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For years, the load-line technique has been applied to the design of negative resistance oscillators¹. The latest implementation of this technique has been improved with the extensive use of nonlinear computer-aided design (CAD) packages supporting harmonic balance algorithms. By inclusion of more complex non linear models, accuracy was improved. Eventually, this was extended to include large-signal analysis to closed loop oscillators².

For mixers, a type of nonlinear circuit, the evolution of design techniques has followed a similar path.

The fundamentals of mixer design are compiled and established³ (specific design goals for sub-harmonic mixers are described by the paper referenced in footnote 4. Software implementations of statistical load-pull techniques are in the document referenced in footnote 5 and a load pull measurement system is presented in the document referenced in footnote 6. In footnote 7, active mixers are studied from the point of view of large signal conversion matrix and load pull.

Although design procedures have been defined for particular cases, there is a lack of a generic method which specifically determines the performance that can be expected from a device and the requirements of the networks that constitute the mixer. With the procedure proposed here, it is possible to take advantage of the non linear CAD potential/resources/libraries to define an analysis/design cycle that uniquely determines the ideal network. Such a network achieves the theoretical optimum performance from the nonlinear device (the optimum mixer of reference).

The method is a new approach to the classical matching problem. If two signals at different frequencies and with different power levels (the first, large signal and the second, small, medium or large signal) are simultaneously applied to a nonlinear device, it is possible to estimate the impedance seen by the second signal and its variation with the first large signal level. One advantage is that it can be applied for large signal RF or IF. Another possibility is the choice of the pumping level according to the easiest-to-implement embedding network.

The procedure will be described for a N-port (N physical accesses referred to ground) nonlinear element with N networks, one attached at each port, with fundamental or sub-harmonic mixing. N should be ≤3, due to the three frequencies involved in the mixing process (see figure 1).

The example will be a two-port nonlinear device (a series diode with ports defined with respect to ground) applying harmonic mixing at 42.5 GHz. This can be applied to balanced topology and be extended, as well, to the design of other basically-nonlinear circuits, such as frequency multipliers. In that case a maximum number of two ports should be considered, one for the input frequency, and the other for the multiplied frequency (N ≤ 2). The results of the method rely on the models (linear and nonlinear), and the accuracy of convergence of the harmonic balance⁸. Its application is limited to the feasible impedances, but provides criteria to establish priorities between frequencies for matching the circuit in the practical implementation of the mixer.

Stability of the mixer is of concern, especially with active mixing. Non linear stability methods, such as those detailed in the document referenced in footnote 9, could also be applied. Mixer noise figure goals can be added to the simulations¹⁰.

It is advisable not to try and optimize everything from the beginning. Requirement priorities must be defined. The method proposed here is an iterative process focused mainly on the conversion gain for the desired frequencies.

The mixer is divided into two parts (see figure 2). The first is a non-linear element (such as a diode, pair of diodes or a transistor in various configurations.), which may contain linear elements in its equivalent circuit model. This element will be represented by an N-port. N may be typically a grounded diode, a common source MESFET, (a non-parallel pair of diodes in series, for example), or a resistive MESFET ring mixer (see figure 1).

The nonlinear behavior is present in the relations between currents and voltages at the access ports or in an inner level (if extrinsic and intrinsic parts are considered). If the nonlinear block consists of unbiased diodes, it is not necessary to consider the bias problem. But if biased diodes or transistors are used, the optimum bias point for maximum conversion gain should be applied. This bias point is also affected by the self-biasing effect due to the local oscillator's (LO) injected power.

The other part of the mixer is described by N linear networks, which perform the matching and filtering functions necessary to get the maximum conversion gain (minimum conversion loss, if diodes or resistive mixers are used) and the rejection of unwanted signals.

If N = 1, only one four-port network is attached to the nonlinear device to apply LO and RF and to extract IF (receiver case) or apply LO and IF and extract RF (transmitter case). If N equals 3, one two-port is used for each frequency. For N = 2, different combinations are possible, depending on the frequency values (i.e. one two-port-network for LO and a three-port for RF + IF).

Several points are detailed, to constitute an iterative feedback loop during the design process. They are:

• Optimum bias of the non linearity (DC + LO) — To find the optimum bias point, the mixing nonlinearity must be taken into account. If a non-biased diode is being used for mixing, there will be minimum LO power to achieve reasonable conversion efficiency to achieve the most nonlinear region of the characteristic (usually, I/V). In general, LO matching will be easier as the LO level is higher.

