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Fig. 1. General view of the structure

Fig. 2. S-parameters versus frequency

With a wider frequency band from 10 to 30 GHz we excite the circuit at a higher resonant frequency (about 27 GHz) and also below the waveguide cutoff.

Fig. 3. Results of simulation in a wider frequency band

|Gam1| and |Gam2| denote the absolute values of propagation constants in the transmission lines terminated by ports 1 and 2, respectively. It can be seen that at port 1 (waveguide input) the propagation constant drops to 0 at the waveguide cutoff (15 GHz), while at the TEM output |Gam2| is naturally proportional to the frequency.

Let us have a closer look at the values of |S11| and |S21| below the cutoff frequency. Naturally |S21| drops fast with decrease of the frequency. But what may seem less intuitive, there is also a fast decrease of |S11|. This is because even below the cutoff a long section of waveguide is reflectionless and thus its |S11| must tend to zero with the length of the section increasing. Let us also note that zero reflection does not imply any transmission of a real power into the guide since the characteristic impedance of the guide (equal to the reference impedance for S-parameter definition) is imaginary below the cutoff frequency.

Let us have a closer look at the values of |S11| and |S21| below the cutoff frequency. Naturally |S21| drops fast with decrease of the frequency. But what may seem less intuitive, there is also a fast decrease of |S11|. This is because even below the cutoff a long section of waveguide is reflectionless and thus its |S11| must tend to zero with the length of the section increasing. Let us also note that zero reflection does not imply any transmission of a real power into the guide since the characteristic impedance of the guide (equal to the reference impedance for S-parameter definition) is imaginary below the cutoff frequency.

Fig. 4. List of available characteristics with standard and extended results options

We have 6 items (left on Fig. 4), which provide complete information for most applications. However, in some specific applications the user may require additional information about the phase angle of the propagation constant and about the changes in the characteristic impedances of the port lines (which are also reference impedances for the calculated S-matrix).

Fig. 5. Display of characteristic input/output impedances available in extended results options

Let us have a look at the characteristic impedances as displayed in Fig. 5. At the TEM output we have a constant and real impedance in the entire frequency band. At the waveguide input the impedance is real (<Z2=0) above the cutoff and imaginary (<Z2=90^{o}) below the cutoff. The absolute value of the impedance changes in a way typical to a hollow waveguide TE mode that is it rises when approaching the cutoff frequency. It is well known that a characteristic impedance of a waveguide is not uniquely definable and it is a subject of arbitrary normalisation. In QW-3D we normalize the characteristic impedance of a waveguide mode in such a way that it is equal to unity at the frequency at which the mode template has been calculated. In the considered example the input mode template has been calculated at 22.5 GHz and that is why |Z1|=1 at that frequency. Note however that in the case of TEM line for which the unique definition of impedance is possible QW-3D indicates the actual impedance being Z2=54.4 Ohm in our case.

The S-parameters are extracted with respect to reference planes situated at some distance from the ports and that we can virtually move the reference plane in a postprocessing operation. Let us return to the results of analysis already considered in Fig. 3. We save them on disk. Let us now try to move the virtual reference planes of both ports to the position of the coaxial line antenna inserted into the waveguide. Thus we set in Port 1 New location=25 and in Port 2 New Location=3. To compare the transformed results with the original ones we read the saved results from disk. Comparison of them is shown in Fig. 7. As expected, moving the reference plane makes the phase characteristics more flat (green line versus red line) and does not change |S21| above the waveguide cutoff (where the blue line is hidden under the yellow line). It is interesting to note that the software takes into account the real part of the propagation constant below the cutoff and appropriately corrects |S21| for low frequencies. That is why for example for 12 GHz |S21| rises from 0.0178 to 0.698.

Fig. 6. Postprocessing dialogue for changing virtual location of the reference planes of ports

Fig. 7. Results of calculations of the considered example with the original reference planes (yellow + red)

and the reference planes virtually moved to the position of the coax antenna (blue + green)

and the reference planes virtually moved to the position of the coax antenna (blue + green)

Sometimes we are interested in the power balance of the analysed structure. In other words, we would like to know what part of the power injected into the circuit is dissipated in all defined ports. Let us consider the results like those shown in Fig. 3. Either Pow.SK1 or Pow.Bal. can be displayed, with Extended Results are off or on, respectively.

Fig. 8. Results of simulation including the power balance characteristic with standard

(non-extended) S-parameters

(non-extended) S-parameters

Fig. 9. Results of simulation including the power balance characteristic with extended S-parameters

There may be several reasons for the power balance not to be equal unity. In each case we can draw interesting conclusions from this fact.

