RF-Microwave Multi-band Design Solutions for Multilayer Organic System on Package Integrated Passives

I. INTRODUCTION
Emerging applications in the RF/microwave/ millimeter wave regimes require miniaturization, portability, cost and performance as key driving forces in this evolution. Multi-band applications are also becoming extremely important within passive development to realize multiple frequency bands for various wireless and optical sub-carrier multiplexing (OSCM) systems. Investigations on System on Package approach for module development [1] have become a primary focus due to the real estate efficiency, cost-saving and performance improvement potentially involved in this integral functionality.
In most of the presently used microwave integrated circuit technologies, it is difficult to integrate the passives efficiently while maintaining the desired performance. Another critical obstacle in efforts to reduce the module size is the design of passive components, which occupy the highest percentage of integrated circuit and circuit board real estate. Design flexibility and optimized
integration can be achieved with multilayer substrate technology in which free vertical real-estate is taken advantage of. Various highly intergrable multilayer technologies such as multilayer low-temperature co-fired ceramic (LTCC) [2]-[3] and multilayer organic (MLO)
[4]- [5] are thus being studied to achieve complete System on Package solutions.

II. MULTILAYER INDUCTORS
High Qs at the frequency range of interest can be obtained by designing CPW inductors and HGP [9] series and cascade inductors using multilayer organic technology. The CPW spiral inductor, Fig. 1, avoids via losses, has reduced dielectric losses and increased SRF. The advantage of the HGP implementation includes shunt parasitic capacitance eddy current reduction resulting in
higher Q. Fig. 2 shows a measured Q of 182, SRF 20GHz, effective inductance (Leff) of 2 nH. The series inductor is designed as one continuous turn; however, the turn on the second layer is offset from the turn on the top layer. This offset helps decrease the parasitic capacitance
between the turns and improves SRF. The top metal and bottom metal of the cascade inductor spiral separately and are connected at the center of the spiral. Both inductors are illustrated in Fig. 3. The top and bottom spiral are overlapped and strongly coupled yielding an impressive Q
and effective inductance, Leff. The Q, L and SRF of the HGP series and cascade inductors are 122, 2.5nH, 10GHz and 165, 3.4nH, and 11.5GHz, respectively, Fig 4. Another benefit of the HGP configuration is that the Leff can be adjusted by increasing or decreasing the shunt
parasitic capacitance due to the ground plane. This is achieved by decreasing or increasing the hollow, respectively.


III. EMBEDDED FILTERS
Several front-end RF filters were designed in various topologies. Two LPF were designed for 750MHz and C band, respectively; two BPF were designed for C GHz and Ku bands, respectively. For RF and low microwave applications, this filter can be implemented by combinations of capacitive and inductive lumped passive components. Fig. 5 shows the 2 nd order Bessel lumped element lowpass filter with cutoff frequency at 750MHz.
The simulated and measured return loss and insertion loss are shown in Fig, 6. It is used to filter 1Gb/s header data stream in a 10Gb/s OSCM system operating at 14GHz. A stepped impedance LPF was designed with cutoff frequency at 7GHz, Fig. 7 . It is used to filter 10Gb/s data stream in a 14GHz OSCM transmitter as well. The series inductors represent the high impedance sections(94? ) and the shunt capacitance represent the low impedance sections(7.2? ). The return and insertion loss are shown in Fig. 8.
The first bandpass filter design for C band applications consists of a square patch resonator[10] with inset feed lines, Fig. 9. The inset gaps act as small capacitors and cause the filter to have a pseudo-elliptic response with transmission zeros on either side of the passband. This structure also has a tunable bandwidth. The length of the insets and the distance between them are the main controlling factors, effectively setting the size of the mode-splitting perturbation in the field of the resonator. The length of the feed lines is determined by the input and output matching requirements. Fig 10 shows a center frequency of 5.8 GHz, bandwidth of 1.5 GHz and a minimum insertion loss of 3 dB.


The second BPF desi gn for Ku band applications uses a broadside-coupled microstrip dual-mode square ring resonator[11]. This consists of a microstrip ring resonator that is perturbed by inserting a small metal polygon at one corner, Fig. 11. The outputs are taken symmetrically with respect to this perturbation, which causes the resonator modes to split into two degenerate coupled modes, giving a second-order bandpass filter response. The advantages of this design are compactness (nominal edge length is l/4) and controllability of the bandwidth by varying the size 'd' of the perturbation that causes mode splitting. Larger perturbations cause more mode splitting and result in a larger bandwidth. The strength of the capacitive coupling also controls the bandwidth. Strong coupling lowers the external Q factor of the resonator and increases the bandwidth. Fig. 12 shows a measured center frequency of 14.5GHz, bandwidth of 1 GHz and an insertion loss of 4 dB.


IV. CONCLUSION
In this paper we have reported embedded passive inductor and filter designs and measurements for C, S, and Ku bands implemented in a multilayer organic-based packaging environment. A compact inductor with a measured Q of as high as 182 and SRF as high as 20GHz is presented. The filters demonstrate potential for compact designs in multiple filter bands.


RF-Microwave Multi-band Design Solutions for Multilayer
Organic System on Package Integrated Passives
http://shunpike.mit.edu/writings/MTT2001final.pdf
CRF_Zambrano C. Jaydi D.

RF and Microwave/Design Challenges in PCBs

INTRODUCTION
Technology and automation are the driving forces to use more complex devices. This in itself promotes consumer electronics providers to include more and more functionality into traditional devices that were used for only one purpose. Let's just look at a few familiar devices such as cell
phones, PDAs, MP3 players, digital cameras, and GPS units. All of these are now mature products and most consumers own one and some cases all of them.
Now looking at the "Road Warrior" of a couple years back, we remember a person with multiple devices attached to their belt representing those devices. In the airport, during those days, there always was a competition for power outlet to charge those gadgets. Now we don't want to carry all of these individualmarvels and electronics companies have answered the challenge by performing an amazing fit of engineering, miniaturization and manufacturing of electronics wizardry to satisfy consumer demand for newer toys.
A cell phone is no longer a device that is used to just talk on. Instead it is an amazing technology that allows users do so much more. These new wonders are email capable, manage contacts, update calendars, provide entertainment, take pictures, and even in some cases provide a GPS device that keeps you from getting lost or tracks the whereabouts of your
kids.
What does that all mean to engineers that need to face all the challenges to provide a device that we all use? I want to examine only one aspect of this challenge for supporting PCB design that incorporates RF and microwave devices.