  When an externally biased device (such as a diode or FET) is used as a mixing element, the DC bias point is chosen for optimum conversion efficiency. In some cases, the optimum conversion loss could be achieved with less LO power. The LO level must be high enough to ensure that the peak voltage provides the maximum transconductance in a transistor, or to turn on the device in the case of the diode.

  Although the input and output impedance of a diode depend more strongly upon the LO power for an unbiased mixer than for a biased one, in both cases, the matching networks must be designed under LO pumping. Different procedures have been described to establish the optimum bias and LO power, according to the device and the specific configuration and requirements (see footnotes 11 through 15).

• Local oscillator matching — After determining the bias point for efficient conversion, some estimation of the most suitable LO power level should be set. However, LO matching must be considered when determining the level as well as (IF and RF matching variation with LO power). The scheme to implement this simulation is shown in figure 2.

  An LO source is placed in the LO access port of the nonlinear block. The rest of the ports should be isolated at that frequency (for example, grounded). Such isolation is inherently achieved due to the nonlinear device or by the addition of stubs to present a short circuit at the LO frequency. It is also necessary to short certain harmonics of the LO. These conditions should be provided by the networks attached to each port of the nonlinear block. It is necessary to take them into account when evaluating LO match. The conditions related to the LO and its harmonics should be implemented with a virtual network in a first point.

  A one-port box may be used to present a short at the LO and an open at the rest, defining its reflection coefficient magnitude as 1 and its phase frequency-dependent (a phase of 180° should be defined for this LO frequency and 0° for the rest of frequencies).

  By plotting the voltage-current ratio at the LO frequency for different values of LO power on the Smith chart, estimation of the most suitable matching levels are obtained. These should agree with the previous estimations for maximum conversion gain. If they match, the LO network is synthesized.

  It is important to take into account that RF and IF filtering conditions must be provided by this network (for example, it should show short or open to the nonlinear device at RF and IF frequencies).

• RF matching — The nonlinear block must be LO-pumped to evaluate the RF impedance. This way the bias point shift of the nonlinearity under LO pumping is the same as it will be in real operation. Under these conditions RF power (usually small signal) is introduced through the corresponding port of the nonlinearity. If the RF level is known and not so small, it is taken into account using this method.

  The ratio between the RF voltage current flowing into the nonlinear device can be evaluated and its dependence with the LO power level can be plotted on the Smith chart. The conjugate impedance should be synthesized and plotted on the Smith chart, as well, to show the degree of RF matching under LO pumping. Its difficult to specify in a general description, but for a particular design, critical frequencies terminations should be included as well.

  For example image frequency matching could be checked, but with the intention to get the maximum mismatch. It is also important to consider the presence of a load at the IF port of the nonlinearity. Since the most suitable value has not been determined, one can start with a mathematically defined band pass filter at IF terminated in 50Ω. This ending impedance will be swept, using an iterative process, to find the optimum process vs. RF impedance after the next step (Figure 2 shows the topology of the simulation corresponding to point 3.

• IF matching — IF matching can be analyzed with the same criteria as RF matching. However, simultaneous RF and IF sources in the same harmonic balance analysis cannot be used.

  To evaluate the IF impedance, an IF signal is injected in the circuit (a global analysis injecting RF would generate additional IF by the mixing process. This would constitute a source of error in the computed IF impedance).

  Now, the previously obtained RF conjugated impedance to terminate the RF port of the nonlinearity is introduced. The IF network is implemented according to the simulated impedance. It should present the conjugate impedance to the nonlinearity. Now it is possible to use the RF matching technique to refine the simulation by including a more precise value for the IF termination impedance. With the new value obtained for RF impedance the IF matching step is repeated. It can be checked by performing separate sweeps of the RF and IF impedance to determine if the fixed values provide optimum conversion for the fixed level of LO.

• The optimum mixer of reference — The optimum mixer of reference (OMR) is defined as the mixer that contains the nonlinearity and the linear blocks mathematically defined (N networks). This shows the desired impedances to the nonlinearity in order to transfer maximum power in every port.

  To compute the OMR the starting point is the set of optimum values of impedances for RF and IF (fixed level of the LO), the value of N and the topology (i.e. N = 2, one two-port network for LO and a three-port for RF + IF). The OMR is defined for the nonlinearity and for a value of the LO. If the LO power level is changed, the OMR also changes.

