Network Theorems

Why it matters

Network theorems are essential for simplifying and analyzing complex electrical circuits, making it easier to understand their behavior and design efficient systems. They are widely used in practical applications such as power distribution, electronic device design, and signal processing.

Key ideas

• Superposition Theorem: In a linear circuit with multiple independent sources, the response (voltage or current) in any element is the algebraic sum of the responses caused by each independent source acting alone, with all other independent sources turned off (replaced by their internal impedances).
• Thevenin’s Theorem: A linear two-terminal network with a well-defined equivalent can be replaced by an equivalent circuit consisting of a single voltage source (Thevenin voltage) in series with a resistance (Thevenin resistance) connected to the load (use impedance for sinusoidal AC).
• Norton’s Theorem: Similar to Thevenin’s Theorem, but the equivalent circuit consists of a current source (Norton current) in parallel with a resistance (Norton resistance).
• Maximum Power Transfer Theorem: Maximum power is transferred to the load when the load resistance equals the Thevenin resistance of the network supplying the power.
• Reciprocity Theorem: In a linear, bilateral network, the current due to a single source in one branch is equal to the current in the original branch when the source is moved to the branch where the current was originally measured.

Keep dependent sources active. Deactivate an ideal independent voltage source by shorting it and an ideal independent current source by opening it; use a test source to find equivalent resistance when needed. Superpose voltages/currents, not powers. Maximum-power matching assumes positive source resistance; with a freely variable complex AC load, Z_L = conjugate(Z_th). Reciprocity additionally needs a reciprocal network and consistent source/response definitions.

Formulas

• Superposition: V_total = V1 + V2 + ... + Vn
  • V_total: Total voltage (V)
  • V1, V2, ..., Vn: Voltages due to individual sources (V)
• Thevenin’s Theorem: V_th = V_oc
  • V_th: Thevenin voltage (V)
  • V_oc: Open-circuit voltage (V)
• Norton’s Theorem: I_n = I_sc
  • I_n: Norton current (A)
  • I_sc: Short-circuit current (A)
• Maximum Power Transfer: R_L = R_th
  • R_L: Load resistance (Ω)
  • R_th: Thevenin resistance (Ω)

Worked example

Given: A circuit with a 10 V voltage source, a 5 Ω resistor, and a 10 Ω resistor in series. Find the Thevenin equivalent across the 10 Ω resistor.

1. Remove the 10 Ω resistor and find the open-circuit voltage (V_oc).
   • With the load removed, current through 5 Ω is zero, so its drop is zero and V_oc = 10 V.
2. Find the Thevenin resistance (R_th) by deactivating the voltage source (replace with a short circuit).
   • R_th = 5 Ω
3. Thevenin equivalent circuit: V_th = 10 V, R_th = 5 Ω

Answer: The Thevenin equivalent is 10 V in series with 5 Ω.

Common mistakes

• Forgetting to turn off all independent sources except one when applying the superposition theorem.
• Incorrectly calculating open-circuit or short-circuit conditions for Thevenin and Norton equivalents.
• Not matching load resistance to Thevenin resistance for maximum power transfer.

For GATE EE

Questions often involve finding Thevenin or Norton equivalents, applying superposition, or calculating conditions for maximum power transfer. Practice problems that require step-by-step simplification of complex circuits using these theorems.

Quick check

1. What is the purpose of Thevenin’s Theorem?
2. How do you find the Norton current?
3. What condition is required for maximum power transfer?

Answers: 1. To simplify a complex circuit to a single voltage source and resistance. 2. By finding the short-circuit current across the terminals. 3. Load resistance equals Thevenin resistance.