# Ohms Law Solve for voltage: $\displaystyle V = \frac{I}{R}$ *Solve for resistance:* $R = \frac{V}{I}$ _Solve for current_ $$ \begin{flalign} I & = \frac{V}{R} & \end{flalign} $$ # Resistors in Series $R = R1 + R2 + R3 ...$ # Resistors in Parallel $$ \begin{flalign} &\frac{1}{R} = \frac{1}{R1} + \frac{1}{R2} + \frac{1}{R3} ... &\\ \\ &\textit{For two resistors in parallel:} &\\ \\ &R = \frac{R1 * R2}{R1 + R2} &\\\ \end{flalign} $$ ***Tip:*** If resistors of the same value are in parallel the total resistance is a single resistor divided by the amount if resistors. ## Thevenin’s Theorem States that it is possible to simplify any linear circuit, no matter how complex, to an equivalent circuit with just a single voltage source and series resistance connected to a load. # Kirchhoff's Law ## Conservation of Charge (First Law) All current entering a node must also leave that node $$ \begin{flalign} \sum{I_{IN}} = \sum{I_{OUT}}&& \end{flalign} $$ **Example:** ![](kirchhoffs-law-01.svg) For this circuit kirchhoffs law states that: $\displaystyle i1 = i2 + i3 + i4$ ## Conservation of Energy (Second Law) All the potential differences around the loop must sum to zero. $\displaystyle \sum{V} = 0$ ## Capacitors in Series $\displaystyle \frac{1}{C_{t}} = \frac{1}{C_{1}}+\frac{1}{C_{2}}+\frac{1}{C_{3}} ...$ ### Impedance in a Circuit $$ \begin{flalign} &Z = \sqrt{R^2 + X^2} &\\\ \\ &X = X_{L} - X_{C} \\ \end{flalign} $$ # Capacitive Reactance $\displaystyle X_{c} = \frac{1}{2 \pi fC}$ # Inductive Reactance $\displaystyle X_{l} = 2\pi fL$ # Analog Filters ## Cutoff Frequency for RC Filters $\displaystyle f_{c} = \frac{1}{2\pi RC}$ ## Cutoff Frequency for RL Filters $\displaystyle f_{c} = \frac{R}{2\pi L}$ ## Signal Response of an RC/RL Filter $X_c$ = [[#Capacitive Reactance]] || [[#Inductive Reactance]] $\displaystyle V_{out} = V_{in}(\frac{X_{c}}{\sqrt{R^2+X_{c}^2}})$ ## Cutoff Frequency for multiple Low Pass Filters $\displaystyle f_{(-3db)} = f_{c}\sqrt{2^{(\frac{1}{n})}-1}$ Where $n$ = Number if **identical** filters ## Resonance Frequency for RLC Low Pass Filter $\displaystyle f_{o} = \frac{1}{2\pi \sqrt{LC}}$ ## Center Frequency with Fc and Fh $f_{c} = \sqrt{f_{h}*f_{l}}$ ## Filter Response for RC Filters $V_{out} = V_{in}(\frac{X_c}{\sqrt{R_{1}^2+X_{c}^2}})$ ## Cutoff Frequency $\pi$ Topology Filter When the two capacitors have the same capacitance, it can be calculated like this: $\displaystyle f_c = \frac{1}{4\pi\sqrt{LC}}$ # Voltage Divider $V_{out} = V_{in}(\frac{R_{1}}{R_1+R_2})$ # Angular Frequency ($\omega$) $\omega = 2\pi f = \frac{2\pi}{T}$ ^4ad7fc # RLC Series Response This is basically Ohms Law: $\displaystyle V = IZ$ Where $Z$ is the impedance: $Z = \sqrt{R^2 + (X_L - X_C)^2}$ # Current through a transistor $\displaystyle I_{EQ} = \frac{V_{BB}-{V_{BE}}}{\frac{R_B}{(\beta+1)}+R_E}$ # Gain Bandwidth Product $GBP = A_V * f_c$ $\displaystyle f_c = \frac{GBP}{A_V}$ # Bandwidth of Multiple OpAmps Where $n$ = number of stages and $BW$ = Bandwidth of single op-amp $BW_E = BW\sqrt{2^\frac{1}{n}-1}$