  Each of the N auxiliary networks that will be attached to the non linear device will be defined by its S parameters. These conditions are: maximum transfer of power (no losses) and adequate impedances shown to the nonlinear device and to the ports of the mixer. It means that the S parameter matrix of the networks and power gain for each pair of ports must verify the following relations:

Where G_p is the power gain for a couple of ports, P_L is the power delivered to the load, P_av is the power available from the source and P_in is the power which really flows into the circuit. [S] is the scattering matrix of the network, t means transposed, and * means conjugated. [I] is the identity matrix. This S parameter property applies to lossless networks¹⁶. In the evaluation of G_p the sense of flowing power has to be taken into account. A detailed example will be given in next section.

For an ideal OMR, reciprocity is not imposed on the networks to fulfill the conditions more freely. This defines an important limitation for physical synthesis of the networks because typical passive matching networks are reciprocal. Theoretically, it is possible to use active devices in the non-reciprocal embedded networks of the nonlinear mixing device to implement the ideal OMR. But, at least, in this case it is impractical and increases complexity. Reciprocity could be imposed also, but it would change the meaning of the OMR from ideal performance to practical passive network performance. And, solving the mathematical optimization equation would include some error.

The ideal OMR may be used in a global simulation of the mixer to obtain an estimation of the ideal maximum conversion gain (or minimum conversion loss). Later, an iterative process is required to adjust each of the physically feasible networks with the other attached to the rest of the ports of the nonlinear part.

For a broadband analysis, harmonic balance allows sweeping the RF or the LO frequency.

With N equal to 1 or 2, a single network must perform different functions, simultaneously.

The final topology of the mixer for the most general case is shown in figure 2.

The procedure was applied to a harmonic mixer with a silicon Schottky beam-lead diode mounted on alumina microstrip lines (ε_r = 9.9, h = 0.01 inch) operating around 41.5 GHz. The LO frequency was 14.167 GHz. The mixing occurs with the third harmonic (14.167 × 3 = 42.501 GHz), therefore, the input IF frequency is around 1 GHz. The nonlinear part consists of the diode placed in series configuration (N = 2) and pumped by the LO. The diode is modeled with two nonlinearities: the Schottky C/V and the typical I/V curve, and also some parasitics. Determined by the frequencies involved, a two-port network was designed for the LO and a three-port network for RF and IF.

Referencing figure 3, in this case there is no DC bias, so the only source of biasing is the LO pumping. The pumping must be strong enough to turn the diode on and off. This can be checked by plotting the dynamic load line over the DC curve in a harmonic balance analysis for different levels of LO, finding the minimum value to turn on and off the diode (see figure 3, step 1, plot of the dynamic load line for 6 dBm of LO power).

Next, the LO impedance was evaluated for different levels of LO power. The initial value was 6 dBm of LO power. The LO network was designed with both filtering and matching requirements (see figure 4, the plot of the diode impedance and the impedance shown by the LO network plus diode to the LO source).

The diode was loaded on the LO side with the proposed LO network. A power source was connected to the RF/IF port of the diode to evaluate the V/I ratio at the RF frequency. A frequency selective termination was also placed in the RF/IF port to load the diode at IF, but it presents an open circuit to the rest of frequencies. A harmonic balance analysis of the LO and RF components plots the RF impedance of the diode and the LO network for different values of LO power (see figure 7). In this case (N = 2) the same network will act as an IF filter. Therefore, before synthesizing the three-port RF/IF network, the following must be implemented.

A frequency selective termination was placed in the RF/IF node of the diode, presenting the optimum to the load, at RF and an open circuit to the rest of frequencies. IF power was injected to evaluate IF impedance under LO pumping. This way the optimum IF impedance, which must be seen from the nonlinear perspective, was obtained for several values of LO power. The procedure in the previous paragraph is repeated to refine the RF impedance estimation using the obtained value of optimum IF impedance. Finally, this procedure is repeated to refine the selection of IF impedance. A load pull sweep verified that the RF and IF impedances were optimum for each level of LO (see figure 5, conversion loss versus the magnitude of the RF reflection coefficient for fixed phase), which, in this case is Γ_RF = Γ_RF (optimum) and the fixed IF reflection coefficient. Steps 3 and 4 could be repeated with different levels of LO power looking for an easier implementation of the RF-IF networks if the nominal level of LO is changed.

To calculate the ideal OMR for this case, and apply equation (1) to the RF/IF 3-port network (see figure 6, the LO network was already fixed). the following relations should be held:

At IF:

where:

and:

At RF:

where:

and:

By definition:

S₂₂(IF) = conjugate(optimum reflection coefficient nonlinearity at IF).

