Davis W.A.Radio frequency circuit design.2001

FIGURE 7.12 Class A amplifiers with (a) collector resistor and (b) collector inductor.

I o,max

I o

I Q

FIGURE 7.13 Magnitude of the output current and quiescent current of the class A amplifier.

The quiescent current, IQ, and the output current, Io, is defined in Fig. 7.13. When the load is drawing the maximum instantaneous power,

The maximum average power delivered to the load can now be written in terms of the supply voltage:

However, it should be noted that many times high-power amplifiers do not have high gain, so the power added efficiency given by Eq. (7.89) offers a more useful quality factor for a transistor than if Pi were neglected.

7.8POWER COMBINING OF POWER AMPLIFIERS

Design of power FET amplifiers requires use of large gate periphery devices. However, eventually, the large the gate periphery causes other problems such as impedance matching especially at RF and microwave frequencies. Bandwidth improvement can be obtained by combining several transistors, often on a single chip. An example of combining two transistors is shown in Fig. 7.14 [7,8]. The separation of the transistors may induce odd-order oscillations in the circuit, even if the stability factor of the individual transistors (even-order stability) indicate they are stable. This odd-order instability can be controlled by adding Rodd between the two drains to damp out such oscillations. This resistor is typically less than 400 $. Symmetry indicates no power dissipation when the outputs of the two transistors are equal and in phase. An example of a four transistor combining circuit is shown in Fig. 7.15, which now includes resistors Rodd1 and Rodd2 to help suppress odd-order oscillations.

R odd

FIGURE 7.14 Power combining two transistors [7,8].

Radio Frequency Circuit Design. W. Alan Davis, Krishna Agarwal

Copyright 2001 John Wiley & Sons, Inc.

Print ISBN 0-471-35052-4 Electronic ISBN 0-471-20068-9

CHAPTER EIGHT

Noise

8.1SOURCES OF NOISE

The dynamic range of a communication transmitter or receiver circuit is usually limited at the high-power point by nonlinearities and at the low-power point by noise. Noise is the random fluctuation of electrical power that interferes with the desired signal. There can be interference with the desired signal by other unwanted deterministic signals, but at this point only the interference caused by random fluctuations will be considered. There are a variety of physical mechanisms that account for noise, but probably the most common is thermal (also referred to as Johnson noise or Nyquist noise). This can be illustrated by simply examining the voltage across an open circuit resistor (Fig. 8.1). The resulting voltage is not zero! The average voltage is zero, but not the instantaneous voltage. At any temperature above absolute 0 K, the Brownian motion of the electrons will produce random instantaneous currents. These currents will produce random instantaneous voltages, and this leads to noise power.

Noise arising in electron tubes, semiconductor diodes, bipolar transistors, or field effect transistors come from a variety of mechanisms. For example, for tubes, these include random times of emission of electrons from a cathode (called shot noise), random electron velocities in the vacuum, nonuniform emission over the surface of the cathode, and secondary emission from the anode. Similarly, for diodes, a random emission of electrons and holes produces noise. In a bipolar transistor, there is in addition partition noise. This represents the fluctuation in the path that charge carriers take between the base and the collector after leaving the emitter. There is in addition 1/f, or flicker noise (where f is frequency), that is caused by surface recombination of base minority carriers at the base-emitter junction [1]. Clearly, as the frequency approaches dc, the flicker noise increases dramatically. As a consequence intermediate amplifier stages are designed to operate well above the frequency where 1/f noise is a significant contributor to the total noise. Typically this frequency ranges from 100 Hz to 10 kHz. In a field effect transistor, there is thermal noise arising from channel resistance, 1/f

144

ix

iy

FIGURE 8.1 Voltage across an open circuit resistor.

noise, and a coupling of the channel noise back to the gate where it is of course amplified by the transistor gain. Noise also arises from reverse breakdown in the avalanching of electrons in such devices as Zener diodes and IMPATT diodes. At RF frequencies the two most common noise sources are the thermal noise and the shot noise.

8.2THERMAL NOISE

The random fluctuation of electrons in a resistance would be expected to rise as the temperature increases, since the electron velocities and the number of collisions per second increases. The noise voltage is expressed as an auto correlation of the instantaneous voltage over a time period T:

The expression for thermal noise voltage has been derived in a variety of ways. Harry Nyquist first solved the problem based on a transmission line model. Other approaches included using a lumped element circuit, the random motion of electrons in a metal conductor, and the radiation from a black body. These are all basically thermodynamic models, and each method resulted in the same expression. The black body method is based on quantum mechanics and therefore provides a solution for noise sources at both cryogenic and room temperatures.

8.2.1Black Body Radiation

Classical mechanics is based on the continuity of energy states. When this theory was applied to calculation of the black body radiation, it was found that the radiation increased without limit. This so-called ultraviolet catastrophe was clearly not physical. However, Max Planck was able to correct the situation by postulating that energy states are not continuous but are quantized in discrete states. These energy values are obtained by solving the Schrodinger¨ equation for the harmonic oscillator. The actual derivation is found in most introductory texts on quantum mechanics [2]:

2

148 NOISE

8.3SHOT NOISE

Shot noise arises from random variations of a dc current, I0, and is especially associated with current carrying active devices. Shot noise is most apparent in a current source with zero-shunt source admittance. For the purpose of illustration, consider a current source feeding a parallel RLC circuit (Fig. 8.2). The inductor provides a dc current path and is open to ac variations of the current. Hence the resulting noise voltage appears across the resistor (which is presumed free of any thermal noise). If an instrument could measure the current produced by randomly arriving electrons, the instrument would record a series of current impulses for each electron. If n is the average number of electrons emitted by the source in a given time interval t, then the dc current is

where q is the charge of an electron. Each current pulse provides an energy pulse to the capacitor with the value of

The equipartition theorem, as found in thermodynamics textbooks, states that the average energy of a system of uniform temperature is equally divided among the degrees of freedom of the system. If there are N degrees of freedom, then

FIGURE 8.2 Equivalent circuit for shot noise and for certain thermal noise calculations.