Capacitors

• Stores charge (Current)
• supplies it if it has some stored
• Capacitance ($C$) in Farads ($F$)
• $C=\frac{Q}{V}$
• Also $C=\frac{ϵA}{d}$, where $ϵ={ϵ}_0{ϵ}_r$ is the dielectric permittivity
• Coulomb's Constant
• The permittivity of free space is $8.85\frac{\mathrm{pF}}{\mathrm{m}}$
• The current flowing through a capacitor is $i=\frac{\mathrm{d}q}{\mathrm{d}t}=C\frac{\mathrm{d}v}{\mathrm{d}t}$
• $V(t)=\frac{1}{C}{\int }_{t_0}^{t}i(x)\mathrm{d}x+V(t_0)$
• Energy stored in a capacitor $w(t)=\frac{1}{2}Cv(t{)}^2$ (take power and integrate)
• $\mathbf{I}$ leads $\mathbf{V}$ by $90\text{°}$ and $\mathbf{I}=j\omega C\mathbf{V}$
  • ICE
• $Z_c=\frac{1}{j\omega C}=-\frac{j}{\omega C}$

Parallel Plate Capacitor

• put two metal plates with area A in parallel, with an insulator of thickness d in between
• Now stick voltage across the plates

Equivalent Capacitors

• It's kinda like the reverse of resistors
• Capacitors in series have the same charge
  • Also $\frac{1}{C_{eq}}=\frac{1}{C_1}+\frac{1}{C_2}+\dots$
• Parallel capacitors
  • $Q=V(C_1+C_2+\dots )$
  • $C_{eq}=C_1+C_2+\dots$