S₂₂(RF) = conjugate(optimum reflection coefficient nonlinearity at RF).

S₂₃(IF) = S₃₂(IF) = 0

S₁₂(RF) = S₂₁(RF) = 0

S₃₃(IF) = S₁₁(RF) = 1 (open)

S₁₃(IF) = S₃₁(IF) = S₁₃(RF) = S₃₁(RF) = 0 — the ideal isolation condition for IF and RF.

The system of equations can be solved analytically or by optimization (error zero was reached).

Once the scattering matrix was defined it was placed in the mixer simulation to establish the optimum performance available from the device.

When impedances expected from the networks are known, linear optimization techniques can be used to design the physical elements, achieving the optimum trade-off between the desired impedances for each band of frequencies. This method is described for a fixed frequency. It is important to emphasize that, in general, for the physical implementation, a compromise between the minimum and maximum frequency of operation (in broadband designs) and between conversion loss and other figures of merit, for example, image rejection, has to be achieved.

In the case of this up-converter, RF frequencies are higher than IF frequencies, so RF matching and power transfer are consider a priority. For the example, image frequency is not a concern. But it can be applied to other designs¹⁷ to mismatch the image frequency as well.

Using the defined OMR and the optimum impedances obtained, a set of microstrip lines on alumina substrate are used to make the LO network and the RF/IF network parts of the mixer. Priority was given to the RF power transfer. Microstrip coupled lines are used in the RF path and lines. Open ended stubs and printed inductor were used for the IF path. The proposed LO side network consists of high and low impedance microstrip line sections and a IF short-circuited stub to short circuit the diode at the IF frequency.

Two-dimensional electromagnetic simulation with the method of moments was used⁸. Due to the limitations of the range of microstrip models and the parasitic coupling, a discrepancy was found between the schematic simulation and the electromagnetic simulation. A determination was made to use the electromagnetic as a base line for the design.

Next the goal was to achieve a physical three-port network providing the same scattering parameters as the ideal network of the OMR. The DC return of the diode has to be provided by this network as well.

A preliminary 50Ω matched diplexer designed from the theory detailed in the document referenced in footnote 18 was used as starting point. It consisted of a band pass filter in the RF path and a low pass filter for the IF. The RF band pass filter was implemented with microstrip coupled lines and the IF low pass filter with high and low impedance microstrip line sections. An LO short was implemented with an open ended stub. The DC short required a printed inductor to block IF.

Optimization techniques were used to determine the trade-off between the desired impedances for RF and IF. Electromagnetic simulation was critical in taking into account the real coupling between RF and IF paths.

The RF access proved to have excellent matching to the diode for a LO power of 6 dBm (see figure 7). Priority was given to the RF section so less matching is achieved for the IF network (see figure 8).

The simulated circuit was fabricated and the simulation parameters were revised according to real values (see figure 9). The case is a general purpose test fixture where access lines were connected with bonding wires to the lines of the circuit board. Deviations in the length of these wires (especially in the RF path) arose as a critical factor in the final performance.

Conversion loss measurements are plotted (see figure 10) with optimum conversion loss (according to the OMR) and with the simulated physical circuit. There is reasonable agreement between the simulations and the measurements, but it has to be taken into account that the OMR consists of non reciprocal ideal networks (physical passive networks are reciprocal). LO return loss was measured with a spectrum analyzer that operates up to 50 GHz and directional couplers. The simulated results and the OMR match well. Actual measurements show a shift in the maximum matching due to some dispersion in the wire bonding lengths used to connect the mixer carrier to the test fixture microstrip lines (see figure 11).

A systematic step by step process has been described to establish the optimum performance which can be expected from the mixer using the definition of the OMR. The method requires accurate non-linear models and harmonic balance tools. The method is applied to a harmonic (x3) Schottky diode mixer operating in the range of millimeter wave, showing how to establish design priorities and define adequate performance even in the case of limitations. The design method presents reasonable agreement between the measurements and the simulations, and is, therefore, deemed valid.

1. W. Wagner, “Oscillator Design by Device Line Measurement,” Microwave Journal, February 1979, pp. 43-48.

2. J. P. Pascual, J. Portilla, E. Artal, “Techniques allow simple design of MMIC Oscillators,” Microwaves and Radio Frequency, March 1996, pp. 67-79.

3. S. A. Maas, The RF and Microwave Circuit Design Cookbook, Artech House, 1998.

4. A. Madjar, “A Novel General Approach for the Optimum Design of Microwave and Millimetre Wave Sub-Harmonic Mixers,” IEEE MTT, November 1996, pp. 1997-2000.

