Section 1: Motivation
In the previous lesson, we explored when and why voltage waves reflect (bounce around) on transmission lines, and provided ways to visualize these reflections. The RF concepts in this lesson directly connect to the simulations. This lesson will provide explanations for what will be measured and modeled in the simulations to come.
Section 2: Reflection and transmission in log scale
When a port to an RF circuit is driven, some amount of energy is transmitted into the circuit, and some amount of energy is reflected back to the source. If the source impedance is perfectly matched to the load impedance (for example, a 50Ώ source to a 50Ώ load), then all of the power is transmitted from the source to the load there is no reflection. If the load is a short (0Ώ) or an open (infinite ohms), then all of the power is reflected back to the source; in both of these cases there is no lossy resistance to burn up the power.
For a given circuit, the amount of power that is reflected (or transmitted) is related to the amount of power delivered into the system. If you put in more power, you will have more energy bouncing around in the circuit the voltages will be higher. Therefore, we like to describe circuits by their relative gain or loss. We could describe an amplifier with a gain curve such as in Figure 2.2.1. However, it is easier to simply say that this particular amplifier outputs 20 times as much power as was input it has a power gain of 20. We might have a voltage divider that outputs 20 times less voltage signal than was input it has a voltage gain of 1/20, or 0.05. It is very important when specifying linear gain to say that it is a voltage gain or a current gain. A voltage divider may drop the voltage by 0.707 but the power by 0.5, since P = V^2 / R, and the square of 0.707 is 0.5.
Because the amount of power transmitted or reflected is often very large or very small, rather than using a linear scale (for example, saying that the voltage gain is 0.05), a logarithmic scale is usually used. These are still ratios (comparisons between how much went out, versus how much went in), so while technically these numbers don't have units we use a mathematical unit of decibel (dB). A decibel is a unit, like a percent is a unit it's not really a unit but it tells you that some specific math has been done to this number. For a percent, it's a multiplication times 100. For decibels, the relationship between the linear gain and the log gain depends on whether you have a voltage gain or a power gain.
So our amplifier with a linear power gain of 20 has a logarithmic power gain of 13dBm. Our voltage divider with the voltage gain of 0.05 has a logarithmic gain of 26dBm.
Section 3: SParameters
In the Lesson 1 movies, we saw short pulses bouncing along the lines. These pulses would sum constructively or destructively when they hit other pulses. When we saw movies of DC or digital voltage waves, it was harder to see the actual voltage waves. We saw the sum of the voltage waves along the transmission lines. But we actually have discrete incident (transmitted) and reflected waves.
SParameters are relationships of the voltage waves entering and leaving the system. They are the ratios of reflected and incident waves, a sort of gain how much leaves the system relative to how much went in. Recall the classic voltage waves traveling down a transmission line:
V_{1}^{+} is the wave going into this system, and V_{1}^{} is the wave going out of the system. In the movie above, the red plot was V_{1}^{}, and the blue plot was the sum of V_{1}^{ }plus V_{1}^{+}. It turns out that the ratio of the two voltage waves is related to the load (Z_{L}) and transmission line (Z_{0}) impedance. We are going to define the ratio of the voltage waves to be a "scattering parameter", or SParameter.
SParameters are defined as
relationships between the incident and the reflected waves. The
nomenclature is: S_{ri}, where 'r' is the port where energy is
coming out of the system and 'i' is the port where energy is entering the
system. For a 1port system, there is only one SParameter: S_{11}. Energy enters and leaves out of port 1, the only port.
Things get more interesting for a twoport network. For a two port systems, there are four SParameters: S_{11}, S_{21}, S_{12}, and S_{22}. The following definitions and points extend for whenever you have more than one port. While perhaps not intuitive, the voltage wave leaving the system (V_{1}^{ or }V_{2}^{)} is always referred to as the "reflected" wave, even if the V_{2}^{} wave is the result of (was excited by) V_{1}^{+}. Since V_{1}^{+} sends a wave to the right (as in Figure 2.3.2), and the V_{2}^{ } wave is also going to the right, it might not be obvious to call that a reflection but that's how it is defined. Consider the following twoport network with four relevant voltage waves.
As in the oneport case, the incident and reflected voltage waves are related to the SParameter.
It's not too difficult to see what we need to do to define the SParameter: one port has to be driven
(for example, V_{1}^{+ }=^{ }1) and the rest of the ports need to be turned off (for example, V_{2}^{+ }=^{ }0). But, not only can we not excite any voltage waves on port 2, we can't let any wave that leaves port 2 reflect back port 2 needs to be perfectly terminated. Recall in our movies in Lesson 1, if the right side is not loaded correctly, we get a lot of voltage bounces. But when we had our 50ohm transmission line terminated with 50ohms, we got the same effect as we are looking for when we say V_{2}^{+ }=^{ }0. So, port 2 must be matched typically to 50ohms.
At this point we have defined the relationship between the incident and reflected voltage waves to a single SParameter. The actual voltage is the sum of the incident and reflected voltage, V_{1} = V_{1}^{+}+ V_{1}^{}, as in Figure 2.3.3 (as copied from Lesson 1 Section 4).
Main Point
SParameters are relationships between voltage waves entering and leaving the circuit. SParameters are individually defined with one port driven, and all other ports perfectly terminated. Driving more than one port? SParameters can't be applied directly. Port has some mismatch? Expect to see changes in system gain and match. Fortunately, the relationships aren't hard to derive (or find, down below).
Section 4: Voltage versus Power Gain
SParameters are ratios of voltage waves, and describe reflections
