1.Varactor
Símbolo Varicap esquemática
En electrónica, Un diodo varicap, diodo varicap, diodo de capacidad variable, diodo de reactancia variable o Tuning diodo es un tipo de diodo que tiene una variable capacitancia de que es una función de la tensión en sus terminales.
Los diodos varactores, también llamados varicap, VVC (capacitancia variable con el voltaje) son capacitores de semiconductores variables y dependientes del voltaje. Su modo de operación depende de la capacitancia que existe en la unión p-n cuando el elemento se polariza de forma inversa. Bajo condiciones de polarización inversa se estableció que existe una región de carga no cubierta sobre ambos lados de la unión, que junto con las regiones forman la región de agotamiento y define el ancho del agotamiento Wd. La capacitancia de transición (CT) establecida por las cargas aisladas no cubiertas está determinada por:
Donde ε es la permitividad de los materiales semiconductores, A es el área de la unión p-n y Wd es el ancho de agotamiento. Conforme el potencial de polarización inversa se incrementa, el ancho de la región de agotamiento se incrementa, lo que a su vez reduce la capacitancia de transición. En la figura se presentan las características de un diodo varicap típico, disponible comercialmente. Observe el declive agudo inicial en CT con el incremento en la polarización inversa. El rango normal de VR para los diodos VVC se limita a cerca de 20V. En términos de la polarización inversa aplicada, la capacitancia de transición está dada de forma aproximada por:
Donde K = constante determinada por el material semiconductor y la técnica de fabricación
VT = potencial en punto de inflexión
VR = magnitud del potencial de polarización inversa aplicado
n = ½ para uniones de aleación y 1/3 para uniones de difusión
En términos de la capacitancia en la condición de cero polarización C(0), la capacitancia como una función de VR está dada por:
En la figura 19.8 se muestran los símbolos que se utilizan con mayor frecuencia para el diodo varicap, así como una primera aproximación a su circuito equivalente en la región de polarización inversa.
Dado que nos encontramos en esta región de polarización inversa, la resistencia en el circuito equivalente es muy grande en magnitud (por lo regular de 1MΩ o mayor), mientras que RS la resistencia geométrica del diodo, es, como se señala en la figura 19.8 muy pequeña. La magnitud de CT variará desde cerca de 2 hasta 100 pF, según el varicap considerado. Para asegurar que RR es lo más grande posible (para un mínimo de corriente de fuga), se utiliza normalmente silicio en los diodos varicap. EL hecho de que el dispositivo se utilizará a frecuencias muy altas requiere que se incluya la inductancia Ls incluso cuando ésta se mida en nanohenrios. Recuerde que XL =2πfL y una frecuencia de 10 GHz dará por resultado que XLS = 2πfL = (6.28)(1010 Hz)(10-9F)= 62.8Ω. Obviamente existe un límite de frecuencia asociado con el uso de cada diodo varicap. Al asumir el rango de frecuencia apropiado y un valor bajo para RS y XLS en comparación con los otros elementos en serie, entonces el circuito equivalente para el varicap de la figura 19.8.a puede remplazarse sólo por el capacitor variable.
Aplicaciones
Operación
Estructura interna de una capacidad variable
Funcionamiento de un varicap
Todos los diodos presentan este fenómeno en cierta medida, pero especialmente hizo diodos varactor aprovechar el efecto de impulsar la capacidad y la variabilidad del rango logrado - la mayoría de los intentos de fabricación de diodos para lograr lo contrario.
En la figura se puede ver un ejemplo de un crossection de un varactor con el agotamiento de la capa formada por una unión pn. Pero el agotamiento de la capa también puede ser hecho de una MOSo un diodo de Diodo Schottky. Esto es de gran importancia en CMOS y MMIC tecnología.
El diodo de capacidad variable o Varactor (Varicap) es un tipo de diodo que basa su funcionamiento en el fenómeno que hace que la anchura de la barrera de potencial en una unión PN varíe en función de la tensión inversa aplicada entre sus extremos. Al aumentar dicha tensión, aumenta la anchura de esa barrera, disminuyendo así la capacidad del diodo. De este modo se obtiene un condensador variable controlado por tensión. Los valores de capacidad obtenidos van desde 1 a 500 pF. La tensión inversa mínima tiene que ser de 1 V.
La capacidad formada en los extremos de la unión PN puede resultar de gran utilidad cuando, al contrario de lo que ocurre con los diodos de RF, se busca precisamente utilizar dicha capacidad en provecho del circuito en el cual se está utilizando el diodo. Al polarizar un diodo de forma directa se observa que, además de las zonas constitutivas de la capacidad buscada, aparece en paralelo con ellas una resistencia de muy bajo valor óhmico, lo que conforma un capacitor de elevadas pérdidas. Sin embargo, si polarizamos el mismo en sentido inverso la resistencia en paralelo que aparece es de un valor muy alto, lo cual hace que el diodo se pueda comportar como un capacitor con muy bajas pérdidas. Si aumentamos la tensión de polarización inversa las capas de carga del diodo se esparcían lo suficiente para que el efecto se asemeje a una disminución de la capacidad del hipotético capacitor (el mismo efecto producido al distanciar las placas del un capacitor estándar). Por esta razón podemos terminar diciendo que los diodos de capacidad variable, varían su capacidad interna al ser alterado el valor de la tensión que los polariza de forma inversa. La utilización más solicitada para este tipo de diodos suele ser la de sustituir a complejos sistemas mecánicos de capacitor variable en etapas de sintonía en todo tipo de equipos de emisión y recepción, ejemplo, cuando cambiamos la sintonía de un receptor antiguo, se varía mecánicamente el eje de un capacitor variable en la etapa de sintonía; pero si por el contrario, pulsamos un botón de sintonía de un receptor de televisión moderno, lo que hacemos es variar la tensión de polarización de un diodo varicap que se encuentra en el módulo sintonizador del TV.