RF AND MICROWAVE
RF and microwave implementation in PCB designs is not a new concept. PCB designers have been doing it for many years if not couple of decades. The question is how hard engineers had to work to get RF/Microwave elements incorporated then and how can it be done today. Electronics Design Automation (EDA) has bought us the use of computers for PCB design and has increased designer productivity. Most advances in automation typically were aimed at digital
circuitry and auto-routing. To help designers that need to manipulate RF and microwave shapes, the EDA industry needs to understand how these type designs are created. Typically electrical engineers use RF simulators to model the circuit that they design. Once the desired electrical performance is achieved, the simulator can produce a representation of the copper shape for this circuit in most cases. The most common way this is achieved is when copper shapes are defined in DXF format. DXF is a standard that AutoCAD introduced forexchanging data between different CAD systems.
Most CAD systems today can import the DXF files, butthe bigger question is how the DXF file is interpreted during the import. If the DXF file is not convertedproperly to become an intelligent copper shape, the designer still has to do a lot of manual manipulation of the imported file. The image may just be a compilation of disjointed lines that mean nothing to CAD tool.
At this time the designer may try to trace over unintelligent shapes to recreate this shape in native CAD format. Any time the retracing happens, it introduces human error and the shape may not be exactly the same size. This is unacceptable as any variations, no matter how small they are, lead to poor or incorrect performance excepted from electrical simulation.
Electrical CAD systems should provide control during the import of the DXF complex Copper shape with as little human interaction as possible. Designers need to maintain control of the layers coming from DXF and re-map them to electrical CAD system layers as well as possess the ability to convert DXF into exact and usable copper shapes (see Figure 1).

DESIGN PROCESS TECHNIQUES
The next important aspect of designing copper shapes for RF and microwave is the ability to create Gerber files with sharp corners. Usually if a designer is using a 50 mils line to draw a shape, during Gerber output the design ends up with a small Radius from plotting with a 50 mils round line. A good PCB CAD system knows how to implement this important aspect of line thicknesses in Gerber format (see Figure 2).

Now let's talk about chamfered corners that are routinely are used to in RF and microwave circuits. Designers need an automated way to specify the ratio of the chamfering that needs to be produced based on design. The distance between the 90 degree corner and chamfer is critical (see Figure 3.1 and 3.2).
Coplanar/wave guide is consistently used in RF and Microwave designs as well. Designers may do this manually and it is a laborious, long and error prone proposition because as the designer needs to control the specific distance between the Trace and Vias as well as the distance between one via and another. If these distances are not maintained, the circuit will not perform as designed. Again a good CAD system can provide control and automate the creation of coplanar/wave guide shielding with vias (see Figure 4).
The last but not the least important part of ensuring the RF or microwave design works is shielding areas with vias. Today as before, designers can manually produce this using elbow grease and a lot of time. Yet automation brings shortened design cycle times and most importantly, adherence to rules and the batch checking of these rules. It is important for designer to specify the rules for via pattern generation and CAD system to the rest of the work. The rules may include:
What type of via to use for what Copper area?
What net the via needs to attach itself to?
What is the distance that needs to be maintained from the edge of the Copper area to a Via?
What is a distance from one via to the next one?
What type of pattern does the via use?
Can Faraday Cage be generated by adding Vias only to outer edge of the Copper Area?
More (see Figure 5.1 and 5.2)

Automation for above techniques has raised the bar for EDA vendors to eliminate manual processes that designers have resorted to create designs with RF andmicrowave elements.

CONCLUSION
Without the support of EDA CAD systems, engineers are bogged down by manually creating complex copper shapes, chamfered corners, via patterns, etc. That is time that can be spent innovating by working smarter not just harder. Improvements in ability to manipulate RF and microwave elements by EDA vendors allow designers to concentrate more on implementing increased functionality and reducing the size of devices with wireless capabilities.

RF and Microwave/Design Challenges in PCBs
fuente:http://www.mentor.com/products/pcb-system-design/techpubs/rf-and-microwave-design-challenges-in-pcbs-32889
CRF_Zambrano C. Jaydi D.

Power Amplifiers and Transmitters for RF and Microwave (4/4)/VI. LINEARIZATION

Linearization techniques are used both to improve linearity and to allow more efficient, but less linear methods of operation. The three principal types of linearization are feedback, feedforward,
and predistortion.

A. Feedback
Feedback linearizes the transmitter by forcing the output tofollow the input. It can be applied either directly to the RF amplifier (RF feedback) or indirectly to the modulation (envelope,
phase, or and components). In RF feedback, a portion of the RF-output signal from the amplifier is fed back to and subtracted from the RF-input signal without detection or down-conversion. The delays involved must be small to ensure stability, and the loss of gain at RF is a more significant design issue. The use of RF feedback in discrete circuits is usually restricted to HF and lower VHF frequencies, but it can be applied within MMIC devices well
into the microwave region [33]. Envelope feedback reduces distortion associated with amplitude
nonlinearity. It can be applied to either a complete transmitter or a single PA [33]. The RF input signal is sampled by a coupler and the envelope of the input sample is detected. The resulting envelope is then fed to one input of a differential amplifier, which subtracts it from a similarly obtained sample of the RF output. The difference signal, representing the error between the input and output envelopes, is used to drive a modulator in the main RF path. This modulator modifies the envelope of the RF signal, which drives the RF PA. The envelope of the resulting output signal is, therefore, linearized to a degree determined by the loop gain of the feedback process. For a VHF BJT amplifier in which amplitude nonlinearity is dominant, two-tone IMD is
typically reduced by 10 dB. The polar loop overcomes the fundamental inability of envelope feedback to correct for AM-PM distortion by adding a phase-locked loop to the envelope feedback system. Envelope detection and phase comparison generally take place at the IF. For a narrow-band VHF PA, the improvement in two-tone IMD is typically around 30 dB. The envelope bandwidth must be at least twice the RF bandwidth, but the phase bandwidth must be
at least ten times the RF bandwidth.
The Cartesian-feedback technique overcomes the problems associated with the wide bandwidth of the signal phase by applying modulation feedback in and (Cartesian) components. Since the and components are the natural outputs of a modern DSP, the Cartesian loop is widely used in mobileradio systems. Two identical feedback processes operate independently
on the and channels (Fig. 19). The inputs are applied to differential integrators (in the case of a first-order loop) and the resulting difference (error) signals are quadrature-upconverted
to drive the PA. A sample of the output from the PA is attenuated and down-converted in quadrature and synchronously with the up-conversion process. The resulting quadrature feedback signals then form the second inputs to the input differential integrators, completing the two feedback loops. The phase shifter shown in the up-converter local-oscillator path is used
to align the phases of the up- and down-conversion processes. The use of Cartesian feedback with a class-C PA amplifying an IS-136 (DAMPS) signal improves the first ACPR by 35 dB and
the allows the signal to be produced with an efficiency of 60%.

