add some transmission line stuff

book/01_tx_lines.md

@@ -25,7 +25,7 @@ $$
 
 ## Reflections
 
-At the end of a transmission line, voltage and current are related by the impedance of the load.
+At the end of a transmission line, voltage and current are related by the impedance of the load $Z_L$.
 Assuming the load is situated at $z=0$ the following boundary conditions are imposed:
 
 $$
@@ -94,21 +94,51 @@ An imperfectly matched but lossy ($R \ne 0$) load results in a reduced amplitude
 ![Reflection with arbitrary complex Z_L](../assets/tx_lines/reflection_complex.gif)
 
 
-You should notice that not all locations along the transmission line see the same peak voltages (shaded region in the above plots). This is a useful property for determining how well matched a device is. A peak detector can be physically moved along a transmission line to find the peaks and troughs of this standing wave. This ratio is called the Voltage Standing Wave Ratio (VSWR). These days VSWR is usually calculated from other measurements.
+### Standing Wave Ratio
+
+You should notice that not all locations along the transmission line see the same peak voltages (shaded region in the above plots). This is a useful property for determining how well matched a device is. A peak detector can be physically moved along a transmission line to find the peaks and troughs of this standing wave. This ratio is called the Voltage Standing Wave Ratio (VSWR). In practice, VSWR is generally calculated from other measurements.
+
+Peaks in the standing wave will occur wherever the peaks of $V_f$ and $V_r$ align. Similarly, the minimum voltage occurs wherever one wave is at its positive peak and the other is at the negative peak.
 
 $$
 \begin{align}
+V_{max} &= \left|\hat{V}_f\right| + \left|\hat{V}_r\right| \\
+V_{min} &= \left|\hat{V}_f\right| - \left|\hat{V}_r\right|
+\end{align}
+$$
+
+VSWR is defined as the ratio of these values
+
+
+$$
+\begin{align}
+\text{VSWR} &= \frac{\left|\hat{V}_f\right| + \left|\hat{V}_r\right|}{\left|\hat{V}_f\right| - \left|\hat{V}_r\right|} \\
+\left|\hat{V}_r\right| &= \left|\hat{V}_f\right| \left|\Gamma\right| \\
 \text{VSWR} &= \frac{1 + |\Gamma|}{1 - |\Gamma|}
 \end{align}
 $$
-TODO: derive VSWR
 
 VSWR is often reported as $N:1$. I'm not sure why since this carries no additional information than just $N$ but its a convention.
 For passive networks (no power is added), $0 < |\Gamma| < 1$ so $1 < \text{VSWR} < \infty$ where 1 is perfectly matched. VSWR does a good job of highlihting poorly-matched networks while making well-matched and decently-matched networks appear similar.
-In general, unless dealing with very high-Q structures (which can be de-tuned by additional resonances) you can consider $\text{VSWR} < 2$ to be well matched.
+In general, $\text{VSWR} < 2$ can be considered well matched.
 
-TODO: move these comments somewhere else
-I have mostly seen VSWR used for antenna or system interface specifications. It is particularly useful for worst-case estimates of cascaded performance since it is additive. We will get to this later.
+I have mostly seen VSWR used for antennas and system interface specifications. It is particularly useful for bounding worst case mismatch when creating a system cascade but we'll get to that later.
+
+## Return Loss
+
+Another way to look at reflections is with return loss. This is really just a decibel way to look at reflected power. Remember that $\Gamma$ is a voltage ratio, not power ratio so return loss is:
+
+$$
+\begin{align}
+\text{Return Loss} = 20 \log_{10}(|\Gamma|)
+\end{align}
+$$
+
+It is not uncommon to see this expressed as either negative (the right way) or positive values. In practice nothing has $|\Gamma| > 1$ so there really isn't any ambiguity regardless of which sign this is presented with.
+
+## Example
+
+TODO: add example plots and such
 
 ## Impedance Transformation