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191 |
lines are 50-V microstrip lines. In this realization we have not taken into account the fringing components of the microstrip T-junctions and open ends [12]. Another often used matching method in transistor amplifiers is an alternating high-low-impedance line. A manual design of amplifiers and other circuits becomes excessively laborious if all parasitic effects are taken into account and the circuit has to operate over a broad band. Then use of a computer-aided design (CAD) package significantly facilitates the design.
8.4.2 Effect of Nonlinearities and Design of Power Amplifiers
In the previous study we assumed that the transistor characteristics did not depend on the signal level, that is, the transistor operates in linear, small-signal conditions. However, as the input power level increases, the nonlinearities of the transistor cause gain compression and generation of spurious frequency components.
As the input power level increases, the gain of an amplifier is constant at small signal levels but finally starts to decrease. Often the bias voltages limit the maximum output voltage. This saturation or gain compression is characterized by the 1-dB compression point, which is the output power at which the gain has decreased by 1 dB from its small-signal value.
Because of the compression, the output waveform is distorted. This distortion produces harmonics nf of a single input signal at frequency f and intermodulation products mf 1 + nf 2 (m , n = ±1, ±2, . . . ) of a two-tone
input |
signal. The order of intermodulation products is defined to be |
| m | + |
| n | . Especially important are the third-order intermodulation products |
at frequencies 2f 1 − f 2 and 2f 2 − f 1 because they are close to frequencies f 1 and f 2 , if f 1 ≈ f 2 . At low input signal levels, the power of third-order intermodulation products increases by 3 dB as the power of input signals having equal magnitudes increases by 1 dB. If there were no compression, the output powers of the desired signals at f 1 and f 2 and the third-order products would be equal at an output power level called the third-order intercept point, IP 3 .
The dynamic range of an amplifier is that operating power range over which the amplifier has desirable characteristics. Noise usually sets the lower limit of dynamic range. The upper limit of the linear dynamic range may be defined as the 1-dB compression point. The spurious-free dynamic range is limited by the power level, which produces unacceptable intermodulation products.
In the design of a power amplifier, the theory presented here for getting proper rS and r L is valid only if the scattering parameters are measured in
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large-signal conditions, that is, at a power level corresponding to the real operation. A power amplifier is designed so that both input and output is conjugate matched, because this provides the maximum gain. Power amplifiers are divided into different classes. A class-A amplifier operates linearly. The bias voltages and signal amplitude are chosen so that output current flows during the full signal period, as shown in Figure 8.18. In a class-B amplifier the output current flows only during one-half of the period, and in a class-C amplifier less than one-half of the period. These amplifiers operate very nonlinearly but they transform dc power more effectively into RF power than a class-A amplifier. A good efficiency is obtained also with class-D, class-E, and class-F amplifiers in which the transistors operate as switches.
In comparing different power amplifiers, an often used figure-of-merit is the power-added efficiency
PAE = |
P L − P in |
(8.33) |
|
Pdc |
|||
|
|
where P L is the RF power coupled to the load, P in is the RF power coupled to the amplifier, and Pdc is the dc power absorbed by the amplifier. When a very high power level is needed, several amplifiers may be combined parallel using a power combiner (see Section 6.1).
8.4.3 Reflection Amplifiers
The reflection-type amplifier is based on the negative resistance of, for example, a Gunn or IMPATT diode. The power gain of a reflection-type amplifier is
Figure 8.18 Current waveforms in class-A, class-B, and class-C amplifiers.
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193 |
|||
G = | r |2 = | |
Z d − Z |
0 |
|2 |
(8.34) |
Z d + Z |
0 |
|||
where Z 0 is the characteristic impedance of the transmission line and Z d is the diode impedance. If for example Z 0 = 50V and Z d = −25V, gain is G = (75/25)2 = 9 = 9.5 dB. The input and output signals can be separated by using a circulator according to Figure 8.19.
8.5Frequency Converters (Mixers) and Frequency Multipliers
The output signal from a linear circuit has shape similar to that of the input signal; however, its amplitude may be higher (amplifier) or lower (attenuator). In a nonlinear circuit the signal is distorted, and the output signal (voltage Vo ) is a nonlinear function of the input signal (voltage Vi ) and can be presented as a power series
Vo = f (Vi ) = AVi + BVi |
2 + CVi |
3 + . . . |
(8.35) |
Figure 8.20 illustrates the difference between a linear and nonlinear transfer function. If, in the case of a nonlinear transfer function, the input signal is weak and causes only a small perturbation in the vicinity of the operating point, the circuit can be considered linear for the signal; that is, dVo = A ′dVi , where A ′ is the slope of the curve f (Vi ) in the operating
Figure 8.19 A reflection-type amplifier.
194 Radio Engineering for Wireless Communication and Sensor Applications
Figure 8.20 Transfer function of (a) a linear and (b) a nonlinear circuit, and (c) a linear small-signal condition.
point. We call this situation the small-signal condition. In a large-signal condition, several terms of (8.35) must be taken into account.