B. Feedforward
The very wide bandwidths (10–100 MHz) required in multicarrier applications can render feedback and DSP impractical. In such cases, the feedforward technique can be used to reduce distortion by 20–40 dB. In its basic form (Fig. 20), a feedforward amplifier consists of two amplifiers (the main and error amplifiers), directional couplers, delay lines, and loop control



networks [34]. The directional couplers are used for power splitting/ combining, and the delay lines ensure operation over a wide bandwidth. Loop-control networks, which consist of amplitudeand phase-shifting networks, maintain signal and distortion cancellation within the various feedforward loops. The input signal is first split into two paths, with one path going to the high-power main amplifier, while the other signal path goes to a delay element. The output signal from the main amplifier contains both the desired signal and distortion. This signal is sampled and scaled using attenuators before being combined with the delayed portion of the input signal, which is regarded as distortion free. The resulting "error signal" ideally contains only the distortion components in the output of the main amplifier. The error signal is then amplified by the low-power high-linearity error amplifier, and then combined with a delayed version of the main amplifier output. This second combination ideally cancels the distortion components
in the main-amplifier output while leaving the desired signal unaltered. Successful isolation of an error signal and the removal of distortion components depend upon precise signal cancellation over a band of frequencies. For a 30-dB cancellation depth, the amplitudes must be matched within 0.22 dB and the phases

within 1.2 [34]. For manufactured equipment, realistic values of distortion cancellation are around 25–30 dB. The limiting factor is nearly always the bandwidth over which a given accuracy can be obtained. The outputs of the main and error amplifiers are typically combined in a directional coupler that both isolates the PAs from each other and provides resistive input impedances. For a typical 10-dB coupling ratio, 90% of the power from the main PA reaches the output. For the same coupling ratio, only 10% of the power from the error amplifier reaches the load, thus the error amplifier must produce ten times the power of the distortion in the main amplifier. The peak-to-average ratio of the error signal is often much higher than that of the desired signal, making amplification of the error signal inherently much less efficient than
that of the main signal. As a result, the power consumed by the error amplifier can be a significant fraction (e.g., one-third) of that of the main amplifier. In addition, it may be necessary to operate one or both amplifiers well into backoff to improve linearity. The overall average efficiency of a feedforward transmitter


may, therefore, be only 10%–15% for typical multicarrier signals. Since feedforward is inherently an open-loop process, changes in device characteristics over time, temperature, voltage, and signal level degrade the amplitude and phase matching and, therefore, increase distortion in the transmitteroutput. An automatic control scheme continuously adjusts the gain and phase to achieve the best signal cancellation and output linearity. The first step is to use FFT techniques, direct power measurement, or pilot signals to determine how well the loop is balanced. Both digital and analog techniques can be used for loop control and adjustment.

C. Predistortion
The basic concept of a predistortion system (Fig. 21) involvesthe insertion of a nonlinear element prior to the RF PA such thatthe combined transfer characteristic of both is linear. Predistortion
can be accomplished at either RF or baseband. An RF predistorter typically creates the expansive predistortion characteristic by subtracting a compressive transfer function (such as that of a diode) from a linear transfer function. Improvements in the ACPR by 10 dB are typical. As with feedforward, the operating bandwidth is limited by the gain and phase flatness of the predistorter itself and of the RF PA. In addition, memory effects in the PA and the predistorter limit the degree of cancellation. Better performance can be achieved with more complex forms of RF predistortion such as Adaptive Parametric Linearization (APL), which is capable of multiorder correction [33]. Most RF-predistortion techniques are capable of broad-band operation with practical operational bandwidths similar to, or greater than, those of feedforward.

D. Digital Predistortion
Digital predistortion techniques exploit the considerable processing power now available from DSP devices, which allows them both to form and to update the required predistortion characteristic. They can operate with analog-baseband, digital-baseband, analog-IF, digital-IF, or analog-RF input signals. Digitalbaseband and digital-IF processing are most common. The two
most common types of digital predistorter are termed "mapping
predistorters" and "constant-gain predistorters." A constant-gain predistorter (Fig. 22) requires only a singledimensional lookup table, indexed by the signal envelope to generate the expansive predistortion characteristic. It is simple to implement and requires only a modest amount of memory for a given level of performance and adaption time. A mapping predistorter utilizes two lookup tables, each of which is a function of the and components of the input. This type of predistorter is capable of excellent performance. However, it requires a significant storage and/or processing overhead for the lookup tables and their updating mechanism, and has a low speed of convergence.
An example of linearization of a PA with two 3G W-CDMA signals by a digital baseband-input predistorter is shown in Fig. 23. The linearized amplifier meets the required spectral
mask with a comfortable margin at all frequency offsets. The noise floor is set by the degree of clipping employed on the waveform, which limits the ACPR improvement obtained. It
clearly demonstrates, however, that digital predistortion can be used in broad-band, as well as narrow-band applications. Fig. 24 shows an example of a commercial 3G transmitter with digital predistortion.

Power Amplifiers and Transmitters for RF and Microwave (4/4)
fuente:http://www.ee.pucrs.br/~decastro/pdf/RF&MicrowavePowerAmp&XMTRs.pdf
CRF_Zambrano C. Jaydi D.

Power Amplifiers and Transmitters for RF and Microwave (3/4)/V. TRANSMITTER ARCHITECTURES

Transmitters use as building blocks not only PAs, but a variety of other circuit elements including oscillators, mixers, low-level amplifiers, filters, matching networks, combiners, and circulators.
The arrangement of building blocks is known as the architectureof a transmitter. The classic transmitter architecture is based upon linear PAs and power combiners. More recently, transmitters are being based upon a variety of different architectures including stage bypassing, Kahn, envelope tracking, outphasing, and Doherty.

A. Linear Architecture
The conventional architecture for a linear microwave transmitter consists of a baseband or IF modulator, an up-converter, and a power-amplifier chain (Fig. 13). The amplifier chain consists of cascaded gain stages with power gains in the range of 6–20 dB. If the transmitter must produce an amplitude-modulated or multicarrier signal, each stage must have adequate linearity. This generally requires class-A amplifiers with substantial power backoff for all of the driver stages. The final amplifier (output stage) is always the most costly in terms of device size and current consumption, hence, it is desirable to operate the output stage in class B. In applications requiring very high linearity, it is necessary to use class A in spite of the lower efficiency.

B. Power Combiners
Whether to use a number of smaller PAs versus a single larger PA is one of the most basic decisions in selection of an architecture [14]. Even when larger devices are available, smaller devices often offer higher gain, a lower matching factor (wider bandwidth), better phase linearity, and lower cost. Heat dissipation is more readily accomplished with a number of small devices, and a soft-failure mode becomes possible. On the other hand, the increase in parts count, assembly time, and physical size are significant disadvantages to the use of multiple, smaller devices.
In the corporate architecture (Fig. 14), power is split and combined in steps of two. Hybrid combiners isolate the two PAs from each other and allow one to continue operating if the other fails. Quadrature combiners insert a 90 phase shift at the input of one PA and a 90 phase shift at the output of the other.
This provides a constant input impedance, cancellation of odd harmonics, and cancellation of backward-IMD (IMD resulting from a signal entering the output port). In addition, the effect of
load impedance upon the system output is greatly reduced (e.g., to 1.2 dB for a 3 : 1 SWR). TheWilkinson combiner is fabricated using quarter-wavelength lines and can be extended to include more than two inputs or outputs.