La aplicación de estos diodos se encuentra, sobre todo, en la sintonía de TV, modulación de frecuencia en transmisiones de FM y radio y en los osciladores controlados por voltaje (oscilador controlado por tensión).
En tecnología de microondas se pueden utilizar como limitadores: al aumentar la tensión en el diodo, su capacidad varía, modificando la impedancia que presenta y desadaptando el circuito, de modo que refleja la potencia incidente.
Frequency Multipliers
Varactor frequency multipliers are extensively used to provide LO power to sensitive millimeter- and submillimeter-wavelength receivers. Today, frequency multipliers are the main application of varactors. Solid-state multipliers are relatively inexpensive, compact, lightweight, and reliable compared to vacuum tube technology, which makes them suitable for space applications at these frequencies. State-of-the-art balanced Schottky doublers can deliver 55 mW at 174 GHz8 and state-of-the-art four-barrier HBV triplers deliver about 9 mW at 248 GHz.9 Frequency multiplication or harmonic generation in devices occur due to their nonlinearity. Based on whether the multiplication is due to a nonlinear resistance or a nonlinear reactance, one can differentiate between the varistor and varactor type of multipliers. Varactor type multipliers have a high potential conversion efficiency, but exhibit a narrow bandwidth and a high sensitivity to operating conditions. According to the Page-Pantell inequality, multipliers that depend upon a nonlinear resistance have at most an efficiency of 1/n2, where n is the order of multiplication.10,11 The absence of reactive energy storage in varistor frequency multipliers ensures a large bandwidth. For the ideal varactor multiplier, i.e., a lossless nonlinear reactance, the theoretical limit is a conversion efficiency of 100% according to the Manley-Rowe formula. However, real devices exhibit properties and parameters that are a mixture of the ideal varistor and the ideal varactor multiplier (see Figure 1.4). The following set of parameters is used to describe and compare properties of frequency multipliers:
• Conversion loss, Ln, is defined as the ratio of the available source power, PAVS, to the output harmonic power, Pn, delivered to the load resistance. It is usually expressed in decibels. The inverted value of Ln, i.e., the conversion efficiency, ηn, is often expressed as a percent.
• In order to minimize the conversion loss, the optimum source and load embedding impedances, ZS and ZL, should be provided to the diode. Optimum source and load impedances are found from maximizing, e.g., the conversion efficiency, and they depend on each other and on the input signal level. In a nonlinear circuit, such as a multiplier, it is not possible to define a true impedance. However, a "quasi-impedance", Zn, can be defined for periodic signals as
where Vn and In are the voltage and the current, respectively, at the nth harmonic. Basic Principles of Single Diode Frequency Multipliers — Single diode frequency multipliers can either be shunt or series mounted. In both cases the input and the output filter should provide optimum embedding impedances at the input and output frequencies, respectively. The output filter should also provide an open circuit for the shunt-mounted varactor and a short circuit for the series-mounted varactor at the pump frequency. The same arguments apply to the input filter at the output frequency. Analysis and design of conventional doublers and high order varactor multipliers are described well in the book by Penfield et al.1 and in Reference 12.
In addition to the above conditions, the correct impedances must be provided at the idler frequencies for a high order multiplier (e.g., a quintupler). In general, it is hard to achieve optimum impedances at the different harmonics simultaneously. Therefore, a compromise has to be found. Performance of Symmetric Varactor Frequency Multipliers — In Fig. 1.5 a calculation of the minimum conversion loss for a tripler and a quintupler is shown. To systematically investigate how the tripler and quintupler performance depends on the shape of the S-V characteristic, a fifth degree polynomial model was employed by Dillner et al.13 The best efficiency is obtained for a S-V characteristic with a large nonlinearity at zero volts or a large average elastance during a pump cycle. The optimum idler circuit for the quintupler is an inductance in resonance with the diode capacitance (i.e., maximized third harmonic current).
Practical Multipliers — Since frequency multipliers find applications mostly as sources at higher millimeter and submillimeter wave frequencies, they are often realized in waveguide mounts14,15 (see Fig. 1.6). A classic design is the arrangement of crossed rectangular waveguides of widths specific for the input and output frequency bands. The advantages are:
• The input signal does not excite the output waveguide, which is cut off at the input frequency.
• Low losses.