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in the case of lossy circuits power balance indicates efficiency of energy transmission

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in the case of transient states during the simulation power balance can be a good indicator of convergence, showing how much energy is still inside the circuit and thus whether we need to prolong the FDTD analysis

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if the power balance does not converge to unity in a case of a lossless structure, this may indicate that some of the energy is dissipated in the source or/and load at the modes not considered at the port. Displays of Fig. 8 and Fig. 9 show a good example of such a case. We can see that at the frequency 27.2 GHz power balance drops to the value of about 0.5. The reason is that at this frequency much of the energy transmitted to the output is coupled to a waveguide mode in the coaxial line. This energy is ignored during calculation of |S21| since Port 2 has been defined at the TEM mode. To have full description of the circuit at that frequency we would need to define the structure as a three-port with the third port associated with the first waveguide mode in the output coax. This kind of multi-port approach will be considered in further examples

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it is interesting to note the difference in power balance display in the case of standard and extended S-parameters below the cutoff frequency of the waveguide. In the case of standard S-parameters (Fig. 8) the power balance is not correctly calculated below 15 GHz. It can be seen that the power balance calculated with extended S-parameters (Fig. 9) displays a value close to 1 even below 15 GHz, despite the fact that characteristic impedance of the input line is imaginary and that the real power injected into the circuits drops very fast with diminishing frequency. This confirms consistency of the S-parameter extraction even below the cutoff frequency of the waveguide. It can be seen in Fig. 9 that below 12.6 GHz the power balance shows 0. This indicates that the real power injected into the circuit at these frequencies is so small that the software cannot calculate the power balance with acceptable accuracy and sets the result to zero to avoid displaying a curve of irrelevant shape

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please also note that there may be a difference between the power balance calculated with standard and extended S-parameters also above the waveguide cutoff frequency in the case of |S11| very close to unity. With the standard S-parameters the power balance calculated as sqrt(|S11|^{2}+|S21|^{2}) firmly indicates unity. Physically, we can say that the reference power is the available power of the source and thus the power balance is calculated accurately even with high input reflections. In the case of power balance calculated with extended S-parameters the software takes as reference the net power actually entering the circuit, which is small due to a high reflection coefficient. Thus it tries to numerically compare very small power dissipated in the output to the very small power entering the input. The result may be somewhat different than obtained in the previous case. Let us note however that each of the results has clear physical interpretation and availability of each of them may be useful for the circuit designer

We can see in Fig. 10 that we have been calculating S-parameters with an option *Sk1 at reference planes*. *Sk1* means that we will use only one excitation from Port 1. *At reference planes* is compatible with the fact of using differential method of S-parameter extraction. The alternative is *Smn at reference planes*. In the latter case the software will calculate the entire S-matrix after consecutive excitations from Port 1 and Port 2. Such a way of calculating S-parameters is normally sequential, which means that the software will first calculate S11 and S21 after excitation from Port 1 and simulation over the declared number of FDTD iterations. Then it will excite the structure from Port 2 and calculate S12 and S22. At the end it will perform matrix operations to correct mutually the results obtained with excitation from each port, and display the final result.

Fig. 10. Processing/Postprocessing dialogue

The two other possibilities are *Multisimulator* and *Multithread*. They operate as follows:

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in the* multisimulator* case the software will create two *simulators* being objects within *QW-Simulator*. Note that *QW-Simulator* has been developed in such a way that many *simulator* objects can exist and operate concurrently. In this case, one *simulator* will be running the FDTD analysis with excitation applied from *Port 1*, and thus extracting S11_{ }and S21. The other *simulator* will be running the FDTD analysis with excitation applied from *Port 2*, and thus extracting S22_{ }and S12. The *Smn postprocessing* functions will be importing data from both *simulators* on-line, and also on-line performing mutual corrections. Therefore the intermediate results watched during the analysis will already be mutually corrected; this is an important advantage with respect to the *sequential* mode where only the final result is mutually corrected. The other advantage is that the number of iterations per port does not have to be predefined, and the decision about terminating the analysis is made dynamically by the user as in the case of single-port excitation. However, memory occupation increases by a factor of two, which is a disadvantage in the case of analysing large problems on low-memory computers. The computing time remains unchanged: although the total number of iterations is reduced by a factor of two, the number of operations per iteration is doubled

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the *multithread *mode is an extension of *multisimulator* for multiprocessor PCs. In this mode each simulator is considered as a separate thread and the operating system will try to assign separate threads to separate processors. In our example, if at least two processors are available in the computer, the computing time will be reduced by a factor of two. The *multithread* operation requires a special *QW-MultiSim* licence

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the final remarks in this Section will concern the options of *S-corrections assuming*... When using the option *Smn at reference planes* we obtain the complete information about the circuit, which permits to compensate possible errors due to imperfect matching of the ports. Such complete information is not available when we use the option *Sk1 at reference planes*. Thus some additional assumptions about the circuit are very helpful in the error compensating procedures. First of all, we use the information about the reciprocity. Since in principle all the calculated structures are reciprocal the option *S-corrections assuming*: *Reciprocal N-port* should always be checked. Moreover, if we know that we are considering a lossless reciprocal two-port we should also check the next option. It helps in accurate extraction of S11* *and* *S21.

Fig. 10. List of characteristics calculated with the option *Smn at reference planes*

Fig. 11. Results of S-matrix calculations with the option *Smn at reference planes*