C. Stage Bypassing and Gate SwitchingStage-bypassing and gate-switching techniques reduce power consumption and increase efficiency by switching between large and small amplifiers (e.g., the driver) according to peak signal level. This can significantly increase the transmitter efficiencywhen operating well into backoff, as shown in Fig. 8 ("GS") for
ideal class-B PAs. These techniques are particularly effective
for mobile handsets that operate over a large dynamic range,
and improvement of the average efficiency from 2.1% to 9.5%
has been demonstrated [20].

D. Kahn Technique
The Kahn envelope elimination and restoration (EER) technique(Fig. 15) combines a highly efficient, but nonlinear RFPA with a highly efficient envelope amplifier to implement a high-efficiency linear RF PA. In its classic form, a limiter eliminates the envelope, allowing the constant-amplitude phase modulated carrier to be amplified efficiently by class-C, class-D, class-E, or class-F RF PAs. Amplitude modulation of the final RF PA restores the envelope to the phase-modulated carrier creating an amplified replica of the input signal. EER is based upon the principle that any narrow-band signal can be produced by simultaneous amplitude (envelope) and phase modulations. In a modern implementation, both the


envelope and phase-modulated carrier are generated by a DSP. In contrast to linear amplifiers, a Kahn-technique transmitter operates with high efficiency over a wide dynamic range and, therefore, produces a high average efficiency for a wide range of signals and power (backoff) levels. Average efficiencies three to five times those of linear amplifiers have been demonstrated from HF to -band [21]. Transmitters based upon the Kahn technique generally have excellent linearity because linearity depends upon the modulator rather than RF-power transistors. The two most important factors affecting the linearity are the envelope bandwidth and
alignment of the envelope and phase modulations. The envelope bandwidth must be at least twice the RF bandwidth and the misalignment must not exceed one-tenth of the inverse of the RF bandwidth [22]. In practice, the drive is not hard limited andfollows the envelope, except at low levels [23]. At higher microwave frequencies, the RF-power devices exhibit softer saturation characteristics and larger amounts of amplitude-to-phase conversion, necessitating the use of predistortion.
The most widely used high-level modulator is class S (Fig. 16). A transistor and diode or a pair of transistors act as a two-pole switch to generate a rectangular waveform with a switching frequency several times that of the output signal. The width of pulses is varied in proportion to the instantaneousamplitude of the desired output signal, which is recovered by a low-pass filter. Class S is ideally 100% efficient and, inpractice, can have high efficiency over a wide dynamic range. The switching frequency must typically be six times the RF bandwidth. A switching frequencies of 500 kHz is readily achieved with discrete components, and 10 MHz is achievable
in IC implementations. Class-G and split-band modulators canbe used in wide-band applications.
E. Envelope Tracking
The envelope-tracking architecture is similar to that of the Kahn technique. The supply voltage is varied dynamically to conserve power, but with sufficient excess ("headroom") to allow the RF PA to operate in a linear mode. The RF drive contains both amplitude and phase information, and the burden of providing linear amplification lies entirely on the final RF PA. Typically, the envelope is detected and used to control a dc–dc converter. While both buck (step-down) oboost (step-up) converters are used, the latter is more common as it allows operation of the RF PA from a supply voltage higher than the dc-supplyvoltage. This configuration is also more amenable to the use ofn-p-n or n-channel transistors for fast switching. The result is a minimum corresponding to the dc-supply voltage and tracking of larger envelopes with a fixed headroom. If the RF PA is operated in class A, its quiescent current can also be varied.
The efficiency is significantly better than that of a linear RF
PA operating from a fixed supply voltage, but lower than that of the Kahn technique. The efficiency of a system based upon an ideal converter and class-B RF PA with headroom that is
10% of peak is included in Fig. 7 ("ET"). In practice, power consumption by the converter and other circuits further reduces the efficiency at lower output amplitudes.
A high switching frequency in the dc–dc converter allows both a high modulation bandwidth and the use of smaller inductors and capacitors. Converters with switching frequencies of 10–20 MHz have recently been implemented using MOS ASICs[24], GaAs HBTs [25], and RF-power MOSFETs [26]. The average efficiency for CDMA signals is typically increased from that of a conventional linear amplifier by a factor of 1.5–2.

F. Outphasing
Outphasing was invented by Chireix during the 1930s as a means of obtaining high-quality AM from vacuum tubes with poor linearity and was used through about 1970 in RCA "ampliphase" AM-broadcast transmitters. In the 1970s, it came into use at microwave frequencies under the name LINC (i.e., linear amplification using nonlinear components). An outphasing
transmitter (Fig. 17) produces an amplitude-modulated signal by combining the outputs of two PAs driven with signals of different time-varying phases. Basically, the phase modulation
causes the instantaneous vector sum of the two PA outputsto follow the desired signal amplitude. The inverse sine of envelope phase modulates the driving signals for the two
PAs to produce a transmitter output that is proportional to . In a modern implementation, a DSP and synthesizer produce the inverse-sine modulations of the driving signals. Virtually all microwave outphasing systems in use today employ hybrid combiners to isolate the two PAs from each other and to allow them to see resistive loads at all signal levels. However, both PAs deliver full power all of the time. Consequently, the efficiency of a hybrid-coupled outphasing transmitter varies with the output power (as in a class-A PA), resulting in an average efficiency that is inversely proportional to peak-to-average ratio (as in class A). Recovery of the power from the dump port
of the hybrid combiner offers some improvement in the efficiency. Summation of the out-of-phase signals in a nonhybrid combiner inherently results in variable reactive PA-load impedances.
If the combiner is untuned, the current drawn from the PAs is proportional to the transmitter output voltage, resulting in an efficiency characteristic that varies with signal amplitude, as in a similar class-B PA. The Chireix technique uses shunt reactances on the inputs to the combiner to tune out the drain reactances at a particular amplitude, which, in turn, maximizes the efficiency
in the vicinity of that amplitude. In the classic implementation, the efficiency is maximized at the level of the unmodulated AM carrier and remains high over the upper 6 dB of the output range
(Fig. 7) and for about 8 dB into backoff (Fig. 8).With judicious choice of the shunt susceptances, the average efficiency can be maximized for any given signal [27]. For example, the average efficiency for a multicarrier signal with a 10-dB peak-to-average ratio can be boosted from the 28% of class B to 52.1%. Simulations suggest that nonhybrid combining of microwave PAs increases both efficiency and distortion [28].

G. Doherty Technique
The classical Doherty architecture (Fig. 18) combines two PAs of equal capacity through quarter-wavelength lines or networks. The "carrier" (main) PA is biased in class B, while the "peaking" (auxiliary) PA is biased in class C. Only the carrier PA is active when the signal amplitude is half or less of the PEP amplitude. Both PAs contribute output power when the signal amplitude is larger than half of the PEP amplitude. Operation of the Doherty system can be understood by dividing it into low-power, medium-power (load-modulation), and peak-power regions [29]. In the low-power region, the
speaking PA remains cut off and appears as an open circuit. The carrier PA, therefore, sees a 100- load and operates as an ordinary class-B amplifier. The instantaneous efficiency increases linearly with output, reaching the 78.5% of ideal class B at saturation of the carrier PA at 6 dB from transmitter PEP. As the signal amplitude increases into the medium-power region,
the peaking PA becomes active. The additional current sent to the load by the peaking PA causes the apparent load
impedance at to increase above the 25 of the low-power region. Transformation through the quarter-wavelength line results
in a decrease in the load presented to the carrier PA.
carrier PA remains in saturation and acts as a voltage source. It operates at peak efficiency and delivers an increasing amount of power. At PEP output, both PAs see 50- loads and each delivers
half of the system output power. The PEP efficiency is ideally the 78.5% of class-B PAs.