Let us assume that Vi = V 1 cos v1 t + V 2 cos v2 t . Then the term AVi contains signal components at frequencies f 1 and f 2 , term BVi 2 contains components at frequencies 0 (dc), 2f 1 , 2f 2 , and f 1 ± f 2 , and term CVi 3 contains components at frequencies f 1 , f 2 , 3f 1 , 3f 2 , 2f 1 ± f 2 , and 2f 2 ± f 1 (a frequency may be also negative). We note that when a sinusoidal signal at frequency f 1 is fed to a nonlinear circuit, the output contains harmonics at frequencies mf 1 , and when two sinusoidal signals at frequencies f 1 and f 2 are fed to a nonlinear circuit, the output contains components at frequencies mf 1 + nf 2 (m and n are integers). These nonlinear characteristics make possible the operation of a frequency converter, or a mixer and a frequency multiplier. Note that a mixer may also be based on a time-dependent linear circuit. In a circuit meant to be linear, such as a low-noise amplifier, the distortion due to nonlinearity produces unwanted frequency components.
8.5.1 Mixers
A mixer is a circuit that converts the frequency of a signal up or down so that the information contained in the signal is preserved. Upconverters are used in modulators and transmitters, downconverters in heterodyne receivers and demodulators. In Figure 8.21, a signal at frequency f s and a local oscillator signal at f LO are fed to a downconverter; then at the output we have a signal at a low intermediate frequency f IF = | f s − f LO | . Processing of the signal at f IF is much easier than that of the original signal at f s . The conversion loss L c is defined as
L c = |
Ps , av |
= |
Available power at f s |
(8.36) |
|
P IF |
Power coupled to load at f IF |
||||
|
|
|
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195 |
Figure 8.21 Mixer as a downconverter.
As in case of an amplifier, a mixer is linear for a low-power signal (Ps < P LO /100) but at higher powers the output signal will be distorted. Thus we can define a 1-dB compression point and a third-order intercept point for a mixer, too.
The nonlinear element may be a diode or a transistor, most often a Schottky diode, bipolar transistor, or FET. Diode mixers are passive; transistor mixers may operate in an active mode and have some conversion gain. Transistor mixers suit well in integrated circuits. Operation of a diode mixer is based on the exponential I–V characteristic, but frequency conversion takes place also in the nonlinear capacitance. In a bipolar transistor the emitter-base junction forms a diode. Therefore, in a common-emitter connection the collector current depends exponentially on the base voltage. In an FET the drain current IDS is a nonlinear function of the gate voltage VGS . Especially a dual-gate FET (DGFET) is well suited as a mixer, because the RF signal can be fed to one gate and the local oscillator signal to another gate.
In case of a diode mixer, frequency conversion can be analyzed as follows; for equations see [13, 14]. Using the embedding impedances loading the diode at frequencies mf LO we calculate the waveforms of the conductance Gj (t ) = 1/R j (t ) and capacitance Cj (t ) caused by the local oscillator signal and a possible dc bias. The better the diode corresponds to an ideal switch operating at f LO , the more effective frequency conversion is. The signal ( f s ) power is usually very small compared to the LO signal ( f LO ) power; then from the signal’s point of view the mixer is a linear, time-dependent circuit.
The operation of a mixer depends on conditions not only at the signal and IF frequency, but also at the sidebands mf LO ± f IF because power may convert from any sideband to another. Especially important is the image sideband f i = 2f LO − f s . Frequencies mf LO ± nf IF (n ≥ 2) are important only if the signal power level is of the same order as that of the LO. Using
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the Fourier series of Gj (t ) and Cj (t ) we can then calculate a frequency conversion matrix. With this and the load impedances we obtain conversion efficiencies between any two sidebands. Figure 8.22 illustrates the conversion from a frequency f s to different sidebands.
In designing a mixer, it is important to find the correct load impedances at different sidebands. The conversion loss is at minimum when there is a conjugate match between the nonlinear element and the embedding network at signal and intermediate frequencies, and the other sidebands are terminated with proper, purely reactive loads. The noise optimum requires slightly different conditions the same way as in case of an amplifier. In practice, mixer design today is carried out with a CAD package employing a harmonic balance [13, 14] simulator.
Depending on which sidebands are selected as the signal bands, we have different mixers. A single-sideband (SSB) mixer converts the signal only from one sideband, either from the upper sideband f LO + f IF or from the lower sideband f LO − f IF , to the intermediate frequency band. A doublesideband (DSB) mixer converts both sidebands to the IF band. Two DSB mixers can be combined to form an SSB mixer, which is then called an image-rejection mixer: The outputs of the individual mixers are combined in phase in case of the desired sideband while the outputs cancel each other in case of the image sideband. A harmonic mixer converts the sidebands of
an LO harmonic mf LO ± f IF (m ≥ 2) to the IF band.
There are a number of different mixer structures or architectures [14]. Figure 8.23 presents the principle of a single-ended, a balanced, and a doublebalanced diode mixer, as well as of a double-balanced transistor mixer. For simplicity, the matching and bias circuits are omitted in Figure 8.23. At millimeter wavelengths the mixers are often single-ended waveguide mixers, where the signal and LO are fed to the diode along the same waveguide after combining them in a directional coupler. At RF and microwave frequencies most often balanced and double-balanced mixers are used, and signal and LO power are fed to the nonlinear elements using 3-dB hybrids (described
Figure 8.22 Conversion of signal power to different sidebands in a mixer.