Electrical Circuits: Series, Parallel, RC, LR, LCR, and Network Theorems

TAKEAWAYS:

• Understand series vs. parallel circuits using voltage/current rules and equivalent resistance formulas.
• Analyze transient responses in RC/RL circuits (charging/discharging, time constant τ).
• Apply Kirchhoff’s laws (loop and junction rules) to solve complex circuits.
• Use network theorems (Thevenin’s, Norton’s, Millman’s) to simplify circuits.
• Recognize phasor diagrams for AC circuits and resonance in LCR circuits.
• Solve NEB-style problems using step-by-step circuit analysis.

1. Series and Parallel Circuits: Basics

Key Definitions

• Series Circuit: Components connected end-to-end; same current flows through all.
• Parallel Circuit: Components connected across the same voltage; current divides.

Rules

Example: Series vs. Parallel

Problem: Find $R_{eq}$ for:

• Series: $R_1=2\Omega ,R_2=3\Omega$
• Parallel: Same resistors.

Solution:

• Series: $R_{eq}=2+3=5\Omega$
• Parallel: $\frac{1}{R_{eq}}=\frac{1}{2}+\frac{1}{3}\Rightarrow R_{eq}=1.2\Omega$

Visual: Series vs. Parallel

graph LR
    A["Series Circuit"] --> B["Current (I)"] --> C["Same for all"]
    A --> D["Voltage (V)"] --> E["Sum of voltages"]
    F["Parallel Circuit"] --> G["Voltage (V)"] --> H["Same for all"]
    F --> I["Current (I)"] --> J["Sum of currents"]

2. Kirchhoff’s Laws

Rules

1. Junction Rule (Current Law): Sum of currents entering a junction = sum leaving.
2. Loop Rule (Voltage Law): Sum of voltage drops in a loop = 0.

Example: Applying Kirchhoff’s Laws

Problem: Find $I_1,I_2$ in the circuit below (assume $V=10V,R_1=2\Omega ,R_2=3\Omega ,R_3=4\Omega$).

       V
       |
   R1---+---R2
       |
   R3---+

Solution:

1. Junction Rule: $I_1=I_2+I_3$
2. Loop Rule (Left Loop): $-V+I_1{R}_1+I_3{R}_3=0$
3. Loop Rule (Right Loop): $I_2{R}_2-I_3{R}_3=0$
4. Solve simultaneously (use substitution).

3. RC, RL, and LCR Circuits

RC Circuit (Resistor-Capacitor)

• Charging: $V_C(t)=V_0(1-e^{-t/RC})$
• Discharging: $V_C(t)=V_0{e}^{-t/RC}$
• Time Constant (τ): $\tau =RC$ (time to charge to ~63% of $V_0$).

RL Circuit (Resistor-Inductor)

• Current Growth: $I(t)=I_0(1-e^{-t/LR})$
• Time Constant (τ): $\tau =L/R$.

LCR Circuit (Resonance)

• Resonant Frequency: $f_0=\frac{1}{2\pi \sqrt{LC}}$
• Impedance at Resonance: $Z=R$ (purely resistive).

Visual: RC Charging Curve

4. Network Theorems

Thevenin’s Theorem

• Replace a complex circuit with a single voltage source $V_{th}$ and resistance $R_{th}$.

Norton’s Theorem

• Replace a circuit with a current source $I_N$ and resistance $R_N$.

Millman’s Theorem

• For parallel branches: $V=\frac{V_1/Z_1+V_2/Z_2+\dots }{1/Z_1+1/Z_2+\dots }$.

Example: Thevenin Equivalent

Problem: Find $V_{th}$ and $R_{th}$ for the circuit:

   V=10V
     |
   R1=2Ω---+---R2=3Ω
     |       |
     R3=4Ω   Load

Solution:

1. Short the load: $V_{th}=\frac{R_2}{R_1+R_2}V=\frac{3}{5}\times 10=6V$.
2. Deactivate sources: $R_{th}=R_1\parallel R_2=\frac{6}{5}\Omega$.

5. AC Circuits and Phasors

Phasor Representation

• Voltage/current in AC circuits are represented as rotating vectors (phasors).
• Impedance (Z): $Z=\sqrt{R^2+(X_L-X_C{)}^2}$, where $X_L=\omega L$, $X_C=1/\omega C$.

Visual: Phasor Diagram for LCR

graph TD
    A["V_R (In Phase)"] --> B["V_L (Leads by 90°)"]
    A --> C["V_C (Lags by 90°)"]
    B --> D["V_L - V_C"]
    C --> D
    D --> E["V (Net)"]

Exam Tip

1. Series/Parallel: Always check if resistors are in series/parallel before applying formulas.
2. Kirchhoff’s Laws: Label currents/voltages clearly and apply loop/junction rules systematically.
3. RC/RL: Remember $\tau =RC$ or $\tau =L/R$ for time-dependent problems.
4. Thevenin/Norton: Simplify circuits to avoid complex calculations.
5. NEB Questions: Often involve numerical problems (e.g., find $I,V,R_{eq}$) or conceptual questions (e.g., "Why does current divide in parallel?").

NEB-Style Questions

1. Short Answer:

   • Define time constant in an RC circuit.
   • State Kirchhoff’s Voltage Law.

2. Numerical Problem:

   • Three resistors $2\Omega ,3\Omega ,4\Omega$ are connected in parallel to a 12V battery. Find: a) Equivalent resistance. b) Current through each resistor.

3. Theory:

   • Explain how a Wheatstone bridge works and its applications.

Answer Key:

1. Time constant (τ): $\tau =RC$ (time for capacitor to charge to 63% of $V_0$).
2. KVL: Sum of voltage drops in a closed loop = 0.
3. a) $R_{eq}=\frac{1}{1/2+1/3+1/4}=1.09\Omega$. b) $I_{total}=12/1.09\approx 11A$, $I_{2\Omega }=6A$, $I_{3\Omega }=4A$, $I_{4\Omega }=3A$.