Power Amplifiers and Transmitters for RF and Microwave (3/4)
http://www.ee.pucrs.br/~decastro/pdf/RF&MicrowavePowerAmp&XMTRs.pdf
CRF_Zambrano C. Jaydi D.

Power Amplifiers and Transmitters for RF and Microwave (2/4)/IV. PAS

RF PAs are commonly designated as classes A–F [3]. Classes of operation differ in the method of operation, efficiency, and power-output capability. The "power-output capability" ("transistor utilization factor") is defined as output power per transistor normalized for peak drain voltage and current of 1 V and 1 A, respectively. The basic single-ended topology (Fig. 5) includes an active device, dc feed, and output filter/matching network.
Transformer-coupled and complementary topologies are also used. The drain voltage and current waveforms of selected ideal PAs are shown in Fig. 6.

A. RF-Power Transistors
RF PAs utilize a wide variety of active devices, including bipolar-junction transistors (BJTs), MOSFETs, JFETs (SITs), GaAs MESFETs, HEMTs, pHEMTs, and vacuum tubes [6],


The power-output capabilities range from tens of kilowatts for vacuum tubes to hundreds of watts for Si MOSFETs at HF and VHF to hundreds of milliwatts for InP HEMTs at MMW
frequencies. Depending upon frequency and power, devices are available in packaged, chip, and MMIC form. Virtually all RF-power transistors are n-p-n or n-channel types because the
greater mobility of electrons (versus holes) results in better operation at higher frequencies. While the voltages and currents differ considerably, the basic principles for power amplification
are common to all devices.

B. Methods of Amplification
Class A: In class-A amplification, the transistor is in the active region at all times and acts as a current source controlled by the gate drive and bias. The drain–voltage and drain–current waveforms are sinusoids. This results in linear amplification with an output power of , where output voltage on load cannot exceed supply voltage . The dc-power input is constant, hence, the instantaneous efficiency (Fig. 7) is proportional to the power output and reaches 50% at PEP. The average efficiency is inversely proportional to the peak-to-average ratio (e.g., 5% for 10 dB) and backoff (Fig. 8). For amplification of amplitude-modulated signals, the quiescent current can
be varied in proportion to the instantaneous signal envelope. The utilization factor is 1/8. Class A offers high linearity, high gain, and operation close to the maximum operating frequency of the
transistor.

Class B: The gate bias in a class-B PA is set at the threshold of conduction so the transistor is active half of the time and the drain current is a half-sinusoid. Since the amplitude of the
drain current is proportional to drive amplitude, class B provides linear amplification. The instantaneous efficiency varies linearly with the RF-output voltage and reaches (78.5%) at PEP for an ideal PA. For low-level signals, class B is significantly more efficient than class A, and its average efficiency can be several times that of class A at high peak-to-average ratios (e.g., 28% versus 5% for dB). The utilization factor is the same 0.125 of class A. Class B is widely used in broad-band transformer-coupled PAs operating at HF and VHF. It is finding increasing use in microwave PAs, including experimental PAs using complementary devices.

Class C: The gate of a classical (true) class-C PA is biased below threshold so that the transistor is active for less than half of the RF cycle. Linearity is lost, but efficiency can be increased arbitrarily toward 100% by decreasing the conduction angle toward zero. Unfortunately, this causes the output power (utilization factor) to decrease toward zero and the drive power to increase toward infinity. A typical compromise is a conduction
angle of 150 and an ideal efficiency of 85%. When driven into saturation, efficiency is stabilized and the output voltage is locked to supply voltage, allowing linear high-level amplitude modulation. Classical class C is widely used in high-power vacuum-tube transmitters, but is generally impractical for solidstate PAs.

Class D: Class-D PAs use two or more transistors as switches to generate square drain–voltage (or current) waveforms. A series-tuned output filter passes only the fundamental- frequency component to the load, resulting in a power outputs of for the transformer-coupled configuration. Current is drawn only through the transistor that is on, resulting in a 100% efficiency for an ideal PA. The utilization factor is the highest of any PA. If the switching is sufficiently fast, efficiency is not degraded by reactance in the load. Practical class-D PAs suffer from losses due to saturation, switching speed, and drain capacitance. Finite switching speed causes the transistors to be in their active regions while conducting current. Drain capacitances must be charged and discharged once per RF cycle, resulting in power loss that is proportional to [8] and increases directly with frequency. Class-D PAs with power outputs of 100 W to 1 kW are readily implemented at HF, but are seldom used above lower VHF because of losses associated with the drain capacitance. Recently, however, experimental class-D PAs have been tested with frequencies of operation as high as 1 GHz [9].

Class E: Class E employs a single transistor operated as a switch [10]. The drain–voltage waveform is the result of the sum of the dc and RF currents charging the drain-shunt capacitance. In optimum class E, the drain voltage drops to zero and has zero slope just as the transistor turns on. The result is an ideal efficiency of 100%, elimination of the losses associated with charging the drain capacitance in class D, reduction of switching losses, and good tolerance of component variation. Optimum class-E operation requires a drain shunt susceptance of and a drain series reactance . It delivers a power output of for an ideal PA with a utilization factor of 0.098. Variations in load impedance and shunt susceptance cause the PA to deviate from optimum operation, but the degradations in performance are generally no worse than those
for classes A and B.
The capability for efficient operation in the presence of significant drain capacitance makes class E useful in a number of applications. High-efficiencyHFPAswithpower levels to1 kWcan be implemented using low-cost MOSFETs intended for switching rather thanRFuse [11]. ClassEhas been used for high-efficiency amplification at frequencies as high as -band [12].

