DC Circuit Analysis

Voltage and Current Division

Voltage Divider

       R1       R2
Vin ──┤├───────┤├── GND
            │
            Vout

Vout = Vin × R2 / (R1 + R2)

Applications:
- Generating reference voltages
- Level shifting
- Potentiometers (variable voltage division)

Loading Effect

When a load is connected to a voltage divider, the equivalent resistance becomes R2' = R2 ∥ RL.
Vout will drop. Design requires RL >> R2.

Current Divider

      ┌── R1 ──┐
It ───┤        ├── It
      └── R2 ──┘

I1 = It × R2 / (R1 + R2)
I2 = It × R1 / (R1 + R2)

Current tends to take the path of least resistance.

Network Theorems

Superposition Theorem

Linear circuits with multiple independent sources:
1. Keep only one source at a time (short voltage sources, open current sources)
2. Calculate each component separately
3. Sum the results

Applicable to: Linear circuits (R, L, C)
Not applicable to: Power (non-linear)

Thévenin's Theorem

Any linear two-terminal network can be equivalent to a voltage source in series with a resistor:

  Complex Network      Equivalent
  ┌──────┐      ⇔     ┌──────┐
  │ ......│            Vth ─┤├─
  │ ......│            └──┬───┘
  └──┬─┬──┘                Rth
     a   b

Vth = Open-circuit voltage between a-b
Rth = Equivalent resistance between a-b with independent sources turned off

Norton's Theorem

Dual form of Thévenin's theorem — equivalent to a current source in parallel with a resistor:

In = Vth / Rth    (Short-circuit current)
Rn = Rth

Thévenin ⇔ Norton interchangeable

Maximum Power Transfer

When RL = Rth, the load receives maximum power:
Pmax = Vth² / (4 × Rth)

However, efficiency is only 50% at this point.
Power circuits pursue high efficiency (RL >> Rs).
RF circuits often pursue maximum power transfer (impedance matching).

Input/Output Impedance

Source               Load
┌──────┐    ┌──────┐
│ Vs   │    │      │
│   ───┼────┤  RL  │
│   Rs │    │      │
└──────┘    └──────┘

Voltage transfer: VL = Vs × RL/(Rs + RL)

Ideal conditions:
- Voltage amplifier: Rin → ∞, Rout → 0
- Current amplifier: Rin → 0, Rout → ∞

RC Circuit Transients

Charging

        R
Vin ──┤├───┬── Vc
           ┌──┐
           │C │
           └──┘
           │
          GND

Vc(t) = Vin × (1 - e^(-t/RC))
Ic(t) = (Vin/R) × e^(-t/RC)

τ = RC (Time constant)

Time Constant Rules

t = 1τ → 63.2% charged
t = 2τ → 86.5%
t = 3τ → 95.0%
t = 4τ → 98.2%
t = 5τ → 99.3%  ← Usually considered fully charged

Discharging

Vc(t) = V₀ × e^(-t/RC)

Similarly decays with τ = RC

Common Analysis Techniques

Nodal Voltage Analysis

1. Select a reference node (GND)
2. Write KCL equations for other nodes
3. Solve the system of equations

Mesh Current Analysis

1. Define mesh current directions
2. Write KVL equations for each mesh
3. Solve the system of equations

Δ-Y Transformation

Δ (Delta)              Y (Wye)
     Rc                  R1
   ┌──┤├──┐           ┌──┤├─┬──
   │      │           │      │
  Ra     Rb    ⇔     R2     R3
   │      │           │      │
   └──┬┬──┘           └──┬┬──┘

Ra = (R1R2 + R2R3 + R3R1) / R1   (Y→Δ)
R1 = RbRc / (Ra+Rb+Rc)            (Δ→Y)

Keywords: Voltage divider, Current divider, Superposition theorem, Thévenin, Norton, Time constant, RC, Input impedance, Output impedance