to the ideal value. This makes it impossible to pick a single set of errors representative for all possible combinations and thus forcing us to rely on Monte Carlo simulations. For the simulations, we assumed that the magnitude errors and the phase errors are equally distributed between the given error-bounds mentioned in Table 1. This assumption is valid because we were only interested in finding an upper error bound and also assumed that the specifications are not exceeded. For this investigation, we chose a fixed S _Δ of approximately −2 dB and swept the magnitude of S _Σ from −15 to 0 dB. The results of this simulation using 5000 different combinations are presented in Fig. 7. As you can see, magnitude and phase imbalance causes mode conversion and a detuning of the set impedances. What you cannot see in this plot is that, similar to what we described in section “Impact of finite isolation” for poor isolation at the ports facing the tuners, imbalance is causing a dependency of the common and differential mode on both tuner settings as well as load-dependent mode conversion. To compensate these hybrid imperfections, again we would have to tune both tuners jointly.

Fig. 7. Analysis of the probability that the derived rule-of-thumb in (13) and (14) covers all possible combinations assuming the errors are uniformly distributed within the specified range. P(A ⊆ B|S _Δ, S _Σ) being the probability that the results of the Monte Carlo simulation (A) are a subset of the area specified by the formula (B) for a given pair of tuner reflection coefficients. To visualize this relationship, a larger number of 50 000 realizations of the Monte Carlo simulation were used. The x-axis indicates the difference in magnitude between the reflection coefficients set at the two tuners. To demonstrate the full shape of the curve, values down to −100 dB are plotted, although they cannot be realized in a real-world scenario.

Hybrid tuning error estimation

As already mentioned, we deliberately did not always use realistic values for the plots mentioned in the sections above. But in this section, we elaborate on which parameters you need to consider most when purchasing a real 180° hybrid for mixed-mode load-pull measurements. The first discussed parameter was the insertion loss which causes a significant decrease in tuning range. Especially for passive load-pull systems, this is a critical value if you aim to achieve high reflection coefficients. Differently, symmetric return loss does not decrease your tuning range, but shifts it depending on its phase and magnitude. Inside the available tuning range, this shift can be compensated with the corresponding tuner. Asymmetric return loss causes mode conversion, but with average return losses between 15 and 25 dB, the reduction in tuning range is not as severe as in the case of insertion loss. The next discussed parameter was the isolation which may cause impedance shifts, dependency of S _dd and S _cc on both tuners as well as load-dependent mode conversion. However, as average isolation values are rather high in a range from 23 to 35 dB, they can often be neglected compared to the impact of the other imperfections. Other than isolation, realistic values of magnitude and phase imbalance may have a severe impact on the performance of a 180° hybrid. Imbalance may cause a significant dependency of S _dd and S _cc on both tuners as well as load-dependent mode conversion. To compensate for this, a joint tuning of both tuners is needed, making the handling of the tuning setup much more complicated. If a joint tuning is not desired, one needs to analyze if the impact of a hybrids imperfections results in an acceptable tuning range. To assist the reader with this decision, we derived the simplified formulas (13) and (14), describing the impact of a 180° hybrid's imperfections on S _dd and S _cc, using the afore mentioned considerations. Compared to the ideal results of (6), these formulas include the impact of the most important imperfections, which are the IL, the magnitude error Δ_m, and the phase error Δ_φ, and were simplified to achieve an easily useable rule-of-thumb. A validation of these formulas is presented in Figs 7 and 8. To achieve this high degree of simplification, some assumptions were made. At first, isolation and return loss were neglected as the impact of isolation is small and impedance shifts due to return loss can be easily compensated. Also higher order terms and products were eliminated from the equation. At last, the different error terms were combined in a way to derive an upper error bound. Due to these simplifications, the formula does fail to deliver an upper error bound if the ratio between the magnitudes of the set reflection coefficients at the tuners is becoming larger than 20 dB as shown in Fig. 8. This is valid for realistic parameters as mentioned in Table 1. The reader can now utilize these formulas to calculate areas in the Smith chart by using the four combinations of positive and negative magnitude and phase errors. These areas indicate how S _dd might change due to a change of S _cc and vice versa. If the resulting area is small enough for the desired application, the reader may consider to use the corresponding 180° hybrid and independently tune S _dd and S _cc by using one tuner for each. Otherwise, a joint tuning of both tuners needs to be performed which is significantly more complicated and requires automatic tuner systems to be applicable in practical measurement campaigns.

(13)$$\tilde{S}_{dd} = S_{\rm \Delta }\cdot \lpar {1 + 2{\rm \Delta }_m} \rpar \cdot IL^2\cdot e^{i2{\rm \Delta }_\varphi }\comma \;$$

(14)$$\tilde{S}_{cc} = S_{\rm \Sigma }\cdot \lpar {1 + 2{\rm \Delta }_m} \rpar \cdot IL^2\cdot e^{i2{\rm \Delta }_\varphi }.$$

Fig. 8. Load-pull setup with directly connected tuners.

Setup with direct connection

A load-pull setup using directly connected tuners is shown in Fig. 9. This setup does not use a 180° hybrid, but as derived in section “Direct connection to separated tuners”, it is also not able to provide different common mode and differential mode load impedances to the DUT. For the load-pull measurements in [Reference Veit, Gadringer and Leitgeb1], a spectrum analyzer was connected via an attenuator to increase the matching. For the measurements presented in this paper, a VNA was used to measure the load impedance a DUT would see.

Fig. 9. Load-pull setup with 180° hybrid.

Setup with 180° hybrid

The setup in this section utilizes a 180° hybrid as depicted in Fig. 3, similar to what was presented in [Reference Ferrero and Teppati7]. As derived in section “Connection via a 180° hybrid”, in this setup, common and differential mode loads can be set independently with the two tuners. For load-pull measurements, the output of the differential mode load tuner would be fed to a spectrum analyzer via an attenuator for better matching. For the measurement results in this paper, again a VNA was used to measure the load impedance a DUT would see.

Setup with connected tuners

Figures 10 and 11 show the measurement setup with connected tuners proposed by us in [Reference Veit, Gadringer and Leitgeb1]. In contrast to the setup depicted in Fig. 3, no 180° hybrid is used, but the ports of the tuners that do not face the DUT are now connected to each other. This connection can be adjusted in magnitude and phase. Ideally, one would use an adjustable reciprocal phase shifter with low insertion loss or a line stretcher in the connection path between the tuners, but such equipment was not available to us. To circumvent this, we used SMA female–female and male–male adapters to adjust the electrical length of the connection between the tuners. Because we were interested in the behavior of the linear load network, and not in the characterization of some DUT, an R&S ZVA VNA was used (see Fig. 10). To perform load-pull measurements with a DUT, directional couplers would be needed as shown in Fig. 11.

Fig. 10. Measurement setup used to measure the tuning behavior of the presented approach.

Fig. 11. Diagram of a load-pull setup utilizing the proposed load configuration to characterize a DUT.