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Network Theorems (Part II) Dr. Mohamed Refky Amin Electronics and Electrical Communications Engineering Department (EECE) Cairo University elc.n102.eng@gmail.com http://scholar.cu.edu.eg/refky/ OUTLINE • • • • Previously on ELCN102 Network Theorems Norton’s Theorem Maximum Power Transfer Theorem 2 Previously on ELCN102 Definition Electric circuit theorems are always beneficial to help find voltage and currents in multi loop circuits. The network theorems include: • • • • Superposition Theorem Thevenin’s Theorem Norton’s Theorem Maximum Power Transfer Theorem 3 Previously on ELCN102 Superposition Theorem For a linear circuit containing multiple independent sources, the voltage across (or current through) any of its elements is the algebraic sum of the voltages across (or currents through) that element due to each independent source acting alone. 1.5V πΌa 3A πΌπ Total πΌ = πΌπ + πΌπ 4 Previously on ELCN102 Steps of Superposition Theorem 1) Keep one independent source in the circuit and turn off all other independent sources. 2) Find the output (voltage or current) due to that active source using one or more circuit solution methods: ο§ ο§ ο§ ο§ Branch Method. Simplification Method. Loop Analysis Method. Node Analysis Method. 3) Repeat step 1 for each of the other independent sources. 4) Find the total contribution by adding algebraically all the contributions due to all the independent sources. 5 Previously on ELCN102 Thevenin’s Theorem A linear two-terminal circuit, can be replaced by an equivalent circuit consisting of a voltage source ππ‘β in series with a resistor π π‘β . 6 Previously on ELCN102 Steps of Thevenin’s Theorem 1) Identify the load resistance and introduce two nodes π and π 2) Remove the load resistance between node π and π 3) Calculate the open circuit voltage between nodes π and π. This voltage is ππ‘β of the Thevenin equivalent circuit. 4) Set all the independent sources to zero (voltage sources are SC and current sources are OC) and calculate the resistance seen between nodes π and π. This resistance is π π‘β of the Thevenin equivalent circuit. 7 Network Theorems Definition Electric circuit theorems are always beneficial to help find voltage and currents in multi loop circuits. The network theorems include: • • • • Superposition Theorem Thevenin’s Theorem Norton’s Theorem Maximum Power Transfer Theorem 8 Norton’s Theorem Definition A linear two-terminal circuit can be replaced by equivalent circuit consisting of a current source πΌπ in parallel with a resistor π π 9 Norton’s Theorem Definition πΌπ is the short-circuit current through nodes π and π 10 Norton’s Theorem Definition π π is the resistance seen between nodes π and π when all the independent sources are set to zero. The same definition of πΉππ 11 Norton’s Theorem Definition Thevenin equivalent circuit must be equivalent to Norton equivalent circuit ππ‘β ππ‘β π π = π π‘β , ππ‘β = πΌπ π π , πΌπ = → π π‘β = π π‘β πΌπ 12 Norton’s Theorem Solution Steps 1) Identify the load resistance and introduce two nodes π and π 2) Remove the load resistance between node π and π and set all the independent sources to zero (voltage sources are SC and current sources are OC) and calculate the resistance seen between nodes π and π. This resistance is π π of the Norton equivalent circuit. 3) Replace the load resistance with a short circuit and calculate the short circuit current between nodes π and π. This current is πΌπ of the Norton equivalent circuit. 13 Norton’s Theorem Example (1) Find the current in the 1πΎΩ resistor using Norton’s theorem. 14 Norton’s Theorem Example (2) For the shown circuit, use Norton’s theorem to find the load current πΌπΏ . 15 Network Theorems Definition Electric circuit theorems are always beneficial to help find voltage and currents in multi loop circuits. The network theorems include: • • • • Superposition Theorem Thevenin’s Theorem Norton’s Theorem Maximum Power Transfer Theorem 16 Maximum Power Transfer Definition The maximum amount of power will be dissipated by a load resistance when that load resistance is equal to the Thevenin/Norton resistance of the network supplying the power. For maximum power ππ πΏ → π πΏ = π π‘β = π π 17 Maximum Power Transfer Theorem Proof ππ πΏ = πΌ 2 π πΏ ππ‘β = π π‘β + π πΏ 2 = ππ‘β 2 π πΏ π πΏ π π‘β + π πΏ 2 For maximum power 2−2 π +π ×π πππ πΏ π‘β πΏ πΏ 2 1 × π π‘β + π πΏ = ππ‘β =0 ππ πΏ π π‘β + π πΏ 4 18 Maximum Power Transfer Theorem Proof For maximum power 1 × π π‘β + π πΏ 2 − 2 π π‘β + π πΏ × π πΏ = 0 2 π π‘β + 2π π‘β π πΏ + π πΏ2 − 2π π‘β π πΏ − 2π πΏ2 = 0 2 π π‘β = π πΏ2 π πΏ = π π‘β For maximum power ππ πΏ → π πΏ = π π‘β = π π 19 Maximum Power Transfer Power Efficiency πππ = πππ‘β = ππ‘β × πΌ = ππ‘β × ππ‘β π π‘β + π πΏ πππ’π‘ = ππ πΏ 2 = ππ‘β π πΏ π π‘β + π πΏ 2 πππ’π‘ π πΏ × 100% = × 100% Power Efficiency π = πππ π π‘β + π πΏ 20 Maximum Power Transfer Power Efficiency At condition of maximum power transfer (π πΏ = π π‘β ) πππ’π‘ = 2 ππ‘β π πΏ π π‘β + π πΏ π πΏ π= × 100% π π‘β + π πΏ → 2 → 2 ππ‘β π πΏ π πΏ + π πΏ 2 2 ππ‘β = 4π πΏ π πΏ × 100% = 50% π πΏ + π πΏ 21 Maximum Power Transfer Example (3) For the shown circuit, find the value of π πΏ required to be terminated between terminals π and π for maximum power transfer. 22 Network Theorems Definition Electric circuit theorems are always beneficial to help find voltage and currents in multi loop circuits. The network theorems include: • • • • Superposition Theorem Thevenin’s Theorem Norton’s Theorem Maximum Power Transfer Theorem 23
