CHAPTER 2 Basic Concepts of Low-Noise Amplifier Design
2.1 System Specifications of LNA
2.1.1 S-Parameters
Systems can be characterized in numerous ways. Usually, we use the well-known representations of impedance (Z), admittance (Y), hybrid (H), and cascade (ABCD) parameters matrix to characterize N-port network. However, in microwave design (or higher frequencies), it is quite difficult to obtain the parameters matrices mentioned above by providing short or open termination, since we can’t measure the current at each port as it results in the reflected wave from short or open termination. For this reason, we must use Scattering Parameters, also called S-parameters, defined by the variables in terms of incident and reflected voltage waves with characteristic impedance, rather than port voltages or currents, to characterize the two-port network in LNA design [28]-[30].
Chapter 2 Basic Concepts of LNA Design
Figure 2.1 Two-port network with S-parameters.
Consider the two-port network for LNA shown in Figure 2.1, where 0i is the characteristic impedance of the port I, and i and i ＿represent the incident and reflected voltage waves at port i respectively. In order to obtain physically meaningful power relations in terms of wave amplitudes, we must ed fine a new set of wave amplitudes as
i i / 0i , (2.1)
i i / 0i , (2.2) where i and i represents the normalized incident and reflected voltage wave by characteristic impedance ( 0i) at port i respectively.
Then, the two-port S-parameters are defined as
, (2.3) where
Port 2 is terminated., (2.4)
Port 1 is terminated., (2.5)
Port 2 is terminated., (2.6)
Port 1 is terminated.. (2.7) Thus, and are simply the input and output reflection coefficient respectively while we terminated another, and and is simply the forward gain and reverse isolation respectively.
.
Chapter 2 Basic Concepts of LNA Design
2.1.2 Noise Figure (NF)
A useful representation of the noise effect of a system is the noise figure, usually denoted F or NF. The noise figure is a measure of the degradation in signal-to-noise ratio (SNR) that a
system introduces. The noise factor is defined as
total output noise power
, (2.8) and we usually represent the noise figure with logarithm defined as
dB 10log . (2.9) The method for analyzing the effect of noise in MOSFET and the calculation of noise figure are illustrated in reference [28].
Figure 2.2 Cascade of system block diagram.
The overall noise figure of a cascade of systems depends on both the individual noise figures as well as their gains. The dependency on the gain results from the fact that, once the signal has been amplified, the noise of subsequent stages is less important. As a result, system noise figure tends to be dominated by the noise performance of the first several stages in a receiver. Consider the block diagram of Figure 2.2, where each is the noise figure and each is the gain. The total noise factor is the sum of these individual contributions, and is therefore given by
∏ . (2.10)
Chapter 2 Basic Concepts of LNA Design
2.1.3 Harmonics
Figure 2.3 Frequency response of nonlinear system with one tone signal.
In wireless receiver, the low-noise amplifier usually can be approximately treated as linear system for processing small signal. In fact, it is nonlinear and the input-output relationship of a nonlinear system can be described by
, (2.11) Here, as shown in Figure 2.3, S(t) is the input signal, and y(t) is the output signal. Using the function (2.11) with one tone signal at the input, cos , the output of the nonlinear system can be viewed mathematically as
cos cos cos
α cosω cos2ω cos3ω . (2.12)
As can be seen easily from (2.12), harmonic distortion is generated and is defined as the ratio of the amplitude of a particular harmonic to that of the fundamental. For example, third-order harmonic distortion (HD3) is defined as the ratio of amplitude of the tone at 3 to that of the fundamental at , and is therefore given by
. (2.13)
Chapter 2 Basic Concepts of LNA Design
2.1.4 1-dB Gain Compression Point (P1dB)
Figure 2.4 Definition of 1-dB compression point.
As can be seen easily from (2.12), for an amplifier, the output amplitude of the fundamental will not be given with a linear gain (α1) when the input signal amplitude is large.
We rewrite only the term of the fundamental as follows
t α cosω α cosω . (2.14)
At this point, from (2.14), α3 is usually negative, and the gain will be compressed while the input signal amplitude exceeds some value. In RF circuits, this effect is quantified by the
“1-dB compression point,” defined as the input signal level that causes the linear small-signal gain to drop by 1 dB. It is depicted graphically in Figure 2.4. To calculate the 1-dB compression point, we can write from (2.14)
20log α ‐ 20log|α | 1dB. (2.15)
That is,
‐ 0.145 . (2.16)
Chapter 2 Basic Concepts of LNA Design
2.1.5 Inter-Modulation
As more than one tone is applied to a nonlinear system, Inter-modulation (IM) arises.
Assume that there are two strong interferers occurred at the input of the receiver, specified by cos cos . The IM distortion can be expressed mathematically by applying to (2.11)
cos cos
cos cos
cos cos (2.17) Using trigonometric manipulations, we can find expressions for the second and the third-order IM products as follows
cos cos , (2.18)
2 cos 2 cos 2 , (2.19)
2 cos 2 cos 2 . (2.20)
The output spectrum in frequency domain can be determined from (2.18)-(2.20) by evaluating its Fourier transform. In a typical two-tone test, , there are two third-order IM products at 2 and 2 respectively, and then the output signal will be corrupted by one of the two, illustrated in Figure 2.5.
Figure 2.5 Inter-modulation in LNA (nonlinear system).
Chapter 2 Basic Concepts of LNA Design
2.1.6 Third-Order Intercept Point (IP3)
The corruption of signals due to third-order IM of two nearby interferes is so common and so critical that a performance has been defined to characterize this behavior. Called the “third intercept point” (IP3), this parameter is measured by a two-tone test in which the input amplitude ( ) is chosen to be sufficiently small so that higher-order nonlinear terms are negligible and the gain is relatively constant and equal to α1. From (2.19) and (2.20), we note that as A increases, the fundamentals increase in proportion to , whereas the third-order IM products increase in proportion to . Plotted on a logarithmic scale in Figure 2.6, the third-order intercept point IP3 is defined to be the intersection of the two dotted lines elongated from linear region. Therefore, we can see that the amplitude of the input interferer at the third-order intercept point, , is d nefi ed by th r la n e e tio
20 log α 20log α . (2.21) From (2.21), we can solve for AIP3:
. (2.22) For 50-Ω systems, we define the input third-order intercept point (IIP3) as IIP3
IIP /50. (IIP3 is hence interpreted as the power level of the input interferer at the third-order intercept point).
Figure 2.6 The input and output third order intercept point (IIP3 and OIP3).
Chapter 2 Basic Concepts of LNA Design