Handbook of Microwave Component Measurements. Joel P. Dunsmore

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where P^F is called the forward power, and P^R is called the reverse power. To put this in terms of the voltage and current of Figure 1.1, the total voltage at the port can be defined as the sum of the forward voltage wave traveling into the port and the reverse voltage wave emerging from the port.

(1.4)

      The forward voltage wave represents a power traveling toward the load, or transferring from the source to the load, and the reflected voltage wave represents power traveling toward the source. To be formal, for a sinusoidal voltage source, the voltage as a function of time is

      (1.5)

      From this it is clear that

is the peak voltage and the root‐mean‐square (rms) voltage is

      (1.6)

      The

factor shows often in the following discussion of power in a wave, and it is sometimes a point of confusion; but if one remembers that rms voltage is what is used to compute power in a sine wave, and is used to refer to the wave amplitude of a sine wave in the following equations, then it will make perfect sense.

      Considering the source impedance Z_S and the line or port impedance Z₀, and simplifying a little by making Z_S = Z₀ and considering the case where Z₀ is pure‐real, one can relate the forward and reverse voltage to an equivalent power wave. If one looks at the reference point of Figure 1.1 and one had the possibility to insert a current probe as well as had a voltage probe, one could monitor the voltage and current.

      The source voltage must equal the sum of the voltage at port 1 and the voltage drop of the current flowing through the source impedance.

      (1.7)

      Defining the forward voltage as

(1.8)

we see that the forward voltage represents the voltage at port 1 in the case where the termination is Z₀. From this and Eq. (1.4), one finds that the reverse voltage must be

(1.9)

      If the transmission line in Figure 1.1 is long (such that the load effect is not noticeable) and the line impedance at the reference point is the same as the source, which may be called the port reference impedance, then the instantaneous current going into the transmission line is

      (1.10)

      The voltage at that point is same as the forward voltage and can be found to be

      (1.11)

The power delivered to the line (or a Z₀ load) is

      (1.12)

      From these definitions, one can now refer to the incident and reflected power waves using the normalized incident and reflected voltage waves, a and b as (Keysight Technologies 1968).

(1.13)

      Or, more formally as a power wave definition

(1.14)

where Eq. (1.14) includes the situation in which Z₀ is not pure real (Kurokawa 1965). However, it would be an unusual case to have a complex reference impedance in any practical measurement.

For real values of Z₀, one can define the forward or incident power as

and the reverse or scattered power as and see that the values a and b are related to the forward and reverse voltage waves, but with the units of square root of power. In practice, the definition of Eq. (1.13) is typically used, as the definition of Z₀ is almost always either 50 or 75 Ω. In the case of waveguide measurements, the impedance is not well defined and changes with frequency and waveguide type. It is recommended to simply use a normalized impedance of 1 for the waveguide impedance. This does not represent 1 Ω but is used to represent the fact that measurements in a waveguide are normalized to the impedance of an ideal waveguide. In Eq. (1.13) incident and reflected waves are defined, and in practice the incident waves are the independent variables, and the reflected waves are the dependent variables. Consider Figure 1.2, a 2‐port network.

Figure 1.2 2‐port network connected to a source and load.

      There are now sets of incident and reflected waves at each port i, where

(1.15)

The voltages and currents at each port can now be defined as

      (1.16)

      where Z_0i is the reference impedance for the ith port. An important point here that is often misunderstood is that the reference impedance does

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