• The height of the waveguide in the diode mounting plane may be chosen to provide the electrical matching conditions. Assuming a thin planar probe, the output embedding impedance is given by analytical expressions.
Today, whole waveguide mounts can be analyzed and designed using commercially available high frequency electromagnetic CAD tools. They either solve Maxwell's equations in the frequency domain or in the time domain using the FDTD method. The inherently limited bandwidth of varactors can be improved by employing a transmission line periodically loaded with varactors.17,18 These NonLinear Transmission Lines (NLTLs) can currently provide the largest bandwidth and still achieve a reasonable conversion efficiency as a frequency multiplier. Simultaneous effects of nonlinearity and dispersion may also be used for pulse compression (soliton propagation).
Frequency Converters
The varactor is useful as a frequency converter because of its good noise properties, and because gain can be achieved. The nonlinear reactance is pumped at a frequency fp, and a small signal is introduced at a frequency fs. Power will exchange at frequencies of the form nfp + fs (n can be negative). If the output frequency is higher than the input frequency, the varactor acts as an upconverter, otherwise it is a downconverter. Furthermore, one differ between lower (n = –1) and upper sideband (n = 1) converters, and according to whether or not power is dissipated at any other sidebands (idlers).
Assume that the elastance is pumped sinusoidally (i.e., Sk = 0 for _k_ > 1), the varactor is open-circuited at all frequencies except fs, fp, fu = fp + fs, and that the varactor termination tunes out the average elastance. The source resistance is then adjusted to give optimum gain or optimum noise temperature. For the upper sideband upconverter, the minimal noise temperature is
When the source resistance is
As predicted by the Manley-Rowe formula for a lossless varactor, the gain increases as the output frequency, fu, increases. The effect of an idler termination at fI = fp – fs can further increase the gain and reduce the noise temperature. The above expressions for optimum noise and the corresponding source impedance are valid for the lower sideband upconverter as well. However, the lower sideband upconverter may have negative input and output resistances and an infinite gain causing stability problems and spurious oscillations. All pumped varactors may have such problems. With a proper choice of source impedance and pump frequency, it is possible to simultaneously minimize the noise and make the exchangeable gain infinite.
This occurs for an "optimum" pump frequency of fp = or approximately m1 fc if the signal frequency is small. Further information on how to analyze, designs, and optimize frequency converters can be found in the book by Penfield et al.
Parametric Amplifiers
The parametric amplifier is a varactor pumped strongly at frequency fp, with a signal introduced at frequency fs. If the generated sidebands are terminated properly, the varactor can behave as a negative resistance at fs. Especially the termination of the idler frequency, fp – fs, determines the real part of the impedance at the signal frequency. Hence, the varactor can operate as a negative resistance amplifier at the signal frequency, fs. The series resistance limits the frequencies fp and fc for which amplification can be achieved and it also introduces noise. The explanation of the effective negative resistance can be described as follows: The application of signal plus pump power to the nonlinear capacitance causes frequency mixing to occur. When current is allowed to flow at the idler frequency fp – fs, further frequency mixing occurs at the pump and idler frequencies. This latter mixing creates harmonics of fp and fp – fs, and power at fs is generated. When the power generated through mixing exceeds that being supplied at the signal frequency fs, the varactor appears to have a negative resistance. If idler current is not allowed to flow, the negative resistance vanishes. Assuming that the elastance is pumped sinusoidally (i.e., Sk = 0 for _k_ > 1), and the varactor is open circuited at all frequencies except fs, fp, fi = fp – fs, and that the varactor termination tunes out the average elastance, gain can only be achieved if
where Ri is the idler resistance. By terminating the varactor reactively at the idler frequency, it can be shown that a parametric amplifier attains a minimum noise temperature when pumped at the optimum pump frequency, which is exactly the same as for the simple frequency converter. This is true for nondegenerated amplifiers where the frequencies are well separated. The degenerate parametric amplifier operates with fi close to fs, and can use the same physical circuit for idler and signal frequencies. The degenerate amplifier is easier to build, but ordinary concepts of noise figure, noise temperature, and noise measure do not apply.
Voltage Tuning
One important application of varactors is voltage tuning. The variable capacitance is used to tune a resonant circuit with an externally applied voltage. This can be used to implement a Voltage Controlled Oscillator (VCO), since changing the varactor capacitance changes the frequency of oscillation within a certain range. As the bias is increased, the resonant frequency fo increases from fo,min to fo,max as the elastance changes from Smin to Smax. If the present RF power is low, the main limitations are the finite tuning range implied by the minimum and maximum elastance and the fact that the series resistance degrades the quality factor, Q, of the tuned circuit. The ratio of the maximum and minimum resonant frequency gives a good indication of the tenability
However, if the present RF power level is large, the average elastance, which determines the resonant frequency, depends upon drive level as well as bias. Second, the allowed variation of voltage is reduced for large RF power levels. Since the varactor elastance is nonlinear, quite steep at low voltages, and almost flat at high voltages, the VCO tuning range is not naturally linear. However, an external bias circuit can improve the linearity of the VCO tuning range. It is also possible to optimize the doping profile of the varactor in terms of linearity, Q-value, or elastance ratio.
