Circuits

This is a combination of knowledge that I have gained from ENGR 40M and EE 101A, both of which I took during summer quarter in 2025, and self-learning.

• Tellegen's theorem - in an electrical network, sum of instantaneous powers in all branches is 0

• High voltage is used in transmission because
  • High voltage means that the current goes down for the same power because $P=IV$
    • A lower current means that the voltage drop between the generator and substation, $V=IR$, is smaller since the $R$ of the transmission line is constant and the $I$ decreases. The voltage drop across the transmission lines modeled as resistors went down even though the voltage on the line is higher. With a constant R and a lower current with HV, $P_{\text{loss}}=I^2R$ goes down and $P_{\text{loss}}=\frac{V^2}R$ also goes down since V in the power equation measures the voltage drop, which went down.
• Inductor
  • $v=Li'$
  • equivalent combinations like that of a resistor
  • Time constant $\tau=\frac{L}R$
• Capacitors
  • $\epsilon_r=\frac\epsilon{\epsilon_0}$
    • $\epsilon_r$ - relative dielectric constant
    • $\epsilon$ - dielectric constant
    • $\epsilon_0$ - permittivity of free space (9E-12 F/m)
  • $C=\frac{\epsilon A}{d}$
    • C ∝ A and C ∝ 1/d
  • $Q=CV$ and $I=CV'$
  • Types
    • Ceramic - cheap, low C
    • Polymer - HV
    • Electrolytic - polar, large C
    • Surface mount capacitors
    • IC capacitor
• Waves
  • $\frac1f=T=\frac\lambda{v}\to f=\frac{v}\lambda$
• Nodal analysis: write out KCL (for nodes with unknown voltages) in terms of voltages
  • Full steps
    1. Choose reference node and label the nodes' voltages. Use $V_1, V_2, ...$ for unknown. Use voltage divider to find out relevant voltages if possible.
    2. Label current directions
    3. Apply KCL at all non-reference nodes (using Ohm's law integrated in the eqns)

• Current divider ([[#Current Divider|derivation]])
  • $I_{R_1}=I_\text{tot}\frac{R_\text{tot}}{R_1}$. mn current ∝ inverse of %resistance in same component
    • $I_{R_1}=I_\text{tot}\frac{R_2}{R_1+R_2}$ (special case of two resistors)
• Voltage divider ([[#Voltage Divider|derivation]])
  • $V_1=V_\text{tot}\frac{R_1}{R_\text{tot}}$. mn voltage ∝ %resistance in same component

• Resistor Chart
• Double-subscript notation:

Derivations

Derive Equivalent Inductance for Series and Parallel Inductors

How Dielectrics Increase Capacitance

Voltage Divider

For serial resistors, current is the same but voltage is divided across the resistors.

Current Divider

For parallel resistors, voltage is the same but current is divided across the resistors.

Wye-Delta

• Special cases
  • $R_\triangle = 3R_Y$

Deriving RMS Voltage for a Sinusoidal AC

First Principles Thinking about Batteries in Series