1.4 Power Parameters
1.4.1 Incident and Reflected Power
Just as there are a variety of S‐parameters, which are derived from the fundamental parameters of incident and reflected waves a and b, so too are there many power parameters that can be identified with the same waves. As inferred earlier, the principal power parameters are incident and reflected, or forward and reverse, powers at each port, which for Z0 real, are defined as
(1.46)
The proper interpretation of these parameters is that incident and reflected power is the power that would be delivered to a nonreflecting (Z0) load. If one were to put an ideal Z0 directional‐coupler in line with the signal, it would sample or couple the incident signal (if the coupler were set to couple the forward power) or the reflected signal (if the coupler were set to couple the reverse power). In simulations, ideal directional‐couplers are often used in just such a manner.
1.4.2 Available Power
The maximum power that can delivered from a generator is called the available power, or PAvailable, and can be defined as the power delivered from a ZS
where ΓS is computed as in Eq. (1.24) as
(1.48)
This maximum power is delivered to the load when the load impedance is the conjugate of the source impedance,
1.4.3 Delivered Power
The power that is absorbed by an arbitrary load is called the delivered power, and it is computed directly from the difference between the incident and reflected power.
(1.49)
For most cases, this is the power parameter that is of greatest interest. In the case of a transmitter, it represents the power that is delivered to the antenna, for example, which in turn is the power radiated less the resistive loss of the antenna.
1.4.4 Power Available from a Network
A special case of available power is the power available from the output of a network, when the network is connected an arbitrary source. In this case, the available power is only a function of the network and the source impedance and is not a function of the load impedance. It represents the maximum power that could be delivered to a load under the condition that the load impedance was ideally matched and can be found by noting that the available output power is similar to Eq. (1.47) but with the source reflection coefficient replaced by the output reflection coefficient of the network Γ2 from Eq. (1.30) such that
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