Jan 19 class notes

content/notes/UWAA560/03-magnetic-field-diagnostics.md

@@ -77,7 +77,7 @@ V_O (s) = \omega_0 ^2 V_i (s) \left[ s^2 + s \left( \frac{R_C}{L_C} + \frac{1}{C
 \omega_0 ^2 = (L_C C_{eq} )^{-1}
 {{< /katex >}}
 
-If we assume a unit step waveform for the generator voltage (that is, \\( \dot B \\)), then \\( V_i(s) = \frac{1}{s} \\) and
+where \\( \omega_0  ^2 = (L_C C_{eq})^{-1} \\). If we assume a unit step waveform for the generator voltage (that is, \\( \dot B \\)), then \\( V_i(s) = \frac{1}{s} \\) and
 
 {{< katex display >}}
 V_O(s) = \frac{\omega_0 ^2}{s (s^2 + s \alpha + \omega_0 ^2 \beta)}
@@ -146,10 +146,6 @@ If we have some plasma current \\( j_z \\) producing a magnetic field \\( B_\the
 
 The key geometric property of the Rogowski coil is the constant inner coil radius (area).
 
-Fun fact: in the lab, we'll often make these coils by stripping the outer sheath and outer conductor from a length of RG58 coaxial cable. Then, we take a wire and solder it to the end of the inner conductor, then wrap the wire about the insulator of the RG58 back towards the un-stripped end. Finally, connect the wire to the outer conductor of the coaxial cable.
-
-<p align="center"> <img alt="14.png" src="/r/img/560/14.png" /> </p>
-
 The total magnetic flux, \\( \oint \vec B \cdot \dd \vec s \\), linked by the Rogowski coil is
 
 {{< katex display >}}
@@ -162,4 +158,25 @@ From Faraday's law,
 V_0 = - \dot \Phi = \mu_0 (N \int \dd S) \dot I
 {{< /katex >}}
 
-So the voltage we measure is related to the time rate of change of the current enclosed in the Rogowski coil. The signal \\( V_0 \\) is integrated in time to give the total current enclosed by the Rogowski coil, \\( I(t) \\). Just like the B-dot probes, instead of computing the geometry of the coil we calibrate the probe by driving current from a known source and measuring the response. They are easier to calibrate because the exact geometry does not matter, as long as the Rogowski coil encloses the current being measured.
+So the voltage we measure is related to the time rate of change of the current enclosed in the Rogowski coil. The signal \\( V_0 \\) is integrated in time to give the total current enclosed by the Rogowski coil, \\( I(t) \\). Just like the B-dot probes, instead of computing the geometry of the coil we calibrate the probe by driving current from a known source and measuring the response. They are easier to calibrate because the exact geometry does not matter, as long as the Rogowski coil encloses the current being measured.
+
+### Considerations
+
+- The signal is insensitive to the shape of the coil since \\( \oint \vec B \cdot \dd \vec l \\) is path independent. We do start to lose signal if we make the loop much larger than the source where the current source is, but as long as the coil is more or less snug about the current source the measurement error is minimal. However, the coil does need to form a complete loop.
+- Signals can be _very_ large, easily \\( \sim 1 kV \\). So adding an attenuator upstream of your measurement circuit is important.
+- The signal can be contaminated by unwanted \\( \dot{\vec B} \\), so the coils are counter-wound.
+<p align="center"> <img alt="18.png" src="/r/img/560/18.png" /> </p>
+In the lab, we'll often fabricate these coils by stripping the outer sheath and outer conductor from a length of RG58 coaxial cable. Then, we take a wire and solder it to the end of the inner conductor, then wrap the wire about the insulator of the RG58 back towards the un-stripped end. Finally, connect the wire to the outer conductor of the coaxial cable.
+<p align="center"> <img alt="14.png" src="/r/img/560/14.png" /> </p>
+
+# Hall Probe
+
+B-dot probes are not very useful for steady-state magnetic fields (it's right there in the name). For steady-state fields, we can exploit the Hall effect. Passing a known current through a conductor with an embedded magnetic field will generate a transverse potential:
+
+<p align="center"> <img alt="19.png" src="/r/img/560/19.png" /> </p>
+
+{{< katex display >}}
+\frac{ \vec j \cross \vec B}{nq} = - \vec E
+{{< /katex >}}
+
+A simultaneous measurement of the voltage drop and current gives a measurement of the embedded magnetic field. Such a device is usually called a Gaussmeter.

