Thevenin Equivalent Circuit - Detailed Guide
What is Thevenin’s Theorem?
Thevenin’s theorem states that any linear circuit seen from two terminals can be reduced to an
equivalent circuit consisting of a single voltage source (V_th) in series with a resistance (R_th).
---------------------------------------Why use it?
It simplifies complex circuits into a single-source, single-resistor form, making it much easier to analyze
current and voltage, especially for transient analysis (like RL and RC circuits).
---------------------------------------Steps to Find a Thevenin Equivalent:
1. Identify the two terminals where you want to find the Thevenin equivalent.
2. Remove the load connected to those terminals.
3. Find V_th (Thevenin Voltage): This is the open-circuit voltage between the two terminals.
4. Find R_th (Thevenin Resistance): Turn off all independent sources (voltage sources → short circuit,
current sources → open circuit), then calculate the equivalent resistance seen from the two terminals.
5. Reconnect the load to V_th in series with R_th.
---------------------------------------Example Applied to the Given Circuit:
For t < 0, the switch is closed creating a short circuit. The 2 A current flows through the short, meaning
no current flows in the inductor branch, so i(0-) = 0.
For t ≥ 0, the switch opens and the right-side branch with 10Ω + inductor is connected to the left
network.
---------------------------------------Step 1: Find V_th (Open-Circuit Voltage):
- Disconnect the right 10Ω + inductor branch.
- All 2 A flows through the vertical 10Ω resistor to ground.
- Voltage across resistor = 2 A × 10Ω = 20 V = V_th.
---------------------------------------Step 2: Find R_th (Thevenin Resistance):
- Turn off the 2 A current source → replace with open circuit.
- The only resistance between the terminals and ground is the 10Ω resistor.
- So R_th = 10Ω.
---------------------------------------Step 3: Reconnect Load (Inductor + 10Ω):
- Total resistance in series with V_th is R_th + 10Ω = 10Ω + 10Ω = 20Ω.
----------------------------------------
Step 4: Solve for i(t):
- Initial current, i(0+) = 0 A (inductor current is continuous).
- Final current, I_inf = V_th / Total Resistance = 20V / 20Ω = 1 A.
- Time constant τ = L / R_total = 1H / 20Ω = 0.05 s.
- So i(t) = 1 - e^(-20t) A.
---------------------------------------Final Formula:
i(t) = 1 - e^(-20t) for t ≥ 0