Figure 1.5 An amplifier with internal noise sources.
The source termination produces an incident noise wave aNS and adds to the internal noise created in the amplifier, which can be represented as an input noise source aNamp. There are scattered noise waves represented by the noise emitted from the input of the amplifier, bN1, and the noise incident on the load is bN2. From this figure, one can make a direct comparison to the S‐parameters and see that reflected noise power might add or subtract to the incident noise power and affect the total noise power. However, at the input of the amplifier, the noise generated inside the amplifier is in general not correlated with the noise coming from the source termination so that they don't add together in a simple way. Because of this, the noise power at the output of the amplifier, and therefore the noise figure, depends upon the source impedance in a complex way. This complex interaction is defined by two real valued parameters and one complex parameter, known collectively as the noise parameters. The noise figure at any source reflection coefficient may be computed as
(1.70)
where NFmin is the minimum noise figure; ΓOpt, called gamma‐opt, is the reflection coefficient (magnitude and phase) that gives the minimum noise figure; and RN, sometimes called the noise resistance, describes how the noise figure increases as the source impedances varies from the gamma‐opt. The characterization required to determine these values is quite complex and is covered in Chapter 6.
1.6 Distortion Parameters
Up to now, all the parameters described have been under the consideration that the DUT is linear. However, when a DUT, particularly an amplifier, is driven with a large signal, non‐linear transfer characteristics become significant, leading to an entirely new set of parameters used to describe these non‐linear characteristics.
1.6.1 Harmonics
One of the first noticeable effects of large signal drive is the generation of harmonics at multiples of the input frequency. Harmonics are described by their order and either by their output power or, more commonly, by the power relative to the output power of the fundamental, and almost always in dBc (dB relative to the carrier). Second harmonic is short for second‐order harmonic and refers to the harmonic found at two times the fundamental, even though it is in fact the first of the harmonic frequency above the fundamental; third harmonic is found at three times the fundamental, and so on. Surprisingly, there are not well‐established symbols for harmonics; for this book, we will use H2, H3 … Hn to represent the dBc values of harmonics or order 2, 3 … respectively. In Chapter 6, the measurements of harmonics are fully developed as part of the description of X‐parameters and utilize the notation b2, m to describe the output normalized wave power at port 2 for the mth harmonic. A similar notation is used for harmonics incident on the amplifier.
One important attribute of harmonics is that for most devices the level of the harmonics increases in dB value as the power of the input increases and to a rate directly proportional to the harmonic order, as shown in Figure 1.6. In this figure, the x‐axis is the drive power, and the y‐axis is the measured output power of the fundamental and the harmonics.
Figure 1.6 Output power of harmonics of an amplifier.
1.6.2 Second‐Order Intercept
This pattern of increasing power as the input power is increased, but to the slope related to the order of the harmonic, cannot continue indefinitely or the harmonic power would exceed the fundamental power. While theoretically possible, in practice the harmonic power saturates just as the output power does and never crosses the level of the output power. However, if one uses the lower power regions to project a line from the fundamental and each of the harmonics, they will intersect at some power, as shown in Figure 1.6. The level that these lines converge is called the intercept point, and the most common value is the second‐order intercept (SOI), and intercept points beyond third order are seldom used.
There is sometimes confusion in the use of the term SOI; while it is most commonly used to refer to the second harmonic content, in some case, it has also been used to refer to the two‐tone SOI, which is a distortion product that occurs at the sum of the two tones. Most properly, one should always use the term two‐tone SOI if one is to distinguish from the more common harmonic SOI.
1.6.3 Two‐Tone Intermodulation Distortion
While the harmonic measurement provides a direct characterization of distortion, it suffers from the fact that the harmonic frequencies are far away from the fundamental, and in many circuits, the network response is such that the harmonic content is essentially filtered out. Thus, it is not possible to discern the non‐linear response of such a network by measuring only the output signal. Of course, if the gain is measured, compression of the amplifier will show that the value of S21 changes with the input drive level. But it is convenient to have a measure or figure of merit of the distortion of an amplifier that relies only on the output signal. In such a case, two signals of different frequencies can be applied at the amplifier input, at a level sufficiently large to cause a detectible non‐linear response of the amplifier. Figure 1.7 shows a measurement of a two‐tone signal applied to the input of an amplifier (lower trace) and measured on the output of the amplifier (upper trace).
Figure 1.7 Measurement of a two‐tone signal at the input and output of an amplifier.
It is clear that several other tones are present at the output and are the result of higher‐order products mixing in the amplifier due to its non‐linear response and creating other signals. The principal signals of interest are the higher and lower intermodulation (IM) products, PwrN_Hi and PwrN_Lo, where N is the order of intermodulation distortion (IMD). Normally, IM products refer to the power of the IM product relative to the carrier, in dBc, and these terms are called IMN_Hi and IMN_Lo. For example, the power in the lower third‐order tone is Pwr3_Lo; the level of the upper third‐order tone relative to the carrier is called IM3_Hi. The frequencies of the higher and lower tones are found at
(1.71)
And more generally
(1.72)