diff --git a/lectures/l00_diode.md b/lectures/l00_diode.md index fd0185b..be58f29 100644 --- a/lectures/l00_diode.md +++ b/lectures/l00_diode.md @@ -1,25 +1,24 @@ ---- -abstract: | - I try to explain how diodes work. -author: -- Carsten Wulff, 2023-10-26, v0.1.1 -title: Diodes -documentclass: IEEEtran -papersize: a4 +footer: Carsten Wulff 2023 +slidenumbers:true +autoscale:true +theme: Plain Jane, 1 +text: Helvetica +header: Helvetica date: 2023-10-26 -classoption: journal -colorlinks: true ---- +# Diodes + +--- # Why -Diodes are a magical [^1] semiconductor device that conduct -current in one direction. It's one of the fundamental electronics components, -and it's a good idea to understand how they work. +Diodes are a magical [^1] semiconductor device that conduct current in one direction. It's one of the fundamental electronics components, and it's a good idea to understand how they work. + + + + +--- + + - As you hopefully know, the energy levels of an electron around a positive nucleus are quantized, and we call them orbitals (or shells). For an atom far @@ -85,47 +85,79 @@ valence band. Electrons can move both in the conduction band, as free electrons, and in the valence band, as a positive particle, or hole. +--> + # Intrinsic carrier concentration -The intrinsic carrier concentration of silicon, or how many free electrons and -holes at a given temperature, is given by +The intrinsic carrier concentration of silicon, or how many free electrons and holes at a given temperature, is given by + + +--- + + + +$$ +n_i = \sqrt{N_c N_v} e^{-\frac{E_g}{2 k T}} +$$ + +--- -\begin{equation} - n_i = \sqrt{N_c N_v} e^{-\frac{E_g}{2 k T}} -\tag{1} -\label{eq:ni} -\end{equation} + + + The density of states are $$ N_c = 2 \left[\frac{2 \pi k T m_n^*}{h^2}\right]^{3/2} \text{ } N_v = 2 \left[\frac{2 \pi k T m_p^*}{h^2}\right]^{3/2} $$ - + + +--- + + + +In BSIM 4.8 [@bsim] the intrinsic carrier concentration is $$ n_{i} = 1.45e10 \frac{TNOM}{300.15} \sqrt{\frac{T}{300.15} \exp^{21.5565981 - \frac{E_g}{2kT}}} $$ + + + +![right fit Intrinsic carrier concentration versus temperature\label{fig:ni}](../media/ni.pdf) + + At room temperature the intrinsic carrier consentration is approximately @@ -143,6 +175,12 @@ read the next section. If you don't care, and just want to memorize the equations, or indeed the number of intrinsic carrier concentration number at room temperature, then skip the next section. +--> + + + + + +--- ## Density of states + + +[.column] + $$-\frac{\hbar^2}{2m}\nabla^2\psi = E\psi$$ + + $$N(dk) = \frac{2}{(2 \pi)^p} dk$$ + + + $$E(k) = \frac{\hbar^2 k^2}{2 m^*}$$ + + +[.column] + $$m^* = \frac{\hbar^2}{\frac{d^2 E}{dk^2}}$$ + + $$N(E)dE = \frac{2}{\pi^2}\frac{m^*}{\hbar^2}^{3/2} E^{1/2}dE$$ + + +--- + + + +$$f(E) = \frac{1}{e^{(E - E_F)/kT} + 1} $$ + +--- + + + +$$N_e dE = N(E)f(E)dE$$ + + + +--- $$ n_e = 2\left( \frac{2 \pi m^\ast k T}{h^2}\right)^{3/2} e^{(E_F - E_C)/kT} $$ +--- + + + +For intrinsic silicon at thermal equlibrium, we could write + +$$ +n_0 = 2\left( \frac{2 \pi m^\ast k T}{h^2}\right)^{3/2} e^{-E_g/(2kT)} +$$ + + +--- + + + # Doping + + + The number of electrons and holes in a n-type material is $$ n_n = N_D \text{ , } p_n = \frac{n_i^2}{N_D} $$ @@ -402,12 +515,20 @@ and in a p-type material $$ p_p = N_A \text{ , } n_p = \frac{n_i^2}{N_A} $$ + + +--- + # PN junctions + + +--- + ## Built-in voltage + + +[.column] + $$ n = \int_{E_C}^{\infty} N(E) f(E) dE $$ + + $$ n_n = e^{E_{F_n}/kT} \int_{E_C}^{\infty} N_n(E) e^{-E/kT}dE $$ + + $$ n_p = e^{E_{F_p}/kT} \int_{E_C}^{\infty} N_p(E) e^{-E/kT}dE $$ + + $$ \frac{n_n}{n_p} = \frac{ e^{E_{F_n}/kT}}{e^{E_{F_p}/kT}} = e^{(E_{F_n} - E_{F_p})/kT} $$ + + +[.column] + $$ E_{F_n} - E_{F_p} = q\Phi $$ + + $$\frac{N_A N_D}{n_i^2} = e^{q\Phi_0/kT}$$ or rearranged to $$\Phi_0 = \frac{kT}{q} ln\left( \frac{N_A N_D}{n_i^2} \right)$$ +--- + ## Current + + +[.column] + $$ \frac{p_p}{p_n} = e^{-q\Phi_0/kT} $$ + + $$ \frac{p(-x_{p0})}{p(x_{n0})} = e^{q(V-\Phi_0)/kT} $$ + + $$ \frac{p(x_{n0})}{p_n} = e^{qV/kT} $$ + + +[.column] + $$ \Delta p_n = p(x_{n0}) - p_n = p_n\left( e^{qV/kT} -1 \right) $$ + + $$ J(x_n) = -q D_p \frac{\partial \rho}{\partial x} $$ + + $$ \partial \rho(x_n) = \Delta p_n e^{-x_n/L_p} $$ + + $$ J(0) = q\frac{D_p}{L_p} p_n \left( e^{qV/kT} - 1\right) $$ + + +--- + $$ I = q A n_i^2 \left( \frac{1}{N_A}\frac{D_n}{L_n} + \frac{1}{N_D}\frac{D_p}{L_p} \right)\left[ e^{qV/kT} - 1 \right] $$ +--- + + + +--- ## Forward voltage temperature dependence + + $$ V_D = V_T \ln\left(\frac{I_D}{I_S}\right) $$ +--- + + + +[.column] + $$ V_D = V_T \ln{I_D} - V_T \ln{I_S} $$ $$ \ln{I_S} = 2 \ln{n_i} + \ln{Aq\left (\frac{D_n}{L_n N_A} + \frac{D_p}{L_p N_D}\right)} $$ + + $$ n_i = \sqrt{B_c B_v} T^{3/2} e^\frac{-E_g}{2 kT} $$ + + + + $$ B_c = 2 \left[\frac{2 \pi k m_n^*}{h^2}\right]^{3/2} \text{ } B_v = 2 \left[\frac{2 \pi k m_p^*}{h^2}\right]^{3/2} $$ +[.column] + $$ 2 \ln{n_i} = 2\ln{\sqrt{B_c B_v}} + 3 \ln T - \frac{V_G}{V_T}$$ + + $$ V_D = \frac{kT}{q}(\ell - 3 \ln T) + V_G $$ + + $$ \ell= \ln{I_D} - \ln{\left (Aq\frac{D_n}{L_n N_A} + \frac{D_p}{L_p N_D}\right)} - 2 \ln{\sqrt{B_c B_v}} $$ +--- + + + From equations above we can see that at 0 K, we expect the diode voltage to be equal to the bandgap of silicon. Diodes don't work at 0 K though. -Although it's not trivial to see that the diode -voltage has a negative temperature coefficient, if you do compute it as in + + +Although it's not trivial to see that the diode voltage has a negative temperature coefficient, if you do compute it as in [vd.py](https://github.com/wulffern/memos/blob/main/2021-07-08_diodes/vd.py), then you'll see it decreases. The slope of the diode voltage can be seen to depend on the area, the current, doping, diffusion constant, diffusion length and the effective masses. + + -![Diode forward voltage as a function of temperature \label{fig:vd}](../media/vd.pdf) +![right fit Diode forward voltage as a function of temperature \label{fig:vd}](../media/vd.pdf) +--- + ## Current proportional to temperature + + $$ I_S e^\frac{qV_{D1}}{kT} = N I_S e^\frac{qV_{D2}}{kT} $$ Taking logarithm of both sides, and rearranging, we see that $$ V_{D1} - V_{D2} = \frac{kT}{q}\ln{N}$$ + + +![right fit Circuit to generate a current proportional to kT\label{fig:ptat}](../media/l3_ptat.pdf) +--- + + + +#[fit] Thanks! + + +--- + + + [^1]: It doesn't stop being magic just because you know how it works. Terry Pratchett, The Wee Free Men + + diff --git a/lectures/l00_refresher.md b/lectures/l00_refresher.md index c230519..3cf5425 100644 --- a/lectures/l00_refresher.md +++ b/lectures/l00_refresher.md @@ -1,44 +1,66 @@ ---- -title: A refresher of the first 4 years of electronics -layout: post +footer: Carsten Wulff 2023 +slidenumbers:true +autoscale:true +theme: Plain Jane, 1 +text: Helvetica +header: Helvetica date: 2023-10-26 -permalink: refresh ---- + + +# A Refresher + +--- -* TOC -{:toc } +# There are standard units of measurement -# There is a standard unit of measurement +All known physical quantities are derived from 7 base units ([SI units](https://en.wikipedia.org/wiki/International_System_of_Units)) -All known physical quantities are derived from 7 base units ([SI -units](https://en.wikipedia.org/wiki/International_System_of_Units)) -, second (s), meter (m), kg (kilogram), ampere (A), kelvin (K), candela (cd). +- second (s) : time +- meter (m) : space +- kg (kilogram) : weight +- ampere (A) : current +- kelvin (K) : temperature +- candela (cd) : luminous intensity All other units (for example volts), are derived from the base units. -I don't go around remembering all of them, they are easily available online. When you forget the equation for charge (Q), voltage (V) and capacitance (C), look at