---
title: DC Circuit Analysis
url: https://doc.liz6.com/en/hardware/01-basic-electricity/02-dc-circuit-analysis
locale: en
area: hardware
tags:
- hardware
- basic-electricity
date: 2026-06-30
modified: 2026-07-16
description: DC Circuit Analysis Voltage and Current Division Voltage Divider Loading Effect Current Divider --- Network Theorems Superposition Theorem Thévenin's Theorem No…
---

# 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*