Rendering Equations in Mkdocs
Prompt
Prompt
Create a list of the 25 most commonly used equations in an undergraduate circuits course taught within a department of electrical engineering.
For each equation, return the following:
- The name of the equation in markdown level 4. This allows us to send a link to each equation in a chat or email.
- Within the body text, render equation in LaTeX format on a line with blank lines before and after the LaTeX equation.
- Surround the equation with single dollar signs in front and in back
- For each variable in the equation, add a "where:" that defines each variable
- Put blank lines around the word "where:" and list of variables
- Wrap the variables names in single dollar signs
Here are two examples of the format I want.
Slope and Intercept
\(y = m * x + b\)
where:
- \(y\) is the vertical axis
- \(x\) is the horizontal axis
- \(m\) is the slope
- \(b\) is the intercept
Area of a Circle
\(A = 2\pi r^2\)
where:
- \(A\) is the area of the circle
- \(r\) is the radius
ChatGPT o1-preview Response
Note I had to run the following regular expressions fixes.
- Replace "\" with "\"
- Replace "_" with "_"
Ohm's Law
\(V = I R\)
where:
- \(V\) is the voltage across the resistor
- \(I\) is the current through the resistor
- \(R\) is the resistance
Kirchhoff's Voltage Law (KVL)
\(\sum_{k=1}^{n} V_k = 0\)
where:
- \(V_k\) is the voltage across the \(k\)th element in the loop
- \(n\) is the number of elements in the loop
Kirchhoff's Current Law (KCL)
\(\sum_{k=1}^{n} I_k = 0\)
where:
- \(I_k\) is the current entering or leaving a node
- \(n\) is the number of branches connected to the node
Power Formula
\(P = V I\)
where:
- \(P\) is the power
- \(V\) is the voltage
- \(I\) is the current
Capacitor Current-Voltage Relationship
\(I_C = C \dfrac{dV_C}{dt}\)
where:
- \(I_C\) is the current through the capacitor
- \(C\) is the capacitance
- \(V_C\) is the voltage across the capacitor
- \(t\) is time
Inductor Voltage-Current Relationship
\(V_L = L \dfrac{dI_L}{dt}\)
where:
- \(V_L\) is the voltage across the inductor
- \(L\) is the inductance
- \(I_L\) is the current through the inductor
- \(t\) is time
Impedance of a Resistor
\(Z_R = R\)
where:
- \(Z_R\) is the impedance of the resistor
- \(R\) is the resistance
Impedance of a Capacitor
\(Z_C = \dfrac{1}{j \omega C}\)
where:
- \(Z_C\) is the impedance of the capacitor
- \(j\) is the imaginary unit (\(j = \sqrt{-1}\))
- \(\omega\) is the angular frequency (\(\omega = 2\pi f\))
- \(C\) is the capacitance
- \(f\) is the frequency
Impedance of an Inductor
\(Z_L = j \omega L\)
where:
- \(Z_L\) is the impedance of the inductor
- \(j\) is the imaginary unit (\(j = \sqrt{-1}\))
- \(\omega\) is the angular frequency (\(\omega = 2\pi f\))
- \(L\) is the inductance
- \(f\) is the frequency
Total Resistance in Series
\(R_{\text{total}} = R_1 + R_2 + \dots + R_n\)
where:
- \(R_{\text{total}}\) is the total resistance of resistors in series
- \(R_1, R_2, \dots, R_n\) are individual resistances
- \(n\) is the number of resistors
Total Resistance in Parallel
\(\dfrac{1}{R_{\text{total}}} = \dfrac{1}{R_1} + \dfrac{1}{R_2} + \dots + \dfrac{1}{R_n}\)
where:
- \(R_{\text{total}}\) is the total resistance of resistors in parallel
- \(R_1, R_2, \dots, R_n\) are individual resistances
- \(n\) is the number of resistors
Time Constant of an RC Circuit
\(\tau = R C\)
where:
- \(\tau\) is the time constant
- \(R\) is the resistance
- \(C\) is the capacitance