Class F: Class F boosts both efficiency and output by using harmonic resonators in the output network to shape the drain waveforms. The voltage waveform includes one or more odd harmonics and approximates a square wave, while the current includes even harmonics and approximates a half sine wave. Alternately ("inverse class F"), the voltage can approximate a half sine wave and the current a square wave. As the number of harmonics increases, the efficiency of an ideal PA increases from the 50% (class A) toward unity (e.g., 0.707, 0.8165, 0.8656, 0.9045 for two, three, four, and five harmonics, respectively) and the utilization factor increases from 1/8 toward [13]. The required harmonics arise naturally from nonlinearities and saturation in the transistor. While class F requires a more complex output filter than other PAs, the impedances at the "virtual drain" must be correct at only a few specific frequencies.
A variety of modes of operation in-between classes C, E, and F are possible. The maximum achievable efficiency [13] dependsupon the number of harmonics. The utilization factor depends upon the harmonic impedances and is highest for ideal class-F operation.
C. Load–Pull Characterization RF-power transistors are characterized by breakdownvoltages and saturated drain currents. The load impedance for maximum power results in drain voltage and current excursions from near zero to nearly the maximum values. The load impedances corresponding to delivery of a given amount of RF power with a specified maximum drain voltage lie along parallel-resistance lines on the Smith chart. The impedances for a specified maximum current analogously follow a series-resistance line. For an ideal PA, the resultant constant-power contour is football shaped [14].
In a real PA, the "virtual drain" is embedded behind the drain capacitance and bond-wire/package inductance. Transformation of the ideal drain impedance through these elements causes
the constant-power contours to become rotated and distorted. With the addition of second-order effects, the contours become elliptical. As shown in the example of Fig. 9, the power and efficiency contours are not necessarily aligned, nor do maximum power and maximum efficiency necessarily occur for the same load impedance. Sets of such "load–pull" contours are widely used to facilitate design tradeoffs. Load–pull analyses are generally iterative in nature, as changing one parameter may produce a new set of contours. A variety of different parameters can be plotted during a load–pull analysis, including not only power and efficiency, but also gain, distortion, and stability. Harmonic impedances as well as drive impedances can also be varied. The variable impedance required for load–pull testing can be obtained by mechanical, electrical, or active techniques.

D. Microwave PAs
At microwave frequencies, lumped elements (capacitors, inductors) become unsuitable as tuning components and are used primarily as chokes and bypasses. Matching, tuning, and filtering at microwave frequencies are, therefore, accomplished with distributed (transmission-line) networks. Proper operation of PAs at microwave frequencies is achieved by providing the required drain–load impedance at the fundamental and a number of harmonic frequencies.
Class F: Typically, a transmission line between the drain and load provides the fundamental-frequency drain impedance of the desired value. A stub that is a quarter-wavelength at the harmonic of interest and open at one end provides a short circuit at the opposite end. The stub is placed along the main transmission line at either a quarter or a half-wavelength from the drain
to create either an open or short circuit at the drain [15]. The supply voltage is fed to the drain through a half-wavelength line bypassed on the power-supply end or alternately by a lumped-element choke. When multiple stubs are used, the stub for the highest controlled harmonic is placed near the drain. Stubs for lower harmonics are placed progressively further away and their lengths and impedances are adjusted to allow for interactions.

Typically, "open" means 3–10 times the fundamental-frequency impedance, and "shorted" means no more 1/10 to 1/3 of the fundamental- frequency impedance [13]. Dielectric resonators can be used in lieu of lumped-element traps. A wide variety of class-F PAs have been implemented at UHF and microwave frequencies. Generally, only one or two harmonic impedances are controlled. In one -band PA [16], for example, the output circuit provides a match at the fundamental and a short circuit at the second harmonic. The third-harmonic impedance is high, but not explicitly adjusted to be open. The
3-dB bandwidth of such an output network is about 20%, and the efficiency remains within 10% of its maximum value over a bandwidth of 15%.

Class E: The drain–shunt capacitance and series inductive reactance required for optimum class-E operation result in a drain impedance of at the fundamental frequency, at the second harmonic, and proportionately smaller capacitive reactances at higher harmonics. At microwave frequencies, class-E operation is approximated by providing the drain with the fundamental frequency impedance and preferably one or more of the harmonic impedances [17]. An example of a microwave approximation of class E that provides the correct fundamental and second harmonic impedances [16], [17] is shown in Fig. 10. The stub immediately to the right of the FET is a quarter-wavelength long at the second harmonic so that the open circuit at its upper end is transformed to a short at its lower end. The line at the drain in combination with drain capacitance and inductance is also a quarter-wavelength to translate the short on its right end to an open at the virtual drain. The remaining lines provide the desired impedance at the fundamental. This circuit uses an FLK052 MESFET to produce 0.68 W at -band with a drain efficiency of 72% and PAE of 60%.
Methods exist for providing the proper impedances through the fourth harmonic [18]. However, the harmonic impedances are not critical [13], and many variations are, therefore, possible. Since the transistor often has little or no gain at the higher harmonic frequencies, those impedances often have little or no effect upon performance. A single-stub match is often sufficient to provide the desired impedance at the fundamental while simultaneously providing an adequately high impedance at the second harmonic, thus eliminating the need for an extra stub
and reducing a portion of the losses associated with it. Most microwave class-E amplifiers operate in a suboptimum mode. Demonstrated capabilities range from 16Wwith 80% efficiency at UHF (LDMOS) to 100 mW with 60% efficiency at 10 GHz[10], [17], [19].
Comparison: Classes AB and F have essentially the same saturated output power, but class F has about 15% higher efficiency and class E has the highest efficiency [19]. Gain compression occurs at a lower power level for class E than for class F. For a given efficiency, class F produces more power. For the same maximum output power, the third-order IMD products are about 10 dB lower for class F than for class E. Lower power PAs implemented with smaller RF-power devices tend to be more efficient than PAs implemented with larger devices.

Power Amplifiers and Transmitters for RF and Microwave (2/4)
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CRF_Zambrano C. Jaydi D.

Power Amplifiers and Transmitters for RF and Microwave (1/4)

The generation of RF/microwave power is requirednot only in wireless communications, but also in applications such as jamming, imaging, RF heating, and miniature dc/dc converters.
Each application has its own unique requirements for frequency, bandwidth, load, power, efficiency, linearity, and cost. RF power isgenerated by a wide variety of techniques, implementations, and active devices. Power amplifiers are incorporated into transmitters in a similarly wide variety of architectures, including linear, Kahn, envelope tracking, outphasing, and Doherty. Linearity can be improved through techniques such as feedback, feedforward, and predistortion.

I. INTRODUCTION
Apower amplifier (PA) is a circuit for converting dc-input power into a significant amount of RF/microwave output power. In most cases, a PA is not just a small-signal amplifier driven into saturation. There exists a great variety of different PAs, and most employ techniques beyond simple linear amplification. A transmitter contains one or more PAs, as well as ancillary circuits such as signal generators, frequency converters, modulators, signal processors, linearizers, and power supplies. The classic architecture employs progressively larger PAs to boost a low-level signal to the desired output power. However, a wide variety of different architectures in essence

disassemble and then reassemble the signal to permit amplification with higher efficiency and linearity.

