Читать книгу Handbook of Microwave Component Measurements - Joel P. Dunsmore - Страница 20

S‐parameters have been developed in the context of microwave measurements but have a clear relationship to voltages and currents that are the common reference for most electrical engineers. This section will develop the definition of traveling waves and from that the definition of S‐parameters, in a way that is both rigorous and ideally intuitive; the development will be incremental, rather than just quoting results, in hopes of engendering an intuitive understanding.

This signal traveling along a transmission line is known as a traveling wave (Marks and Williams 1992) and has a forward component and a reverse component. Figure 1.1 shows the schematic of a two‐wire transmission structure with a source and a load.

Figure 1.1 Voltage source and two‐wire system.

If the voltage from the source is sinusoidal, it is represented by the phasor notation

(1.1)

The voltage and current at the load are

(1.2)

The voltage along the line is defined as V(z), and the current at each point is I(z). The impedance of the transmission line provides for a relationship between the voltage and the current. At the reference point, the total voltage is V(0) and is equal to V₁; the total current is I(0). The power delivered to the load can be described as

(1.3)

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 not have to be the same as the port impedance or the impedance of the network. It is a “nominal” impedance; that is, it is the impedance that we “name” when we are determining the S‐parameters, but it need not be associated with any impedance in the circuit. Thus, a 50 Ω test system can easily measure and display S‐parameters for a 75 Ω device, referenced to 75 Ω.

The etymology of the term reflected derives from optics and refers to light reflecting off a lens or other object with an index of refraction different from air, whereas it appears that the genesis for the scattering or S‐matrix was derived in the study of particle physics, from the concept of wavelike particles scattering off crystals. In microwave work, scattering or S‐parameters are defined to relate the independent incident waves to the dependent waves; for a 2‐port network they become

(1.17)

which can be placed in matrix form as

(1.18)

where a's represent the incident power at each port, that is, the power flowing into the port, and b's represent the scattered power, that is, the power reflected or emanating from each port. For more than two ports, the matrix can be generalized to

(1.19)

From Eq. (1.17) it is clear that it takes four parameters to relate the incident waves to the reflected waves, but Eq. (1.17) provides only two equations. As a consequence, solving for the S‐parameters of a network requires that at least two sets of linearly independent conditions for a₁ and a₂ be applied, and the most common set is one where first a₂ is set to zero, the resulting b waves are measured, and then a₁ is set to zero, and finally a second set of b waves are measured. This yields

(1.20)

which is the most common expression of S‐parameter values as a function of a and b waves, and often the only one given for their definition. However, there is nothing in the definition of S‐parameters that requires one or the other incident signals to be zero, and it would be just as valid to define them in terms of two sets of incident signals, a_n and , and reflected signals, b_n and .

(1.21)

From Eq. (1.21) one sees that S‐parameters are in general defined for a pair of stimulus drives. This will become quite important in more advanced measurements and in the actual realization of the measurement of S‐parameters, because in practice it is not possible to make the incident signal go to zero because of mismatches in the measurement system.

These definitions naturally lead to the concept that S_nn parameters are reflection coefficients and are directly related to the DUT port input impedance and S_mn parameters are transmission coefficients and are directly related to the DUT gain or loss from one port to another.

Now that the S‐parameters are defined, they can be related to common terms used in the industry. Consider the circuit of Figure 1.3, where the load impedance Z_L may be arbitrary and the source impedance is the reference impedance.

Figure 1.3 1‐port network.

From inspection one can see that

(1.22)

which is substituted into Eq. (1.8) and Eq. (1.9), and from (1.15) one can directly compute a₁ and b₁ as

(1.23)

From here S₁₁ can be derived from inspection as

(1.24)

It is common to refer to S11 informally as the input impedance of the network, where

(1.25)