Varactor Devices
Conventional Diodes
Common conventional varactors at lower frequencies are reverse biased semiconductor abrupt p+-n junction diodes made from GaAs or silicon. However, metal-semiconductor junction diodes (Schottky diodes) are superior at high frequencies since the carrier transport only relies on electrons (unipolar device). The effective mass is lower and the mobility is higher for electrons compared to holes. Furthermore, the metal-semiconductor junction can be made very precisely even at a submicron level. A reverse biased Schottky diode exhibits a nonlinear capacitance with a very low leakage current. High frequency diodes are made from GaAs since the electron mobility is much higher than for silicon. The hyperabrupt p+-n junction varactor diode has a nonuniform n-doping profile and is often used for voltage tuning. The n-doping concentration is very high close to the junction and the doping profile is tailored to improve elastance ratio and sensitivity. Such doping profiles can be achieved with epitaxial growth or by ion implantation.
The Heterostructure Barrier Varactor Diode
The Heterostructure Barrier Varactor (HBV), first introduced in 1989 by Kollberg et al. is a symmetric varactor. The main advantage compared to the Schottky diode is that several barriers can be stacked epitaxially. Hence, an HBV diode can be tailored for a certain application in terms of both frequency and power handling capability. Moreover, the HBV operates unbiased and is a symmetric device, thus generating only odd harmonics. This greatly simplifies the design of high order and broadband multipliers.
The HBV diode is an unipolar device and consists of a symmetric layer structure. An undoped high band gap material (barrier) is sandwiched between two moderately n-doped, low band gap materials. The barrier prevents electron transport through the structure. Hence, the barrier should be undoped (no carriers), high and thick enough to minimize thermionic emission and tunnelling of carriers. When the diode is biased a depleted region builds up (Fig. 1.7), causing a nonlinear CV curve. Contrary to the Schottky diode, where the barrier is formed at the interface between a metallic contact and a semiconductor, the HBV uses a heterojunction as the blocking element. A heterojunction, i.e., two
adjacent epitaxial semiconductor layers with different band gaps, exhibits band discontinuities both in the valence and in the conduction band. Since the distance between the barriers (>1000 Å) is large compared to the de Broglie wavelength of the electron, it is possible to understand the stacked barrier structure as a series connection of N individual barriers. A generic layer structure of an HBV is shown in Table 1.1.
The HBV Capacitance — The parallel plate capacitor model, where the plate separation should be replaced with the sum of the barrier thickness, b, the spacer layer thickness, s, and the length of the depleted region, w, is normally an adequate description of the (differential) capacitance. The depletion length is bias dependent and the layer structure is symmetric, therefore the elastance is an even function of applied voltage and is given by
where Vd is the voltage across the depleted region, Nd is the doping concentration in the modulation layers, b is the barrier thickness, s is the undoped spacer layer thickness, A is the device area, and εb and εd are the dielectric constants in the barrier material and modulation layers, respectively. The maximum capacitance or the minimum elastance, Smin, occurs at zero bias. However, due to screening effects, the minimum elastance, Smin, must include the extrinsic Debye length, LD, as:
To achieve a high Cmax/Cmin ratio, the screening length can be minimized with a sheet doping, Ns, at the spacer/depletion layer interface. The minimum capacitance, Cmin, is normally obtained for punch through condition, i.e., w = l, or when the breakdown voltage, Vmax, is reached.
An accurate quasi-empirical expression for the C-V characteristic of homogeneously doped HBVs has been derived by Dillner et al.19 The voltage across the nonlinear capacitor is expressed as a function of its charge as
Where T is the device temperature, q is the elementary charge, and Q is the charge stored in the HBV. The substrate is either highly doped or semi-insulating (SI), depending on how the device is intended to be mounted. The contact layers (Nos. 1 and 7) should be optimized for low losses. Therefore, the buffer layer (No. 1) must be relatively thick (δ ~ 3 μm) and highly doped for planar HBVs (see Fig. 1.8). The barrier itself can consist of different layers to further improve the blocking characteristic. The spacer prevents diffusion of dopants into the barrier layer and increases the effective barrier height. The thickness of the barrier layer will not influence the cutoff frequency, but it has some influence on the optimum embedding impedances. Hence, the thickness is chosen to be thick enough to avoid tunneling of carriers. Several III-V semiconductor material systems have been employed for HBVs. The best choices to date for HBVs are the lattice matched In0,53Ga0,47As/In0,52Al0,48 As system grown on InP substrate and the lattice matched GaAs/AlGaAs system grown on GaAs substrate. High dynamic cutoff frequencies are achieved in both systems. However, the GaAs/AlGaAs system is well characterized and relatively easy to process, which increases the probability of reproducible results. The In0,53GaAs/In0,52AlAs system exhibits a higher electron barrier and is therefore advantageous from a leakage current point of view. The thickness and doping concentration of the modulation layers should be optimized for maximal dynamic cutoff frequency. In the future, research on wide bandgap semiconductors (e.g., InGaN) could provide solutions for very high power HBVs, a combination of a II-VI barrier for low leakage current and a III-V modulation layer for high mobility and peak velocity. Today, narrow bandgap semiconductors from the III-V groups (e.g., InxGa1–xAs) seem to be the most suitable for submillimeter wave applications.