the units below, and you can see it's $Q = C V$[^1] + +![left](https://www.nist.gov/sites/default/files/images/2021/08/23/NIST.SP_.1247.png) + +--- # Electrons Electrons are fundamental, they cannot (as far as we know), be divided into smaller parts. Explained further in the [standard model of particle physics](https://en.wikipedia.org/wiki/Standard_Model) +![right 60%](../media/standard_model.pdf) + + + An electron cannot occupy the same quantum state as another. This rule that applies to all Fermions (particles with spin of 1/2) The quantum state of an electron is fully described by it's spin, momentum (p) and position in space (r). +--- + # Probability The probability of finding an electron in a state as a function of space and time is @@ -49,6 +71,8 @@ $$ P = |\psi(r,t)|^2 $$ $$ \psi(r,t) = A e^{i( kr - \omega t)}$$ + + +--- + # Uncertainty principle We cannot, with ultimate precision, determine both the position and the momentum of a particle, the precision is $$\sigma_x \sigma_p \ge \frac{\hbar}{2}$$ -From the [uncertainty principle](https://en.wikipedia.org/wiki/Uncertainty_principle) we can actually [estimate the size of the atom](https://wulffern.github.io/aic2023/atom) +From the [uncertainty (Unschärfe) principle](https://en.wikipedia.org/wiki/Uncertainty_principle) we can actually [estimate the size of the atom](https://wulffern.github.io/aic2023/atom) + +--- # States as a function of time and space @@ -72,16 +102,23 @@ $$ i\hbar \frac{d}{dt} \psi(r,t) = H \psi(r,t)$$ , where H is named the Hamiltonian matrix, or the energy matrix or (if I understand correctly) the amplitude matrix of the probability amplitude to change from one state to another. + + +--- + # Allowed energy levels in atoms -Solutions to Schrodinger result in quantized energy levels for an electron bound -to an atom. +Solutions to Schrodinger result in quantized energy levels for an electron bound to an atom. + + -The probability of a state transition (change in energy) can be determined from -the probability amplitude and Schrodinger. +The probability of a state transition (change in energy) can be determined from the probability amplitude and Schrodinger. +--- # Allowed energy levels in solids + + $$ H = \begin{bmatrix} A & 0 \\ @@ -128,78 +170,128 @@ and $$ E_2 = E_0 - A$$ + + -![](https://upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Solid_state_electronic_band_structure.svg/640px-Solid_state_electronic_band_structure.svg.png) +The discrete energy levels of the electron transition into bands of allowed energy states. +![right fit](https://upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Solid_state_electronic_band_structure.svg/640px-Solid_state_electronic_band_structure.svg.png) For a crystal, the allowed energy bands is captured in the [band structure](https://en.wikipedia.org/wiki/Electronic_band_structure) +--- + # Silicon Unit Cell A [silicon](https://en.wikipedia.org/wiki/Silicon) crystal unit cell is a diamond faced cubic with 8 atoms in the corners spaced at 0.543 nm, 6 at the center of the -faces, and 4 atoms inside the unit cell at a nearest neighbor distance of 0.235 -nm. - -![](https://upload.wikimedia.org/wikipedia/commons/f/f1/Silicon-unit-cell-3D-balls.png) +faces, and 4 atoms inside the unit cell at a nearest neighbor distance of 0.235 nm. +![left fit](https://upload.wikimedia.org/wikipedia/commons/f/f1/Silicon-unit-cell-3D-balls.png) -# Valence band and Conduction band - -The full band structure of a silicon unit cell is complicated. For bulk silicon -we simplify, and we think of two bands. In the conduction band ($E_C$) is the -lowest energy where electrons are free (not bound to atoms). The valence band -($E_V$) is the highest band where electrons are bound to silicon atoms. +--- -The difference between $E_C$ and $E_V$ is a property of the material we've named the band gap. +# Band structure -$$ E_G = E_C - E_V$$ +The full band structure of a silicon unit cell is complicated, it's a [3 dimensional concept](http://lampx.tugraz.at/~hadley/ss1/semiconductors/silicon_bandstructure.php) +![right fit](https://lampx.tugraz.at/~hadley/ss1/semiconductors/Si_bandstructure.png) -# Metals -In metals, the band splitting of the energy levels causes the valence band and conduction band to overlap. As such, electrons can easily transition between bound state and free state. As such, electrons in metals are shared over large distances, and there are many electrons readily available to move under an applied field, or difference in electron density. That's why metals conduct well. +--- -# Insulators +# Valence band and Conduction band -In insulating materials the difference between the conduction band and the valence band is large. As a result, it takes a large energy to excite electrons to a state where they can freely move. +For bulk silicon we simplify, and we think of two bands, the conduction band, and valence band -That's why glass is transparent to optical frequencies. Visible light does not have sufficient energy to excite electrons from a bound state. + -In a silicon the bandgap is lower than an insulator, approximately $E_G = 1.12\text{ } eV$ for silicon. +$$ E_G = E_C - E_V$$ -At room temperature, that allows a small number of electrons to be excited into the conduction band, leaving behind a "hole" in the valence band. +--- # Fermi level + + +> In band structure theory, used in solid state physics to analyze the energy levels in a solid, the Fermi level can be considered to be a hypothetical energy level of an electron, such that at thermodynamic equilibrium this energy level would have a 50% probability of being occupied at any given time + + + $$ f(E) = \frac{1}{e^{(E - E_F)/kT} + 1} $$ + + $$ f(E) \approx e^{(E_F - E)/kT} $$ +--- + +# Metals + +In metals, the band splitting of the energy levels causes the valence band and conduction band to overlap. + +![left fit](https://upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Band_filling_diagram.svg/1024px-Band_filling_diagram.svg.png) + + + +--- + +# Insulators + +In insulating materials the difference between the conduction band and the valence band is large. As a result, it takes a large energy to excite electrons to a state where they can freely move. + +That's why glass is transparent to optical frequencies. Visible light does not have sufficient energy to excite electrons from a bound state. + +That's also why glass is opaque to ultra-violet, which has enough energy to excite electrons out of a bound state. + +Based on these two pieces of information you could estimate the bandgap of glass. + +--- + +# Semiconductors + +In a silicon the bandgap is lower than an insulator, approximately + +$$E_G = 1.12\text{ } eV$$ + +At room temperature, that allows a small number of electrons to be excited into the conduction band, leaving behind a "hole" in the valence band. + +--- + # Band diagrams @@ -207,14 +299,18 @@ A [band diagram](https://en.wikipedia.org/wiki/Band_diagram) or energy level diagrams shows the conduction band energy and valence band energy as a function of distance in the material. -![](https://upload.wikimedia.org/wikipedia/commons/4/43/Pn-junction_zero_bias.png) +![right fit](https://upload.wikimedia.org/wikipedia/commons/4/43/Pn-junction_zero_bias.png) The horizontal axis is the distance, the vertical axis is the energy. The figure shows a PN-junction +--- + # Density of electrons/holes + + $$ n_e = \int_{E_C}^{\infty} N(E)f(E) dE$$ + + $$ n_e = e^{E_F/kT} \int_{E_C}^{\infty} N(E) e^{-E/kT}dE$$ + + +--- # Fields -There are equations that relate electric field, magnetic field, charge density -and current density to each-other. +There are equations that relate electric field, magnetic field, charge density and current density