This is clearly true for a 1‐port network and can be extended to a 2‐port or n‐port network if all the ports of the network are terminated in the reference impedance; but in general, one cannot say that S₁₁ is the input impedance of a network without knowing the termination impedance of the network. This is a common mistake that is made with respect to determining the input impedance or S‐parameters of a network. S₁₁ is defined for any terminations by Eq. (1.21), but it is the same as the input impedance of the network only under the condition that it is terminated in the reference impedance, thus satisfying the conditions for Eq. (1.20). Consider the network of Figure 1.2 where the load is not the reference impedance; as such, it is noted that a₁ and b₁ exist, but now Γ₁ (also called Γ_In for a 2‐port network) is defined as

(1.26)

with the network terminated in an arbitrary impedance. As such, Γ₁ represents the input impedance of a system comprised of the network and its terminating impedance. The important distinction is that S‐parameters of a network are invariant to the input of output terminations, providing they are defined to a consistent reference impedance, whereas the input impedance of a network depends upon the termination impedance at each of the other ports. The value of Γ₁ of a 2‐port network can be directly computed from the S‐parameters and the terminating impedance, Z_L, as

(1.27)

where Γ_L computed as in Eq. (1.24) is

(1.28)

or in the case of a 2‐port network terminated by an arbitrary load then

(1.29)

Similarly, the output impedance of a network that is sourced from an arbitrary source impedance is

(1.30)

Another common term for the input impedance is the voltage standing wave ratio, called VSWR (also simply called SWR), and it represents the ratio of maximum voltage to minimum voltage that one would measure along a Z₀ transmission line terminated in some arbitrary load impedance. It can be shown that this ratio can be defined in terms of the S‐parameters of the network as

(1.31)

If the network is terminated in its reference impedance, then Γ₁ becomes S₁₁. Another common term used to represent the input impedance is the reflection coefficient, ρ_In, where

(1.32)

It's also common to write

(1.33)

Another term related to the input impedance is return loss, which is alternatively defined as

(1.34)

with the second definition being most properly correct, as loss is defined to be positive in the case where a reflected signal is smaller than the incident signal. But, in many cases, the former definition is more commonly used; the microwave engineer must simply refer to the context of the use to determine the proper meaning of the sign. Thus, an antenna with 14 dB return loss would be understood to have a reflection coefficient of 0.2, and the value displayed on a measurement instrument might read −14 dB.

For transmission measurements, the figure of merit is often gain or insertion loss (sometimes called isolation when the loss is very high). Typically this is expressed in dB, and similarly to return loss, it is often referred to as a positive number. Thus

(1.35)

Insertion loss or isolation is defined as

(1.36)

Again, the microwave engineer will need to use the context of the discussion to understand that a device with 40 dB isolation will show on an instrument display as −40 dB, due to the instrument using the evaluation of Eq. (1.35).

Notice that in the return loss, gain, and insertion loss equations, the dB value is given by the formula 20log₁₀(|S_nm|), and this is often a source of confusion because common engineering use of dB has the computation as X_dB = 10log₁₀(X). This apparent inconsistency comes from the desire to have power gain when expressed in dB be equal to voltage gain, also expressed in dB. In a device sourced from a Z₀ source and terminated in a Z₀ load, the power gain is defined as the power delivered to the load relative to the power delivered from the source, and the gain is

(1.37)

The power from the source is the incident power |a₁|², and the power delivered to the load is |b₂|². The S‐parameter gain is S₂₁ and in a matched source and load situation is simply

(1.38)

So computing power gain as in Eq. (1.37) and converting to dB yields the familiar formula

(1.39)

A few more comments on power are appropriate, as power has several common meanings that can be confused if not used carefully. For any given source, as shown in Figure 1.1, there exists a load for which the maximum power of the source may be delivered to that load. This maximum power occurs when the impedance of the load is equal to the conjugate of the impedance of the source, and the maximum power delivered is

(1.40)

But it is instructive to note that the maximum power as defined in Eq. (1.40) is the same as |a₁|² provided the source impedance is real and equals the reference impedance; thus, the incident power from a Z₀ source is always the maximum power that can be delivered to a load. The actual power delivered to the load can be defined in terms of a and b waves as well.

(1.41)

If one considers a passive two‐port network and conservation of energy, power delivered to the load must be less than or equal to the power incident on the network minus the power reflected, or in terms of S‐parameters

(1.42)

which leads the well‐known formula for a lossless network

(1.43)