The Si/SiO2/Si Varactor
By bonding two thin silicon wafers, each with a thin layer of silicon dioxide, it is possible to form a structure similar to HBV diodes from III-V compounds. The SiO2 layer blocks the conduction current very efficiently, but the drawback is the relatively low mobility of silicon. If a method to stack several barriers can be developed, this material system may be interesting for lower frequencies where the series resistance is less critical.
The Ferroelectric Varactor
Ferroelectrics are dielectric materials characterized by an electric field and temperature-dependent dielectric constant. Thin films of BaxSr1–xTiO3 have been proposed to be used for various microwave applications. Parallel plate capacitors made from such films can be used in varactor applications. However, the loss mechanisms at strong pump levels and high frequencies have not yet been fully investigated.
2. Schottky Diode Frequency Multipliers
Heterodyne receivers are an important component of most high frequency communications systems and other receivers. In its simplest form a receiver consists of a mixer being pumped by a local oscillator. At lower frequencies a variety of local oscillator sources are available, but as the desired frequency of operation increases, the local oscillator source options become more limited. The "lower frequencies" limit has increased with time. Early transistor oscillators were available in the MHz and low GHz range. Two terminal transit time devices such as IMPATT and Gunn diodes were developed for operation in X and Ka band in the early 1970s. However, higher frequency heterodyne receivers were needed for a variety of communications and science applications, so alternative local oscillator sources were needed. One option was vacuum tubes. A variety of vacuum tubes such as klystrons and backward wave oscillators grew out of the radar effort during the Second World War. These devices were able to produce large amounts of power over most of the desired frequency range. However, they were large, bulky, expensive, and suffered from modest lifetimes. They were also difficult to use in small science packages. An alternative solid state source was needed and the technology of the diode frequency multiplier was developed beginning in the 1950s. These devices use the nonlinear reactance or resistance of a simple semiconductor or diode to produce high frequency signals by frequency multiplication. These multipliers have been a part of many high frequency communications and science applications since that time. As time passed the operating frequencies of both transistors and two-terminal devices increased. Silicon and GaAs transistors have been replaced by much higher frequency HFETs and HBTs with fmax values of hundreds of GHz. Two-terminal IMPATT and Gunn diodes can produce more than 100 milliwatts at frequencies above 100 GHz. However, the desired operating frequencies of communications and scientific applications have also increased. The most pressing needs are for a variety of science applications in the frequency range between several hundred GHz and several THz. Applications include space-based remote sensing of the earth's upper atmosphere to better understand the chemistry of ozone depletion and basic astrophysics to investigate the early history of the universe. Both missions will require space-based heterodyne receivers with near THz local oscillators. Size, weight, and prime power will be important parameters. Alternative approaches include mixing of infrared lasers to produce the desired local oscillator frequency from higher frequencies, and a multiplier chain to produce the desired frequency from lower frequencies. Laser-based systems with the desired output frequencies and powers are available, but not with the desired size and weight. Semiconductor diode-based frequency multipliers have the modest size and weight needed, but as of now cannot supply the required powers, on the order of hundreds of microwatts, needed for the missions. This is the subject of ongoing research.
The goal of this chapter is to briefly described the performance of diode frequency multipliers in order to better understand their performance and limitations. The chapter is organized as follows. The next section will describe the properties of Schottky barrier diodes, the most useful form of a varactor multiplier. The following section will describe the analytic tools developed to predict multiplier operation. Two limitations, the reactive multiplier described by Manley and Rowe and the resistive multiplier discussed by Page will be discussed. The results of these two descriptions can be used to understand the basic limits of multiplier operation. However, these analytic results do not provide enough information to design operating circuits. A more realistic computer-based design approach is needed. This will be discussed in the next section. Limitations on realistic devices and some alternative structures will then be described followed by a look at one future application and a brief summary.
Schottky Diode Characteristics
Multiplier operation depends on the nonlinear properties of Schottky diodes. The diode has both a capacitive and a resistive nonlinearity. Consider the simple representation of a uniformly Schottky diode shown in Fig. 2.1. Figure 2.3(a) shows a semiconductor with a Schottky barrier contact on the right and an ohmic contact on the left. The semiconductor has a depletion layer width w, with the remaining portion of the structure undepleted. The depletion layer can be represented as a capacitor and the undepleted portion can be represented as a resistor. The depletion layer will act as a parallel plate capacitor with a capacitance of:
Where C is the capacitance, is the dielectric constant, A is the area, and W is the depletion width w from Fig. 2.1(a). The electric field vs. distance for this structure is shown in Fig. 2.1(b). For reasonable conditions the electric field in the undepleted region is small. The field in the depletion regions extends over the width of the depletion region and depends linearly on x . The area under the electric field curve in Fig. 2.1(b) is the depletion layer voltage, the sum of the applied voltage Vb and the built-in potential fbi. The resulting depletion width vs. applied reverse voltage is:
Where fbi is the built-in potential of the metal semiconductor or junction, Vbios is the applied reverse bias, q is the electronic charge, and Nd is the uniform semiconductor doping. This width vs. applied bias will result in the capacitance vs. applied voltage of the form:
Where Ci0 is the capacitance at zero applied bias. The resulting capacitance vs. bias voltage is shown in Fig. 2.1(c). This capacitance characteristic is the starting point for the analytic models in the next section. However, other effects are also present. Under realistic pumping conditions, the diode can also be forward biased allowing forward conduction current flow. This can be approximated with a nonlinear voltagedependent resistance. The resulting equivalent circuit then becomes the right 2 nonlinear elements in Fig. 2.1(c). There are also parasitic elements. The undepleted region of the device and various contact and package resistance will appear in series with the nonlinear junction. Although the undepleted region width is voltage dependent, this series resistance is usually modeled with a constant value. However, at very high frequencies the current through the undepleted region can crowd to the outside edge of the material due to the skin effect, increasing the resistance. This frequency-dependent resistance is sometimes included in multiplier simulations [9]. The varactor diode must be connected to the external circuit. This resulting physical connection usually results in a parasitic shunt capacitance associated with the connection and a parasitic series inductance associated with the current flow through the wire connection.
