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When a 2‐port network is connected to a generator with arbitrary impedance, the output scattered wave into matched load is
(1.51)
Here the incident wave is represented as aS rather than a1 as an indication that the source is not matched, and Γ1 is defined by Eq. (1.27). The output power incident to the load is
Combining Eqs. (1.52) and (1.50), the available power at the output from a network that is driven from a generator with source impedance of ΓS is
(1.53)
With Γ2 defined as in Eq. (1.30).
1.4.5 Available Gain
Available gain is the gain that an amplifier can provide to a conjugately matched load from a source or generator of a given impedance and is computed with the formula
(1.54)
Other derived values such as maximum available gain and maximum stable gain are discussed in detail in Chapter 6.
1.5 Noise Figure and Noise Parameters
For a receiver, the key figure of merit is its sensitivity, or the ability to detect small signals. This is limited by the intrinsic noise of the device itself, and for amplifiers and mixers, this is represented as noise figure. Noise figure is defined as a signal‐to‐noise ratio at the input divided by signal to noise at the output expressed in dB.
(1.55)
Its related value, noise factor, which is unitless, is
(1.56)
Here the signal and noise values are represented as a power; traditionally, this is available power, but incident power can be used as well with a little care. Rearranging Eq. (1.55), one can obtain
In most cases, the input noise is known very well, as it consists only of thermal noise associated with the temperature of the source resistance. This is the noise available from the source and can be found from
where k is Boltzmann's constant (1.38 × 10−23 J K−1), B is the noise bandwidth, and T is the temperature in Kelvin. Note that the available noise power does not depend upon the impedance of the source. From the definition in Eq. (1.57) it is clear that if the temperature of the source impedance changes, then the noise figure of the amplifier using this definition would change as well. Therefore, by convention, a fixed value for the temperature is presumed, and this value, known as T0, is 290 K.
This is the noise power that would be delivered to a conjugately matched load. Alternatively, the noise power can be represented as a noise wave, much like a signal, and one can define an incident noise (sometimes called the effective noise power), which is defined as the noise delivered to a nonreflecting nonradiating load and is found as
which is consistent with the definition of Eq. (1.47). Since the available noise at the output of a network doesn't depend upon the load impedance, the available gain from a network similarly doesn't depend upon the load impedance, and the available noise at the input of the network can be computed as Eq. (1.58), the measurement of noise figure defined in this way is not dependent upon the match of the noise receiver. One way to understand this is to note that the available gain is the maximum gain that can be delivered to a load. If the load is not conjugately matched to Γ2, both the available gain and the available noise power at the output would be reduced by equal amounts, leaving the noise figure unchanged and independent of the noise receiver load impedance. Thus, for the case of noise measurements, the available noise power and available gain have been the important terms of use historically.
Recently more advanced techniques have been developed and made practical based on incident noise power and gain. If the impedance is known, the incident noise power can be computed as in Eq. (1.59); and if the output incident noise power NOE can be measured, then one can compute the output available noise as
(1.60)
Substituting into Eq. (1.57) to find
When the source is a matched source, this simplifies to
(1.62)
Thus, for a simple system of an amplifier sourced with a Z0 impedance and terminated with a Z0 load, the noise factor can be computed simply from the noise power measured in the load and the S21 gain. However, Eq. (1.61) defines the noise figure of the amplifier in terms of the source impedance, and this is a key point. In general, although the 50 Ω noise figure is the most commonly quoted, it is measured only when the source impedance provided is exactly 50 Ω. In