5. J. F. Villemazet, M. Soulard, “A Statistical Load Pull for Mixer Design Using a Commercial Circuit Simulator,” IEEE MTT-S Digest, 1996, pp. 757-760.

6. Di-Luan Le and F. M. Ghannouchi, “Multitone Characterization and Design of FET Resistive Mixers Based on Combined Active Source Pull/Load-pull Techniques,” IEEE Trans. at M.T.T., Sept. 1998. pp. 1201-1208.

7. F. Giannini, G. Leuzzi, E. Limiti, F. Morgia, “Optimum Nonlinear Design of Active Microwave Mixers,” GaAs 2000, Paris, pp. 346-348.

8. Hewlett Packard's Microwave Design System Manual. Vers. B.07.10.

9. J. Jugo, J. Portilla, A. Anakabe, A. Suárez, J. M. Collantes, “Closed Loop Stability Analysis of Microwave Amplifiers, Electronic Letters,” Vol. 37, no. 4, 2001, pp. 226-228.

10. HP product Note 85150-4, “Simulating Noise in Non-linear Circuits Using the HP RF and Microwave Design System,” Literature number 5091-9582E.

11. M^a L. de la Fuente, J. P. Pascual, E. Artal, “Analyse and Design a Cascode MESFET Mixer,” Microwaves and Radio Frequency, May 1998, pp. 129-138.

12. M^a L. de la Fuente, J. Portilla, J. P. Pascual, E. Artal, “Low Noise Ku-Band MMIC Balanced pHEMT Up-converter,” IEEE Journal of Solid State Circuits, February 1999. pp. 259-263.

13. M^a L. de la Fuente, J. P. Pascual, F. J. Gonzalez, E. Artal, “Broad Band GaAs MMIC Down Converter With Very Low Intermodulation,” European GaAs and related III-V Compounds Application Symposium, Munich, October, 1999.

14. J. A. García, J. C. Pedro, M^a L. de la Fuente, N. B. Carvalho, A. Mediavilla, A. Tazón, “Resistive FET Mixer Conversion Loss and IMD Optimisation by Selective Drain Bias,” IEEE Microwave Theory and Techniques symposium. 1999, California. pp. 2382-2392.

15. F. Filicori, A. Santarelli, P. Traverso, G. Vannini, “Large Signal Modelling and Cold FET Mixer Design,” ISRAMT 99, pp. 235-238.

16. R. E. Collin, Foundations for Microwave Engineering, Mc. Graw Hill International Editions, 1992.

17. J. P. Pascual, M. Rodriguez-Girones, M^a L. De La Fuente, F. López, C. I. Lin, A. Simon, H. L. Hartnagel, “Subharmonically Pumped MMIC Mixer Operating at 150 GHz,” ISRAMT 99, pp. 248-251.

18. J. A. G. Malherbe, Microwave Transmission Line Filters, Artech House, 1979.

Juan Pablo Pascual was born in Santander, Spain, in 1968. He received an M.S. degree with Honours in Electronics in 1990 and a Ph.D. in electronics engineering in 1996, both from the University of Cantabria, where he currently works as an associate professor‥ He can be contacted at pascual@dicom.unican.es

M^a Luisa de la Fuente was born in Reinosa, Spain, in 1968. She graduated from the University of Cantabria, Santander, Spain in 1991 and received the doctoral degree in electronics engineering in 1997.

Eduardo Artal received a doctoral engineering in telecommunication, 1982. He was a teacher at the Polytechnical University of Catalonia (Barcelona) from 1976 to 1990. Since 1990 he has been a professor at the University of Cantabria in Santander (Spain).

Manuel Rodríguez-Gironés Arbolí was born in Madrid, Spain, in 1971. He received his degree in Telecommunications Engineering from the Politechnical University of Madrid, Spain, in 1997, and his doctoral degree from the Technical University of Darmstadt in 2002. He is currently working as an assistant researcher at the Institute for Microwave Engineering at the Technical University of Darmstadt.

Hans L. Hartnagel received the Dipl.-Ing. degree from the Technical University of Aachen, Aachen, Germany, in 1960, and the Ph.D. and Dr. Eng. degrees from the University of Sheffield, Sheffield, U.K., in 1964 and 1971, respectively. After having worked for a short period with Telefunken, Ulm, Germany, he joined the Institute National des Sciences Appliquées, Villeurbanne, Rhône, France, and then the Department of Electronic and Electrical Engineering, University of Sheffield.