and transmission of voltages. We can define these ratios of voltage waves as gains and losses, in the same way we described voltage and power gain in Lesson 1. The S_{21} parameter is really a voltage gain. The S_{11} parameter relates to the voltage wave that returns back to the source, rather than going on to the output. The S_{11} is often called the return loss, because it relates to energy that returns to the source, instead of going to the load as far as the load is concerned, that energy was lost.
Because power is related to voltage, one can describe the power gain of a circuit as a function of the SParameters. People often use the following equation:
Optional Note on Power Gains:
Equation 2.4.1 is not always correct, because it assumes that all of the ports are terminated in the same impedance. Similarly, people use 10*log10 for power gain, and 20*log10 for voltage gain. Often, we deal in systems that are all 50Ώ inputs and outputs. The general equation when the port impedances are not equal (but are real) for the power gain relating the SParameters to gain is:
Note that all we have done is preserved the power equation P=V^2 / R. As a side note, if Mason is run with different power impedances, it will adjust the SParameters accordingly.
Main Point
A device can be said to have a gain (or a loss) associated with the voltage waves that go through it. The SParameters are the gain of the voltage waves, either going through the system (such as an S_{21}) or reflecting off of the device back to the source (such as the S_{11}). When calculating the gain (or loss) of a circuit, take care to account for the actual impedances of the system.
Section 5: Interesting Scattering Parameter Properties
Scattering Parameters are very useful to RF designers. For one, it turns out that it is easier to measure voltage waves than actual voltages in an RF circuit. When you have pulses and sine waves going through an RF device, with lots of different reflections in the circuit due to mismatches, trying to describe what's going on with a single voltage or current becomes somewhat arbitrary. SParameters also have many interesting properties which can help provide insight.
System of SParameter Equations
Consider a power divider many houses have an RF power divider which splits the cable signal to different TVs around the house. Consider specifically a fourway power divider, where port 1 is the input (the cable TV signal goes into here) and ports 25 are the output (these signals are sent to TVs around the house). If port 1 is driven, it makes sense that port 2 will have a quarter of the power coming out (as a voltage wave, 6dB less than the input). The actual SParameter equations for this system are:
Now, writing out equations like this can be cumbersome, so we are going to short hand these equations by putting the equivalent variables in matrix formulation. If you aren't familiar with linear algebra, don't worry about the mechanics, just notice that the variables follow the same pattern... and we're going to be talking about patterns. I assure you, these are equivalent sets of equations.This means the device has symmetry to it. Let's consider the above device where port 1 is driven by the cable signal from outside your house, and port 2 is one of four ports where power is split out to your TV. Equation 2.5.3 says that if S_{21} is 6dB (25% of the power goes to port 2 from port 1), then S_{12} is also 6dB. If you put 1W into port 1, then 0.25W will come out of port 2 (assuming everything is matched, no bad reflections). If you put 1W into port 2, then 0.25W will come out of port 1 (assuming everything is matched, no bad reflections). So, S_{21} = S_{12} for a reciprocal device.
If we consider a transistor amplifier, the device has gain. The voltage wave coming out of port 2 will be larger than the voltage wave coming out of port 1, for example we described a 13dB gain amplifier above: S_{21} = 13dB. However, if you connect the device backwards, if you drive power into the output, not much energy will come out of the input: S_{21} ≠ S_{12}._{ }The S_{12} is often called the isolation of the amplifier, this is how well what the input is isolated from what happens on the output, and is commonly on the order of 20dB (or only 1% of the power gets through).
_{Figure 2.5.2: Amplifier with 13dB gain and 20dB isolation }
Section 6: (Advanced topic) SParameters and multiple driven ports
SParameters assume one port is driven and the rest are terminated if this condition isn't met, then these gain numbers won't directly apply to the observation. However, you can still use SParameters to calculate what will happen when multiple ports are driven. Consider the fourway power divider in Figure 2.5.1, where port 1 is the input (the cable TV signal goes into here) and ports 25 are the output (these signals are sent to TVs around the house). If port 1 is driven, it makes sense that port 2 will have around a quarter of the power coming out (6dB less than the input). But if we turn this around, and Ports 25 are driven, obviously the output from Port 1 is not going to go down to quarter power (6dB gain): if we drive multiple ports, we expect more power coming out then we put into any single port we expect power to be conserved. We expect four times the power of any single port. The problem is we are breaking that SParameter condition. Two approaches can be taken.
Excess Loss
The quick approach is the concept of
excess loss: how much loss do we see beyond the expected power
divider loss. For example, the output of a fourway divider should be 6dB lower than the input. In reality, the device will have some additional loss in the metal wires and the dielectrics; the actual
divider might measure 7dB. The 1dB difference between the theoretical
loss and the measured loss is the excess loss, and it is the power
burned up inside the device and/or reflected by the inputs (that S_{11} return loss keeps
some power from even entering the divider).
A quick way to approximate the combined power coming out of Port 1 with Ports 25 driver is to simply sum the
power going into the combiner (e.g., 1W into four ports is 4W, or
6dBW) and subtract out the 1dB of excess loss (so 5dBW is at the
combiner port, or 3.2W).
Real Math
Perhaps you want a more exact answer, or don't trust "engineering" math. Doing the math for real isn't hard.
The voltage wave coming out of port 1 (the summed port) is equal to the weighted sum of all of the
incident waves. V_{1}^{+} is equal to zero (this is the output port), and assuming all ports are driven equally then.... that's just the standard power / voltage / resistance relationship.
So, doing it for real really isn't so hard. If your intuition is throwing a red flag, go through the math:

