# Inductance and Inductors, Capacitance and Capacitors - ==Capacitors:== passive elements that store electric field energy. Have capacitance $C$, like resistors have resistance $R$, given in farads $F$ - Parallel plates w/ electric field energy running through them. Increasing source voltage raises (+) charge and (-) charge -> $\vec{E}\propto v$ - Capacitors as an element that stores charge in the form of $\vec{E}.$ **Capacitors store charge in the form of voltage** - Dependent on time: $VC=I, \frac{dv}{dt}C=I\to \boxed{ I_{C}=C \frac{dv_{C}}{dt} }^{ }$ - ==Inductors:== passive elements that store magnetic field energy. Have inductance $L$ given in henry $H$ - Store energy in the form of magnetic field, $\vec{B}\propto I,$ or **in the form of current** $\mathbf{I_{L}}$ $\boxed{ v_{L}=L \frac{dI_{L}}{dt} }^{ }$ - Steady state happens when capacitors/inductors can't store charge anymore (see future lecs.) | | Inductors | Capacitors | | ------------------------------- | ------------------------------------------------- | ---------------------------------------------- | | $\mathbf{V-I}$ **relationship** | $I_{L}=\frac{1}{L}\int v_{L}dt+I_{L}(0)$ | $v_{C}=\frac{1}{C}\int I_{C}dt+v_{C}(0)$ | | **Power** | $P_{L}=v_{L}I_{L}=\frac{dI_{L}}{dt} \int v_{L}dt$ | $P_{C}=v_{C}I= \frac{dv_{C}}{dt} \int I_{C}dt$ | | **Energy** | $E_{L}=\frac{1}{2}LI_{L}^2$ | $E_{C}=\frac{1}{2}Cv_{C}^2$ | - In an inductor, $I_{L}$ is continuous form, $v_{L}$ is non-continuous, so $I_{L}(t^-)=I_{L}(t^+)$ but $v_{L}(t^-)\neq v_{L}(t^+)$. Vice versa for capacitors - At frozen time, can model an inductor as current source and inductor as voltage source. # L & C Combinations; I-div, V-div; Steady State - ==Steady state:== when capacitor/capacitor charge is not changing - Fully charged: in presence of an independent source in circuit, inductor/capacitor will become fully charged after a certain time, $t=\infty$ - Fully discharged: no independent source active, $t=0$ > [!info] Fully dis/charged capacitors & inductors > - When **capacitor is fully charged**, voltage is no longer changing: $\frac{dv_{C}}{dt}=0$ so $I_{C}=C(0)$ -> no current flows so **open circuit** > - **Discharged capacitor** starts with $v_{C}=0$ so when discharged, since voltage can't instantly jump, right when you connect the source, it still has $0V$ across it -> **short circuit** > ‎ > - When **inductor is fully charged**, its current is constant so $\frac{dI_{L}}{dt}=0\therefore v_{L}=L(0)=0$ -> no voltage drop so **short circuit** > - **Discharged inductor** means that $i_{L}=0$ and current cannot jump instantly, so right when it's connected, has zero current -> **open circuit** - **Series combinations** - $L_{eq}=\sum L_{n}$ - $\frac{1}{C_{eq}}=\sum \frac{1}{C_{n}}$ - **Parallel combinations** - $\frac{1}{L_{eq}}=\sum \frac{1}{L_{n}}$ - $C_{eq}=\sum C_{n}$ - **Voltage division** - $v_{L2}=v_{s} \frac{L_{2}}{L_{1}+L_{2}}$ - $v_{C2}= v_{s} \frac{\frac{1}{C_{2}}}{\frac{1}{C_{1}}+\frac{1}{C_{2}}}$ - **Current division** - $I_{L2}= I_{s}\frac{\frac{1}{L_{2}}}{\frac{1}{L_{1}}+\frac{1}{L_{2}}}$ - $I_{C2}=I_{S} \frac{C_{2}}{C_{1}+C_{2}}$