to each-other. -The equations +--- $$ \oint_{\partial \Omega} \mathbf{E} \cdot d\mathbf{S} = \frac{1}{\epsilon_0} \iiint_{V} \rho \cdot dV$$ ,relates net electric flux to net enclosed electric charge +--- + $$ \oint_{\partial \Omega} \mathbf{B} \cdot d\mathbf{S} = 0$$ ,relates net magnetic flux to net enclosed magnetic charge +--- + $$ \oint_{\partial \Sigma} \mathbf{E} \cdot d\mathbf{\ell} = - \frac{d}{dt}\iint_\Sigma \mathbf{B} \cdot d\mathbf{S}$$ ,relates induced electric field to changing magnetic flux +--- + $$ \oint_{\partial \Sigma} \mathbf{B} \cdot d\mathbf{\ell} = \mu_0\left( \iint_\Sigma \mathbf{J} \cdot d\mathbf{S} + \epsilon_0 \frac{d}{dt}\iint_\Sigma \mathbf{E} \cdot d\mathbf{S} \right)$$ ,relates induced magnetic field to changing electric flux and to current -These are the [Maxwell -Equations](https://en.wikipedia.org/wiki/Maxwell%27s_equations), and are -non-linear time dependent differential equations. +--- + +These are the [Maxwell Equations](https://en.wikipedia.org/wiki/Maxwell%27s_equations), and are non-linear time dependent differential equations. -Under the best of circumstances they are fantastically hard to solve! But it's -how the real world works. +Under the best of circumstances they are fantastically hard to solve! But it's how the real world works. + +--- The permittivity of free space is defined as @@ -276,25 +389,26 @@ $$\mu_0 = \frac{2 \alpha}{q^2}\frac{h}{c}$$ , where $\alpha$ is the [fine structure constant](https://en.wikipedia.org/wiki/Fine-structure_constant). +--- # Voltage -The electric field has units voltage per meter, so the electric field is the -derivative of the voltage as a function of space. +The electric field has units voltage per meter, so the electric field is the derivative of the voltage as a function of space. $$ E = \frac{dV}{dx}$$ +--- # Current -Current has unit $A$ and charge $C$ has unit $As$, so the current is the number -of charges passing through a volume per second. +Current has unit $A$ and charge $C$ has unit $As$, so the current is the number of charges passing through a volume per second. -The current density $J$ has units $A/m^2$ and is often used, since we can multiply -by the surface area of a conductor, if the current density is uniform. +The current density $J$ has units $A/m^2$ and is often used, since we can multiply by the surface area of a conductor, if the current density is uniform. $$ I = A \times J $$ +--- + # Drift current Charges in an electric field will give rise to a drift current. @@ -303,49 +417,93 @@ We know from Newtons laws that force equals mass times acceleration $$ \vec{F} = m \vec{a}$$ +--- + If we assume a zero, or constant magnetic field, the force on a particle is -$\vec{F} = q\vec{E}$ + +$$\vec{F} = q\vec{E}$$ The current density is then $$ \vec{J} = q\vec{E} \times n \times \mu $$ + + +--- + +Assuming + +$$ E = V/m$$ + +, we could write $$ J = \frac{C}{m^3}\frac{V}{m}\frac{m^2}{Vs} = \frac{C}{s}m^{-2}$$ -So multiplying by an area $A = B m^2$ +--- + +So multiplying by an area + +$$ A = B m^2$$ $$ I = q n \mu B V$$ -and we can see that the conductance $G = q n \mu B$, and since $G = 1/R$, where -R is the resistance, we have +and we can see that the conductance + +$$G = q n \mu B$$ + +--- + +, and since + +$$G = 1/R$$ + +, where R is the resistance, we have $$ I = G V \Rightarrow V = RI$$ Or [Ohms law](https://en.wikipedia.org/wiki/Ohm%27s_law) +--- # Diffusion current -A difference in charge density will give rise to a diffusion current, and the -current density is +A difference in charge density will give rise to a diffusion current, and the current density is $$ J = -q D_n \frac{d \rho}{dx}$$ + + +--- + +# Why are there two currents? + + +$$-\frac{\hbar^2}{2 m} \frac{\partial^2}{\partial^2 x}\psi(x,t) + +V(x)\psi(x,t) = i\hbar\frac{\partial}{\partial t} \psi(x,t) $$ + +--- # Currents in a semiconductor @@ -353,6 +511,8 @@ Both electrons, and holes will contribute to current. Electrons move in the conduction band, and holes move in the valence band. + + $$ I = I_{n_{drift}} +I_{n_{diffusion}} + I_{p_{drift}} + I_{p_{diffusion}}$$ + + +--- + # Resistors We can make resistors with metal and silicon (a semiconductor) -In metal the dominant carrier depends on the metal, but it's usually electrons. -As such, one can often ignore the hole current. +In metal the dominant carrier depends on the metal, but it's usually electrons. As such, one can often ignore the hole current. -In a semiconductor the dominant carrier depends on the Fermi level in relation -to the conduction band and valence band. If the Fermi level is close to the -valence band the dominant carrier will be holes. -If the Fermi level is close to the conduction band, the dominant carrier will be -electrons. +In a semiconductor the dominant carrier depends on the Fermi level in relation to the conduction band and valence band. If the Fermi level is close to the valence band the dominant carrier will be holes. If the Fermi level is close to the conduction band, the dominant carrier will be electrons. -That's why we often talk about "majority carriers" and "minority carriers", both -are important in semiconductors. +That's why we often talk about "majority carriers" and "minority carriers", both are important in semiconductors. + +--- # Capacitors @@ -386,8 +550,9 @@ A capacitor resists a change in voltage. $$ I = C \frac{dV}{dt}$$ -and store energy in an electric field between two conductors with an insulator -between. +and store energy in an electric field between two conductors with an insulator between. + +--- # Inductors @@ -397,6 +562,10 @@ $$ V = L \frac{dI}{dt}$$ and store energy in the magnetic fields in a loop of a conductor. +--- + + @@ -436,10 +607,3 @@ charge transport, a built-in electric field. - - - - - - -[^1]: Although you do have to keep your symbols straight. We use "C" for Capacitance, but C can also mean Columbs. Context matters. diff --git a/lectures/l01_intro.md b/lectures/l01_intro.md index 1aa0123..d7af66a 100644 --- a/lectures/l01_intro.md +++ b/lectures/l01_intro.md @@ -16,13 +16,6 @@ date: 2024-01-12 --- - - -# - ---- - - #[fit] Who --- @@ -148,6 +141,14 @@ your circuit for free. --> +--- + + + +# + + + --- @@ -160,7 +161,7 @@ It's not likely that you'll find all the skills in one human, and even if you could, one human does not have sufficient bandwidth to design ICs with all it's aspects in a reasonable timeline -That is, unless we can find a way to ICs easier to make. +That is, unless we can find a way to make ICs easier. The skills needed are @@ -182,33 +183,8 @@ The skills needed are - _Physics_: transistor, pn junctions, quantum mechanics --- -[.background-color: #000000] -[.text: #FFFFFF] - - - -> Find a problem that you really want to solve, and learn programming to solve it. There is no point in saying "I want to learn programming", then sit down with a book to read about programming, and expect that you will learn programming that way. It will not happen. The only way to learn programming is to do it, a lot. --- Carsten Wulff - - - -``` perl -s/programming/analog design/ig -``` - ---- - -### Zen of IC design (stolen from Zen of Python) +## Zen of IC design (stolen from Zen of Python) + +> Find a problem that you really want to solve, and learn programming to solve it. There is no point in saying "I want to learn programming", then sit down with a book to read about programming, and expect that you will learn programming that way. It will not happen. The only way to learn programming is to do it, a lot. +-- Carsten Wulff + + + + +``` perl +s/programming/analog design/ig +``` + --- # My Goal @@ -285,7 +286,7 @@ The "lectures" will be Q & A's on the topic. 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