A major part of multiplier research over the past decade has involved attempts to reduce these parasitic effects. This nonlinear capacitance vs. voltage characteristic can be used as a frequency multiplier. Consider the charge Q in the nonlinear capacitance as a function of voltage
Where a is a constant. This function can be expanded in a series:
(2.5)
The current I(t) is the time derivative of the charge,
(2.6)
If V (t) is of the form Vrf sin(ωt), then the higher order V terms will produce harmonics of the input pump frequency.
Equation (2.6) shows that we can get higher order frequencies out of a nonlinear element. However, it is not clear what the output power or efficiency will be. The earliest research on frequency multipliers were based on closed form descriptions of multiplier operation to investigate this question. This section will discuss the ideal performance of reactive and resistive frequency multipliers. A nonlinear resistor or capacitor, when driven by a pump source, can generate a series of harmonic frequencies. This is the basic form of a harmonic multiplier. The Manley-Rowe relations are a general description of power and frequency conversion relations in nonlinear reactive elements [1, 2]. They describe the properties of frequency conversion and general in nonlinear reactances. The earliest work on these devices sometimes used nonlinear inductances, but all present work involves the nonlinear capacitance vs. voltage characteristic of a reverse-biased Schottky barrier diode. Although the Manley Rowe equations describe frequency multiplication, mixer operation, and parametric amplification, they are also useful as an upper limit on multiplier operation. If an ideal nonlinear capacitance is a pump with a local oscillator at frequency f 0, and an embedding circuit allows power flow at harmonic frequencies, then the sum of the powers into and out of the capacitor is zero,
(2.7)
This expression shows that we can have an ideal frequency multiplier at 100% efficiency converting input power to higher frequency output power if we properly terminate all the other frequencies. Nonlinear resistors can also be used as frequency multipliers [3, 4]. For a nonlinear resistor pumped with a local oscillator at frequency f 0 , the sum of the power is:
(2.8)
For an mth order resistive harmonic generator with only an input and output signal, the efficiency is at best 1/m2, 25% for a doubler and 11% for a tripler. Although Eqs. (2.7) and (2.8) give upper limits on the efficiency to be expected from a multiplier, they provide little design information for real multipliers. The next step in multiplier development was the development of closed form expressions for design based on varactor characteristics [5, 6]. Burckardt [6] gives design tables for linear and abrupt junction multipliers based on closed form expressions for the charge in the diode. These expressions form the starting point for waveform calculations. Computer simulations based on the Fourier components of these waveforms give efficiency and impedance information from 2nd to 8th order. However, these approximations limit the amount of design information available. A detailed set of information on multiplier design and optimization requires a computer-based analysis.
Computer-Based Design Approaches
The analytic tools discussed in the last section are useful to predict the ideal performance of various frequency multipliers. However, more exact techniques are needed for useful designs. Important information such as input and output impedances, the effects of series resistance, and the effect of harmonic terminations at other harmonic frequencies are all important in multiplier design. This information requires detailed knowledge of the current and voltage information at the nonlinear device. Computerbased simulations are needed to provide this information. The general problem can be described with the help of Fig. 2.2. The multiplier consists of a nonlinear microwave diode, an embedding network that
provides coupling between the local oscillator source and the output load, provisions for DC bias and terminations for all other non-input or output frequencies. Looking toward the embedding network from the diode, the device sees embedding impedances at the fundamental frequency and each of the harmonic frequencies. The local oscillator power available is usually specified along with the load impedances and the other harmonic frequencies under consideration. The goal is to obtain the operating conditions, the output power and efficiency, and the input and output impedances of the overall circuit.