In the early days of wireless communication (1895–mid-1920s), RF power was generated by spark, arc, and alternator techniques.With the advent of the DeForest audion in 1907, the thermoionic vacuum tube offered a means of generating and controlling RF signals, and vacuum-tube PAs were dominant from the late 1920s through the mid-1970s. Discrete solid-state RF-power devices began to appear at the end of the 1960s with the introduction of silicon bipolar transistors such as the 2N6093 [(75-W HF single sideband (SSB)] by RCA. Their
dominance in the 1980s brought about the use of lower voltages, higher currents, and relatively low load resistances. The 1990s saw a proliferation of a variety of new solid-state devices
including HEMT, pHEMT, HFET, and HBT, using a variety of new materials such as InP, SiC, and GaN. These devices offer amplification to 100 GHz or more and are in many cases grown to order in MMIC form. The combination of digital signal processing (DSP) and microprocessor control allows widespread use of complicated feedback and predistortion techniques to improve efficiency and linearity.
Modern applications are highly varied. Frequencies from VLF through millimeter wave (MMW) are used for communication, navigation, and broadcasting. Output powers vary from 10 mW in short-range unlicensed wireless systems to 1 MW in long-range broadcast transmitters. Almost every conceivable type of modulation is being used in one system or another.
PAs and transmitters also find use in systems such as radar, RF heating, plasma generation, laser drivers, magnetic-resonance imaging, and miniature dc/dc converters. No single PA or transmitter technique suits all applications. Many techniques that are now coming into use were devised decades ago, butm only recently made possible by advances in signal-processing and control technology.

II. LINEARITY
The need for linearity is one of the principal drivers in the design of modern PAs. Signals such as CW, FM, classical FSK, and GMSK (used in GSM) have constant envelopes (amplitudes) and, therefore, do not require linear amplification. Full-carrier amplitude modulation is best produced by high-level amplitude modulation of the final RF PA. Linear amplification is required when the signal contains both amplitude and phase modulation. Examples include SSB voice,

vestigal-sideband television (both NTSC and HDTV), modern shaped-pulse data modulation (QAM, QPSK, CDMA), and multiple carriers (OFDM).
The requirements for both high data rates and efficient utilization of the increasingly crowded spectrum necessitates the use of shaped data pulses in modern digital signals such as QPSK,
QAM, and CDMA. Most systems use raised-cosine shaping, which eliminates intersymbol interference during detection and allows the spectrum to be shaped arbitrarily close to rectangular
[1]. This requires the transmission of square-root–raised-cosine (SRRC) data pulses that look much like truncated sinc functions. The resultant modulated carrier (Fig. 1) has simultaneous amplitude and phase modulation with a peak-to-average ratio of 3–6 dB.
Applications such as cellular base-stations, satellite repeaters, and active phased arrays require the simultaneous amplification of multiple signals. The signals can, in general, have different amplitudes, different modulations, and irregular frequency spacing. In a number of applications including HF modems and digital broadcasting, it is more convenient to use a large number of carriers with low data rates than a single carrier with a high data rate. Orthogonal frequency division multiplex (OFDM) [2] employs carriers with the same amplitude and modulation, separated in frequency so that modulation products from one carrier are zero at the frequencies of the other carriers.
The resultant composite signal (Fig. 1) has a peak-to-average ratio in the range of 8–13 dB. Distortion of the amplified signal can be caused by both amplitude nonlinearity (such as a variable gain) or amplitude-tophase conversion (produced, for example, by a voltage-variable
capacitance). The result is splatter into adjacent channels and impairment of detection. Linearity is characterized, measured, and specified by various techniques, depending upon the specific
signal and application.
The carrier-to-intermodulation (C/I) ratio, compares the amplitude of the desired output carriers to the intermodulation-distortion (IMD) products [3]. Noise-power ratio (NPR) is the ratio of the notch power to the total signal power when a PA is driven by noise with a spectral notch. Adjacent channel power ratio (ACPR) compares the power in an adjacent channel to that of the signal (Fig. 2). It is currently the most widely used measure of linearity, but defined differently for each application. Error vector magnitude (EVM) is the distance between the desired and actual signal vectors.

III. EFFICIENCY
Efficiency, like linearity, is a critical factor in PA design.
Three definitions of efficiency are commonly used. Drain efficiency is defined as the ratio of RF-output power to dc-input



power, n=Po/Pi. Power-added efficiency (PAE) incorporates the RF-drive power by subtracting it from the output power, (Po-Pdr)/Pi . PAE gives a reasonable indication of PA performance when gain is high; however, it can become negative for low gains. An overall efficiency such as is usable in all situations. This definition can be varied to include driver dc-input power, the power consumed by supporting circuits, and anything else of interest.
The instantaneous efficiency is the efficiency at one specific output level. For most PAs, the instantaneous efficiency is highest at the peak output power (PEP) and decreases as output decreases. When amplifying signals with time-varying amplitudes, a useful measure of performance is the average efficiency, which is defined [4] as the ratio of the average outuput power to the average dc-input power.
The probability-density function (PDF) gives the relative amount of time an envelope spends at various amplitudes (Fig. 3). The PDF of an SRRC signal must generally be determined by simulation or measurement. Multiple carriers produce random-phasor sums and, therefore, have Rayleigh-distributed envelopes. The average input and output powers are found by
integrating the product of the variable of interest and the PDF of the envelope over the range of the envelope.

The need to conserve battery power and to avoid interference to other users operating on the same frequency necessitates the transmission of signals whose peak amplitudes are well below the PEP of the transmitter. Since peak power is needed only in the worst-case links, the "backoff" is typically in the range of 10–20 dB. For a single-carrier mobile transmitter, backoff
rather than envelope PDF is dominant in determining the average power consumption and average efficiency. The PDF of the transmitting power (Fig. 4) depends not only upon the distance, but also upon factors such as attenuation by buildings, multipath, and orientation of the mobile antenna.

Power Amplifiers and Transmitters for RF and Microwave (1/4)
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CRF_Zambrano C. Jaydi D.

NONLINEAR MODELING OF RF/MICROWAVE CIRCUITS FOR MULTI-TONE SIGNAL ANALYSIS

I. INTRODUCTION
The scarcity of spectrum bandwidth available for present and upcoming wireless telecommunications systems has been entailing an increased demand for more linear circuits, stringent distortion figures of merit and progressively involved laboratory characterization procedures [1]-[3].
Microwave/RF nonlinear CAD/CAE now play a dominant role in the telecommunications equipment designprocess, because the inherent circuits' complexity, associated with the impossibility of getting exact closed-formsolutions of even the simplest circuit responses to nonlinear regimes, obviates any attempt of hand-made calculations. This, in turn, asks for more ingenious simulation techniques, but also accurate nonlinear device models and model extraction procedures, since the analysis results can be, at most, as good as the adopted circuit's representation.
Recent advances in the former of these research fields now allow the determination of most circuits' response to a reasonably wide range of stimuli that span from the most simple single-tone or two-tone excitation, to the more complex digital modulated carriers, multi-carrier signals or even band-limited noise. The nonlinear device model field has also sensed some recent advances that can be grouped into the levels of model formulation and model extraction procedures. Although there are still some work going on for proposing mathematical black-box models or 2D/3D physics device models [4], [5], it seams that empirical modeling based in equivalent circuit descriptions has, for this time, won the nonlinear device modeling dispute. Such models associate empirical mathematical functions to nonlinear controlled sources (usually voltage dependent current sources) or charges/fluxes (for quasi-static nonlinear capacitors or inductors), which are then embedded in a linear equivalent circuit network. Therefore, nonlinear device model extraction procedure becomes a two-step process, in which, first, the linear equivalent circuit topology is obtained, and then, the parameter sets of the sources or charges (or fluxes) are determined. Despite any time a new device family comes into use, novel equivalent circuit topologies and element extraction methods are proposed, it seams that equivalent circuits of classic devices like the BJT/HBT or the MESFET/HEMT are almost stabilized. So, the research efforts have been progressively directed to the nonlinear element model formulations and their parameter extraction methodologies.
The present paper addresses this problem by using the theoretical framework of Volterra Series to state some basic conditions of nonlinear system identification. Then, the methodologies commonly in use are discussed under that theory, and the author's approach is proposed. That is finally validated by showing good agreement between nonlinear simulation results and laboratory measurements of a medium power amplifier driven with band-limitedwhite noise.