content/notes/UWAA560/04-electrostatic-diagnostics.md

@@ -0,0 +1,89 @@
+---
+title: Electrostatic Diagnostics
+weight: 40
+bookToc: false
+---
+
+{{< katex display >}}
+
+{{< /katex >}}
+
+# Langmuir Probes
+
+Langmuir probes are the oldest and simplest plasma diagnostic: You simply insert an electrode into the plasma and measure the \\( I \\)-\\( V \\) curve. Doing so is fairly simple, but the probe measurements are often difficult to interpret, since extracting accurate plasma properties from measurements can be challenging.
+
+Consider a probe inserted into a steady-state plasma. We assume that there is an electrode (e.g. the wall of the vacuum chamber) which is essentially in electrical contact with the plasma, and we will treat that as ground.
+
+<p align="center"> <img alt="20.png" src="/r/img/560/20.png" /> </p>
+
+Assuming a thermal plasma (Maxwellian) with \\( T_i \approx T_e \\), ions and electrons strike the probe in all directions. Assume the bulk plasma is static (the bulk velocity is zero). Then the particle flux for each species \\( s \\) is
+
+{{< katex display >}}
+\Gamma_s=\frac{1}{4} n_s \overline{v}_s
+{{< /katex >}}
+
+where \\( \overline{v}_s \\) is the mean particle speed. A probe of area \\( A \\) initially at zero potential inserted in a singly charged, quasi-neutral plasma will emit a current
+
+{{< katex display >}}
+I = - e A \frac{1}{4} (n_i \overline v _i - n_e \overline v _e) \\
+\approx \frac{1}{4} e A n_e \overline v_e > 0
+{{< /katex >}}
+
+A positive current here means the probe absorbs electrons. The probe potential, which started at ground, drops and repels incoming electrons until \\( \Gamma _e = \Gamma _i \\) such that \\( I = 0 \\). When this happens, we can measure the probe voltage to give what we call the "**floating potential**" \\( V_f \\). If the probe potential can be externally controlled, then we can find the floating potential without depleting the plasma of electrons. We can start out with a negative probe \\( V \\) and adjust it until the probe current vanishes.
+
+A thermal plasma by definition has a probability distribution of speed of the form
+
+{{< katex display >}}
+f(v) = \left( \frac{m}{2 \pi k T} \right) ^{3/2} 4 \pi v^2 e^{- \frac{m v^2}{2T}}
+{{< /katex >}}
+
+Since the electron mass is small, the speed distribution is higher than that of the ions for \\( T_i \approx T_e \\):
+
+<p align="center"> <img alt="21.png" src="/r/img/560/21.png" /> </p>
+
+As the probe potential is more positive, eventually no electrons are repelled. This potential is the **plasma potential** (or space potential) \\( V_p \\). Increasing \\( V > V_p \\) further produces no increase in current, because at \\( V_p \\) the probe is already absorbing all of the incoming electrons.
+
+As a point of reference, the plasma potential is approximately 5 times the electron temperature
+
+{{< katex display >}}
+V_p \approx 5 k T_e
+{{< /katex >}}
+
+An idealized I-V characteristic looks like this:
+
+<p align="center"> <img alt="22.png" src="/r/img/560/22.png" /> </p>
+
+Since \\( m_i \gg m_e \\), \\( I_{is} \ll I_{es} \\). If \\( T_i = T_e \\) then
+
+{{< katex display >}}
+\frac{I_{is}}{I_{es}} = \left( \frac{m_e}{m_i} \right)^{1/2}
+{{< /katex >}}
+
+The transition region is where we make most of our measurements of plasma parameters. As \\( V \\) is adjusted between \\( (V_f, V_p) \\), the electron current varies exponentially as
+
+{{< katex display >}}
+I_e = I_{es} e^{\frac{ V - V_p}{k T_e}}
+{{< /katex >}}
+
+where
+{{< katex display >}}
+I_{es} = \frac{1}{4} e A n_e \overline{v}_e = e n_e A \left( \frac{k T_e}{2 \pi m_e} \right)^{1/2}
+{{< /katex >}}
+
+If we measure the electron current \\( I_e \\) vs. \\( V \\), this allows us a measurement to get the electron temperature \\( T_e \\). Note that \\( I_e \\) is not the total current; we need to subtract the ion current \\( I_e = I - I_i \\). So to recover the electron temperature, we plot \\( \ln (I) \\) vs. \\( V \\) and check the slope in the transition region:
+
+<p align="center"> <img alt="23.png" src="/r/img/560/23.png" /> </p>
+
+Since \\( I_{is}  \\) is fairly constant at low \\( V \\), it is often easier to adjust \\( I_i \\) until the linear portion is maximized. With \\( T_e \\) determined, the density could be extracted from the electron saturation current \\( I_{es} \\). Looking at the form of \\( I_{es} \\), you can see that it depends only on the probe geometry, electron temperature, and density. So we could measure the density as
+
+{{< katex display >}}
+n_e = \frac{I_{es}}{e A} \left( \frac{k T_e}{2 \pi m_e} \right) ^{1/2}
+{{< /katex >}}
+
+but besides being difficult to measure, \\( I_{es} \\) can be very large, leading to Bad Things like altering the plasma properties, damaging the probe, or destroying the probe circuitry.
+
+Instead, we take a look at the other end of the \\( I \\)-\\( V \\) curve and determine the ion density \\( n_i \\) from the ion saturation current, which is generally less than \\( 10 \\% \\) of the electron saturation current.
+
+If the electrons have an energetic population (e.g. a beam or other population of high-temperature electrons) in addition to the thermal population, then the distribution is not Maxwellian. Useful information can still be obtained from the \\( I \\)-\\( V \\) characteristic, but it will contain additional inflection points
+
+<p align="center"> <img alt="24.png" src="/r/img/560/24.png" /> </p>

layouts/partials/docs/inject/content-after.html

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