Voltage Across a Charging Capacitor
\(V_C(t) = V_s \left(1 - e^{-t/\tau}\right)\)
where:
- \(V_C(t)\) is the voltage across the capacitor at time \(t\)
- \(V_s\) is the source voltage
- \(t\) is time
- \(\tau\) is the time constant (\(\tau = R C\))
- \(e\) is Euler's number
Voltage Divider Rule
\(V_{\text{out}} = V_{\text{in}} \left( \dfrac{R_2}{R_1 + R_2} \right)\)
where:
- \(V_{\text{out}}\) is the output voltage across \(R_2\)
- \(V_{\text{in}}\) is the input voltage
- \(R_1\) and \(R_2\) are resistors in series
Current Divider Rule
\(I_1 = I_{\text{total}} \left( \dfrac{R_2}{R_1 + R_2} \right)\)
where:
- \(I_1\) is the current through resistor \(R_1\)
- \(I_{\text{total}}\) is the total current entering the parallel network
- \(R_1\) is the resistance through which \(I_1\) flows
- \(R_2\) is the other parallel resistance
RMS Value of a Sinusoidal Voltage
\(V_{\text{RMS}} = \dfrac{V_{\text{peak}}}{\sqrt{2}}\)
where:
- \(V_{\text{RMS}}\) is the root mean square voltage
- \(V_{\text{peak}}\) is the peak voltage
Resonant Frequency of an RLC Circuit
\(f_0 = \dfrac{1}{2\pi \sqrt{L C}}\)
where:
- \(f_0\) is the resonant frequency
- \(L\) is the inductance
- \(C\) is the capacitance
Complex Power
\(S = V_{\text{RMS}} I_{\text{RMS}}^\* = P + j Q\)
where:
- \(S\) is the complex power
- \(V_{\text{RMS}}\) is the RMS voltage
- \(I_{\text{RMS}}^\*\) is the complex conjugate of the RMS current
- \(P\) is the real power
- \(Q\) is the reactive power
- \(j\) is the imaginary unit (\(j = \sqrt{-1}\))
Thevenin's Equivalent Voltage
\(V_{\text{Th}} = V_{\text{oc}}\)
where:
- \(V_{\text{Th}}\) is the Thevenin equivalent voltage
- \(V_{\text{oc}}\) is the open-circuit voltage across the terminals
Thevenin's Equivalent Resistance
\(R_{\text{Th}} = \dfrac{V_{\text{Th}}}{I_{\text{sc}}}\)
where:
- \(R_{\text{Th}}\) is the Thevenin equivalent resistance
- \(V_{\text{Th}}\) is the Thevenin equivalent voltage
- \(I_{\text{sc}}\) is the short-circuit current at the terminals
Norton's Equivalent Current
\(I_{\text{No}} = I_{\text{sc}}\)
where:
- \(I_{\text{No}}\) is the Norton equivalent current
- \(I_{\text{sc}}\) is the short-circuit current at the terminals
Laplace Transform of a Function
\(F(s) = \int_0^\infty f(t) e^{-s t} dt\)
where:
- \(F(s)\) is the Laplace transform of \(f(t)\)
- \(f(t)\) is a time-domain function
- \(s\) is the complex frequency variable
Relationship Between Frequency and Angular Frequency
\(\omega = 2 \pi f\)
where:
- \(\omega\) is the angular frequency
- \(f\) is the frequency
Transfer Function
\(H(s) = \dfrac{Y(s)}{X(s)}\)
where:
- \(H(s)\) is the transfer function
- \(Y(s)\) is the Laplace transform of the output
- \(X(s)\) is the Laplace transform of the input
- \(s\) is the complex frequency variable
Power in Terms of Resistance
\(P = I^2 R\)
where:
- \(P\) is the power
- \(I\) is the current
- \(R\) is the resistance
Maximum Power Transfer Theorem
\(P_{\text{max}} = \dfrac{V_{\text{Th}}^2}{4 R_{\text{Th}}}\)
where:
- \(P_{\text{max}}\) is the maximum power transferred to the load
- \(V_{\text{Th}}\) is the Thevenin equivalent voltage
- \(R_{\text{Th}}\) is the Thevenin equivalent resistance
Fourier Series Representation
\(f(t) = a_0 + \sum_{n=1}^\infty \left\[ a_n \cos(n \omega_0 t) + b_n \sin(n \omega_0 t) \right\]\)
where:
- \(f(t)\) is a periodic function
- \(a_0\) is the average (DC) component
- \(a_n\) and \(b_n\) are Fourier coefficients
- \(\omega_0\) is the fundamental angular frequency