The nonlinear nature of the problem makes the solution more difficult. Within the context of Fig. 2.2, the current as a function of time is a nonlinear function of the voltage. Device impedances can be obtained from ratios of the Fourier components of the voltage and current. The impedances of the diode must match the embedding impedances at the corresponding frequency. A "harmonic balance" is required for the correct situation. Many commercial software tools use nonlinear optimization techniques to solve this "harmonic balance" optimization problem. An alternative approach is the multiple reflection algorithm. This solution has the advantage of a very physical representation of the actual transient response of the circuit. This technique is discussed in References 7, 8, and 9 and will be described here. The basic problem is to calculate the diode current and voltage waveforms when the device is embedded in an external linear circuit. The diode waveforms are best represented in the time domain. However, the embedding circuit consists of linear elements and is best represented in the frequency or impedance domain. One approach is the multiple reflection technique [7, 8, 9]. This technique splits the simulation into two parts, a nonlinear time domain description of the diode multiplier and a linear frequency domain description of the embedding network. The solution goal is to match the frequency domain impedances of the embedding circuit with the time domain impedances of the nonlinear device. The circuit is shown in Fig. 2.3. The initial circuit is modified by including a long transmission line with a length l equal to an integral number of pump frequency wavelengths and an arbitrary characteristic impedance Z0 between the diode and the embedding network. Waves will propagate in both directions on this transmission line depending on the conditions at the ends. When steady-state conditions are reached, the waveforms at the circuit and the diode will be the same, with or without the transmission line. This transmission line allows us to determine the waveforms across the diode as a series of reflections from the circuit. The signals on the transmission line are composed of left- and right-traveling current and voltage waves. The voltage at the diode is the sum of the right- and left-traveling wave voltages,
(2.9)
and the current at the diode is the difference
Since the transmission line is an integral number of pump frequency wavelengths long, the conditions are the same at each end under steady-state conditions
(2.11)
(2.12)
At the start of the simulation, we assume that there is a right-traveling wave associated with the DC bias at frequency 0 and the pump at frequency 1. This wave with a DC component and a component at the local oscillator frequency will arrive at the diode at x = 0 as a Vr (t). The resulting voltage across the diode will produce a diode current. This current driving the transmission line will produce a first reflected voltage,
(
2-13)
This time domain reflected or left-traveling wave will then propagate to the embedding network. Here is can be converted into the frequency domain with a Fourier transform. The resulting signal will contain harmonics of the local oscillator signal due to the nonlinear nature of the diode current vs. voltage and charge vs. voltage characteristic. The resulting frequency domain information can then be used to construct a new reflected voltage wave from the embedding network. This process of reflections from the diode and the circuit or "multiple reflections" continues until a steady-state solution is reached. This computer-based solution has several advantages over the simpler analysis-based solutions. It can handle combinations of resistive and reactive nonlinearities. Most high-efficiency multipliers are pumped into forward conduction during a portion of the RF cycle so this is an important advantage for accurate performance predictions. In practice the varactor diode will also have series resistance, parasitic capacitances, and series inductances. These additional elements are easily included in the multiplier simulation. At very high frequencies the series resistance can be frequency dependent, due to current crowding in the semiconductor associated with the skin effect. This frequency dependence loss can also be included in simulations. Computer programs with these features are widely used to design high performance frequency multipliers. An alternative solution technique is the fixed point method [10]. This technique uses a circuit similar to Fig. 2.3, with a nonlinear device and an embedding network connected with a transmission line. However, this approach uses a fixed point iteration to arrive at a converged solution. The solution starts with arbitrary voltages at the local oscillator frequency at the harmonics. With these starting conditions, a new voltage is obtained from the existing conditions using
(2.14)
for the driven local oscillator frequency and
(2.15)
for the remaining frequencies, where Z nL are the embedding impedances at frequency n, Z0 is the same line characteristic impedance used in the multiple reflection simulation,
Z n,kNL is the nonlinear device impedance at frequency n and iteration number k , Vn,k
, and In,k are the frequency domain voltage and current at iteration k, and VSn is the RF voltage at the pump source. These two equations provide an iterative solution of the nonlinear problem. They are particularly useful when a simple equivalent circuit for the nonlinear device is not available. We now have the numerical tools to investigate the nonlinear operation of multipliers. However, there are some operating conditions where this equivalent circuit approach breaks down. Some of these limitations will be discussed in the next section.
Device Limitations and Alternative Device Structures
The simulation tools and simple device model do a good job of predicting the performance of low frequency diode multipliers. However, many multiplier applications require very high frequency output signals with reasonable output power levels. Under these conditions, the output powers and efficiencies predicted are always higher than the experimental results. There are several possible reasons. Circuit loss increases with frequency, so the loss between the diode and the external connection should be higher.