For a fourway combiner, from the port 1 single input to the ports 25 outputs, the transmission Sparameters are S_{21} = S_{31} = S_{41} = S_{51} = 0.5 (SParameters are voltage ratios, and the square of the voltage ratio is the power ratio, so the power ratio is 0.25)

The sum of incident power is S_{21}^{2} + S_{312} + S_{41}^{2 }+ S_{51}^{2} = 1

The incident voltages are 7.07V for 1W incident power and a 50ohm system.

The output voltage wave (V_{1}^{}) is 28.28V

The output power is 4W.
Main Point
Even though SParameters don't individually apply directly when multiple ports are driven, it's easy to get expected power output from the SParameters. Please remember that these voltage waves are complex vectors, keep track of phase as you do the math!
Section 7: Smith Charts
While the Smith Chart is not the only way to look at input and output impedances of RF circuits, it has strong advantages over alternative methods. The basic Smith Chart is shown in Figure 2.7.1. All of the impedances that are positive and real (0Ώ, 50Ώ, 1000Ώ, etc.) are mapped to the horizontal line that crosses the center (between the short and the open). All of the impedances with a positive real part (0+j50, 50j20, 100+j12) are mapped inside of the circle: above the real line are the positive imaginay parts (see Figure 2.7.2), and below the real line are the negative imaginary parts. All of the negative real values are mapped outside of the circle (10Ώ, 50Ώ, etc).
Being able to visualize the impedance helps is a couple of ways. The closer you are to the center of the Smith Chart circle, the closer you are to 50ohms (well matched). You can see how close
Figure 2.7.3 and Figure 2.7.4 show the same data presented in two different ways, a rectangular plot and a Smith Chart. Both ways of looking at the data have advantages and disadvantages. The rectangular plot has the obvious advantage that one can read off the actual impedance versus frequency directly. The Smith Chart has the obvious disadvantage that there is not frequency axis. The main advantage of the Smith Chart is that one can, with experience, quickly glean qualitative relationships. For example, at a glance one can tell from the Smith Chart that:

the circuit is matched at the center frequency

the circuit has the same amount of mismatch at the edge frequencies

pulling the real impedance lower (shifting the curve to the left) will improve the overall match
These are all observations that can be made from the rectangular plot, just not as quickly.
Section 8: SParameters equations
By understanding the fundamentals of
SParameters, one can better apply them to the design process. Since
Circuit Mason and Mason Plot both use an advanced math parser (it takes human readable text equations and compiles it into machine language byte code that runs efficiently), one
can optimize and plot not just for simple SParameters but instead
potentially more relevant figures of merit. Simple SParameter terms
such as return loss or simple gain are not defined below.
These
are just a few figures of merits commonly found. To make it easier to
copy and paste into Circuit Mason or Mason Plot, all these equations
will be in text form.
Whereas the traditional SParameter gain penalizes the device for mismatch, the insertion loss only looks at the loss (or gain) within the device itself.
IL = 10*log10(pow(abs(S_2_1),2) / (1pow(abs(S_1_1),2)))
VSWR (Voltage Standing Wave Ratio)
Optimizing for VSWR is the same as optimizing for return loss, but one may want to plot VSWR.
VSWR = (1+abs(S_1_1)) / (1abs(S_1_1))
Relative Phase
To compare phasing, it is nice to look at relative phase shifts, and to compare these plots on top of each other you may need to add some fixed phase delay.
arg(S_3_1/S_2_1)+90 [S_3_1 relative
to S_2_1, shifted by 90degrees]
arg(S_5_1/S_2_1)90 [S_5_1
relative to S_2_1, shifted by 90degrees]
Mason Plot does phase unwrapping
in degrees (using the Arg plot option). However, mostly as an
example, this is an alternative way to unwrap by equations (using the
dB plot option).
if(arg(S_4_1/S_2_1)+180<180,arg(S_4_1/S_2_1)+180,arg(S_4_1/S_2_1)180)
[S_5_1 relative to S_2_1, shifted by +/180degrees]
Main Point
Understanding how and when to apply the different figures of merit is an important factor in design.