II. THEORETICAL BASIS FOR THE EXTRACTION OF DYNAMIC NONLINEAR DEVICE MODELS
For the sake of simplicity, let us consider a certain class of single-input/single-output dynamic nonlinear systems that can be represented by a Volterra Series. If their input signal, xi(t), is composed of a combination of various sinusoids:
then, their output response, yo(t), can be represented by the following series:

in which the Hn(ωq1, …, ωqn) are the so-called nonlinear transfer functions, NLTF, that completely identify the system up to order n. The Volterra Series representation then states that the unique identification of an n'th order system requires the determination of n NLTFs, which depend on a number of frequencies equal to their order. That is, 1st order NLTF (or linear transfer function) demands for several tests using one sinusoid, 2nd order NLTF demands for several tests using two independent sinusoids, and the complete n'th order NLTF identification requiresthe use of n independent sinusoids, or n degrees of freedom.
Expressing that in more usual terms, it means that we cannot rely on simple one-tone tests to extract the 2nd order system behavior, since 2nd harmonic do not contain all information needed, nor we should rely on one-tone or twotone tests to extract even the simplest 3rd order system's characteristics. This has already been widely recognized by the scientific community dedicated to the modeling extraction field, which led to first use one-tone tests in nonlinear regimes, like the so-called large-signal S-parameters [6], and more recently two-tone excitations of large amplitude.
Indeed, that has been driven by the need to specify and build what could be assumed as an ideal nonlinear network analyzer [7]-[10]. Unfortunately, expression (2) discourages further attempts in this way since it states that the extraction of an accurate dynamic nonlinear system of n order requires laboratory measurements of system responses to n independent sinusoids. And this seams to be unavoidable, unless it is assumed that the system is mildly nonlinear (and thus requiring only a small number of NLTFs) or it exhibits a rapidly vanishing memory compared to the timescales of signals it is intended to operate.
The first hypothesis should be discarded in general, as it is widely recognized that, e.g., 3rd order Volterrarepresentations are insufficient to describe common amplifiers driven close to, or above, their 1dB compression point. It is, thus, only utilized for simulating quasi-linear distortion regimes. The second hypothesis is even more unreasonable as microwave devices are continuously driven to their frequency limits.
There is, however, an alternative that can be explored. Since SPICE-like or harmonic-balance simulators require equivalent circuit models where the elements are either dynamic, but linear, or nonlinear, but quasi-static, the proposed way could be to first wipe out the device from its memory by extracting the linear reactive elements, and then simply extract the reminding memoryless nonlinear model descriptions of the dependent sources or charges.
For the concrete case of two-port nonlinear devices, like microwave/RF transistors, that could be implemented following the approach described in Werthof et al. [11]. There, the linear sub-network was considered as a four-port, described, as in Fig. 1, by a set of linear parameters, e. g. a 4x4 [Y] matrix, in which the 1st and 2nd ports are the access ports, and the 3rd and 4th are connections to the nonlinear four-pole.
Since it is assumed that the required linear equivalent circuit extraction procedure has been previously performed, the 4x4 [Y] matrix is uniquely known, as are the 1st order linear transfer functions of the nonlinear sub-network.
Thus, it is always possible to deduce (I3, I4, V3, V4) from the access currents and voltages (I1, I2, V1, V2), and the nonlinear two-port becomes de-embedded.

III. ILLUSTRATIVE MODEL EXTRACTION EXAMPLE
For extracting the memoryless nonlinear model descriptions, several considerations must also be taken into account, which will determine the selection of the optimum functional descriptions and the sets of measurements to be performed. In particular, if multi-tone distortion simulations are desired, then a good mathematical function is necessary that is able to approximate, not only the overall device I/V characteristics, but also their first three derivatives. That is especially important when usual approximation routines (like minimum squares) or more involved approaches (like neural networks) are used [12].
Our best results were obtained by beginning with a study of the detailed device characteristics from physicssimulations. From that, we proposed an approximating function [13], [14], and then extracted the model parameters
from DC and small-signal S-parameter data (0'th and 1st order model characteristics), but also from 2nd and 3rd
order derivatives obtained from 2nd and 3rd order harmonic and intermodulation distortion (2nd and 3rd order model characteristics) [15], [16].
Fig. 2 represents the equivalent circuit model extracted for our sample device - a general-purpose GaAs MESFET- using the now classic method of Dambrine [17].
The adopted nonlinear drain-source current model and gate-source charge model are shown in (3-6) and (7), respectively,
while Fig. 3 and Fig. 4 present a comparison between the correspondent extracted and model predicted 1st, 2nd and 3rd order derivatives.


IV. NONLINEAR DEVICE MODEL VALIDATION
In order to validate the proposed model extraction methodology, a microwave amplifier was then built around our GaAs MESFET and tested under common one-tone, two-tone and also very demanding band-limited white noise excitations. Fig. 5 represents a comparison between measured and simulated Adjacent Channel Power Ratio, ACPR, power sweep data, corresponding to this latter stimulus. Fig. 6, depicts the whole spectrum of one of these power points where not only the adjacent-channel distortion, but also the co-channel distortion is shown. The distortion measurements were made by the authors' Co-Channel Distortion Ratio, CCPR, measurement set-up [2], and the simulations were obtained with the extracted model inserted in an in-house developed artificial frequency mapping based harmonic-balance simulator [18].

IV. CONCLUSIONS
This paper presented a well-defined methodology for the extraction of nonlinear device models. It starts by first deembedding the memoryless nonlinear elements from the linear sub-circuit network of the equivalent circuit. Then, it proposes convenient approximating functions for the nonlinearities, and extracts their parameter sets from largesignal DC I/V, small-signal S parameters and intermodulation measurements. The validity of that approach was shown by comparing measured and simulated distortion characteristics of a microwave amplifier driven by bandlimited white noise.

NONLINEAR MODELING OF RF/MICROWAVE CIRCUITS FOR MULTI-TONE SIGNAL ANALYSIS
fuente:http://www.av.it.pt/nbcarvalho/docs/ci23.pdf
CRF_Zambrano C. Jaydi D.