Measurements are less accurate at these frequencies, so the differences between the desired designed circuit embedding impedances and the actual values may be different. Parasitic effects are also more important, degrading the performance. However, even when all these effects are taken into account, the experimental powers and efficiencies are still low. The problem is with the equivalent circuit of the diode in Fig. 2.1(d). It does not correctly represent the high frequency device physics [11]. The difficulty can be explained by referring back to Fig. 2.1(a). The device is a series connection of the depletion layer capacitance and a bulk series resistance. The displacement current flowing through the capacitor must be equal to the conduction current flowing through the undepleted resistive region. The capacitor displacement current is
(2.16)
If we approximate V(t) with Vrf cos (ωt), then the current becomes
(
2.17)
For a given device, the displacement current and the resulting current through the resistor increase with the frequency and the RF voltage. At modest drive levels and frequencies the undepleted region can support this current and the equivalent resistance remains constant. However, the current density through the undepleted region is
(2.18)
where Jdep is the conduction current density in the undepleted region and v(E) is the carrier velocity vs. electric field. At low electric fields the slope of the velocity field curve is constant with a value equal to the carrier mobility μ. However at higher electric fields, the velocity begins to saturate at a constant value. Additional increases in the electric field do not increase the conduction current through the varactor. This current saturation is a fundamental limit on the multiplier performance. A more physical explanation also uses Fig. 2.1(a). A nonlinear capacitor implies a change in the depletion layer width with voltage. However, changing the depletion layer width involves moving electrons from the depletion layer edge. These electrons are limited by their saturated velocity, so the time rate of change of the capacitance is also unlimited. This simple theory allows a modified device design. Equations (2.18) and (2.17) are the starting point. Our goal is usually to optimize the power or efficiency available from a multiplier circuit. Clearly one option is to increase the doping Nd in Eq. (2.18) to increase the available Jdep. However, increasing the doping decreases the breakdown voltage and thus the maximum RF voltage that can be present across the reverse biased depletion layer. The device doping design becomes a parameter in the overall multiplier design. The optimum efficiency and optimum power operating points for the same input and output frequency are usually different. Although this simple description provides useful information, a detailed physical model is usually needed for best results [12]. The discussion so far has been based on a uniformly doped abrupt junction varactor. However, other doping or material layer combinations are possible [13]. One option is to tailor the doping or material profile to obtain a capacitance vs. voltage that is more nonlinear than the 1/ dependence in Eq. (2.2). Two options are the hyperabrupt varactor and the BNN structure. These devices are shown in Fig. 2.4. The doping profile of a hyperabrupt varactor is shown in Fig. 2.4(a). Instead of a uniform doping, this structure has a much smaller doping over most of the structure with a high doping or doping spike near the metal semiconductor junction. The corresponding capacitance vs. voltage characteristic is shown in Fig. 2.4(c). At modest reverse biases the depletion layer extends from the metal contact to the doping spike. The resulting narrow depletion layer produces a high capacitance. Higher applied voltages begin to deplete the charge in the doping spike. When the spike charge is depleted, there is a rapid increase in the depletion layer width through the second lightly doped region and a corresponding decrease in the capacitance. This structure can produce a more nonlinear capacitance variation than a uniformly doped device. However, the structure combines both lightly doped and heavily doped regions, so saturation
effects can occur in the lightly doped portion. An alternative structure is the BNN or BIN device. This structure uses combinations of Barriers Intrinsic and N doped regions formed with combinations of different epitaxial materials and doping to produce optimized capacitance structures. This structure can have either ohmic or Schottky contacts. A typical structure is shown in Fig. 2.4(b). Notice that conduction band energy rather than doping is being plotted. The structure consists of an n+ ohmic contact on the right followed by an n region that can be depleted, a wide bandgap barrier region, and a second n+ ohmic contact on the left. This structure can have a highly nonlinear capacitance characteristics as shown in Fig. 2.4(d), depending on the choice of layer doping and energy band offsets. These BNN structures have potential advantages in monolithic integration and can be fabricated in a stacked or series configuration for higher output powers. Since the major application of frequency multipliers is for high frequency local oscillator sources, it would be reasonable to try to fabricate higher order multipliers, triplers for example, instead of doublers. Although based on Eq. (2.7), this is possible, there are some problems. Efficient higher order multiplication requires currents and voltages at intermediate frequencies, at the second harmonic in a tripler for example. This is an "idler" frequency or circuit. However, in order to avoid loss, this frequency must have the correct reactive termination. This adds to the complexity of the circuit design. Some details of doublers and triplers are given in Reference 14. An alternative that avoids the idlers is an even or symmetrical capacitance voltage characteristic. One possibility is the single barrier or quantum barrier varactor [15]. The structure and associated capacitance voltage characteristic are shown in Fig. 2.5. This structure, shown in Fig. 2.5(a), is similar to the BNN expect the barrier is the middle with lightly doped regions on either side. This will be a series connection of a depletion layer, a barrier and a depletion region, with ohmic contacts on each end. The capacitance is maximum at zero applied bias, with a builtin depletion layer on each side of the barrier. When a voltage is applied, one to the depletion layers will become forward biased and shrink and the other one will be reverse biased and expand. The series combination capacitance will become smaller. Reversing the applied voltage will produce the same capacitance. The resulting symmetrical capacitance is shown in Fig. 2.5(b). This capacitance characteristic is a useful starting point for odd order multipliers.
Fuentes:
http://en.wikipedia.org/wiki/Varicap
http://es.wikipedia.org/wiki/Diodo_Varicap
http://www.electronica2000.com/temas/diodostipos.htm
http://www.scribd.com/doc/22383992/diodo-varactor
¡Nuevo MSN Entretenimiento! Todos los trailers, series de tv y videoclips, los mejores juegos online y lo último sobre tus estrellas favoritas.
No hay comentarios:
Publicar un comentario