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TECHNIQUES OF DC CIRCUIT ANALYSIS: Superposition Principle Thevenin’s Theorem Norton’s Theorem Source Transformation Maximum Power Transfer 1 • Applies only for LINEAR CIRCUIT Circuit containing linear elements, linear dependent and independent sources A LINEAR relationship between voltage and current What do we mean by a linear relationship? 2 When the relationship fulfilled 2 properties: • Homogeneity (scaling) f(x) = y f(kx) = ky = kf(x) • Additivity f(x) = y f(x1 + x2) = f(x1) + f(x2) = y1 + y2 What do we mean by a linear relationship? 3 Superposition Principle: The voltage across an element ( or the current through an element) of a linear circuit containing more than one independent source, is the algebraic sum the voltage across that element (or the current through that element) due to each independent source acting alone. All other independent sources are KILLED • voltage sources are shorted • current sources are opened Dependent sources CANNOT be killed ! 4 Superposition Principle: The voltage across an element ( or the current through an element) of a linear circuit containing more than one independent source, is the algebraic sum the voltage across that element (or the current through that element) due to each independent source acting alone. 5 Superposition Principle: The voltage across an element ( or the current through an element) of a linear circuit containing more than one independent source, is the algebraic sum the voltage across that element (or the current through that element) due to each independent source acting alone. • may involve MORE work • cannot be applied to power calculation – find i or v first (using superposition) before calculating power ! • most suitably used when involved with sources of different properties or types, e.g. different frequencies, mixture of DC and AC, etc. 6 Thevenin’s : A linear two-terminal circuit can be In 1883, M.L. Theorem Thevenin proposed a theorem ……. replaced by an equivalent circuit consisting of a voltage source in series with a resistor I Linear twoterminal circuit + V Load RTh I VTh= ? + VTh V Load RTh= ? 7 Thevenin’s Theorem: A linear two-terminal circuit can be replaced by an equivalent circuit consisting of a voltage source in series with a resistor To determine VTh RTh VTh Load = Linear twoterminal circuit Load 8 Thevenin’s Theorem: A linear two-terminal circuit can be replaced by an equivalent circuit consisting of a voltage source in series with a resistor To determine VTh RTh + VTh Loadvoltage = Voc open circuit = VTh Linear twoterminal circuit Load 9 Thevenin’s Theorem: A linear two-terminal circuit can be replaced by an equivalent circuit consisting of a voltage source in series with a resistor To determine VTh RTh + VTh open circuit voltage = Voc = VTh Linear twoterminal circuit + Load open circuit voltage = Voc 10 Thevenin’s Theorem: A linear two-terminal circuit can be replaced by an equivalent circuit consisting of a voltage source in series with a resistor To determine VTh RTh + VTh open circuit voltage = Voc = VTh VTh = Voc = Open circuit voltage Linear twoterminal circuit + = VTh voltage (Since=the open circuit Voc circuit is equivalent) 11 Thevenin’s Theorem: A linear two-terminal circuit can be replaced by an equivalent circuit consisting of a voltage source in series with a resistor To determine RTh Case 1 Network with NO dependent sources • Kill all the independent sources Linear circuit – independent sources killed Rin = RTh • Find the equivalent R looking between the terminals 12 Thevenin’s Theorem: A linear two-terminal circuit can be replaced by an equivalent circuit consisting of a voltage source in series with a resistor To determine RTh Case 2 Network with dependent sources • Kill all the independent sources - dependent sources stay as they are Linear Circuit – ONLY dependent sources killed • io vo + - Introduce a voltage (or current) source. RTh is calculated as: R Th vo io 13 Norton’s Theorem: A linear two-terminal circuit can be replaced 43byyears later, E.L. Norton proposed a similar theorem. …. an equivalent circuit consisting of a current source in parallel with a resistor I + Linear twoterminal circuit Load V IN = ? I IN RN + V Load RN = ? 14 Norton’s Theorem: A linear two-terminal circuit can be replaced by an equivalent circuit consisting of a current source in parallel with a resistor To determine IN IN RN IN Linear circuit 15 Norton’s Theorem: A linear two-terminal circuit can be replaced by an equivalent circuit consisting of a current source in parallel with a resistor To determine IN IN Linear circuit RN IN= Short circuit current Short circuit current = IN 16 Norton’s Theorem: A linear two-terminal circuit can be replaced by an equivalent circuit consisting of a current source in parallel with a resistor To determine IN IN RN IN= Short circuit current IN = Isc = Short circuit current Linear circuit Short circuit current = IN 17 Norton’s Theorem: A linear two-terminal circuit can be replaced by an equivalent circuit consisting of a current source in parallel with a resistor To determine RN SIMILAR METHOD AS HOW TO OBTAIN RTh RN = RTh 18 Relationship between Norton’s and Thevenin’s equivalents a IN RN b Linear twoterminal circuit a OR b VTh RTh a b 19 Relationship between Norton’s and Thevenin’s equivalents a + IN v oc INR N RN b Since both circuits are equivalent, VT h vocvmust oc VT h INR N Rbe R the same N Th is c IN + v oc VTh VTh RTh a b 20 Source Transformation: A tool used to simplify circuit; a process of replacing a voltage source in series with a resistor by a current source in parallel with a resistor or vice versa R a vs a is b R b voc = isR voc = vs If the circuit is equivalent at terminal a-b, their open-circuit and short-circuiti characteristics are similar isc = is = v /R sc s 21 Source Transformation: A tool used to simplify circuit; a process of replacing a voltage source in series with a resistor by a current source in parallel with a resistor or vice versa R a a vs is R b b voc = isR voc = vs isc = is isc = vs/R is vs R or v s i sR 22 Maximum Power Transfer Linear circuit RL What would be the value of RL for power delivered to it become MAXIMUM? 23 Maximum Power Transfer RTh VTh Linear circuit RL What would be the value of RL for power delivered to it become MAXIMUM? 2 RL VTh 2 VTh R Th R L R L PL RL R Th R L 24 Maximum Power Transfer Maximum power 2.4 p Rl=linspace(1,60,500); Vth=10; Rth=12; p=((Vth./(Rl+Rth)).^2).*Rl; 2.2 2 1.8 1.6 2 R L 1.4 plot(Rl,p,'r'); V 1.2 2 Th grid; VTh R Th R L 1 R L PL 0.8 RL R Th R L 0.6 0.4 RL = 12 0 10 20 30 40 50 60 RL 25 Maximum Power Transfer 2 RL VTh 2 R R VTh L Th R L PL RL R Th R L Mathematically, we evaluate RL when dPL 0 dRL dPL 2VTh2 VTh2 R 0 3 L 2 dRL (R Th RL ) (R Th RL ) dPL VTh2 2R L R Th R L 0 3 dRL (R Th R L ) RL RTh 26 Using PSpice to verify Norton’s and Thevenin’s Theorems Find Thevenin equivalent at terminals a-b 27 Using PSpice to verify Norton’s and Thevenin’s Theorems 28 Using PSpice to verify Norton’s and Thevenin’s Theorems + - E + - E2 R9 R8 2 2 I3 1Aac TRAN = 0 R6 R7 4 6 0 29 Using PSpice to verify Norton’s and Thevenin’s Theorems + - E + - E2 R9 R8 2 2 I3 1Aac TRAN = 0 R6 R7 4 6 0 30 Using PSpice to verify Norton’s and Thevenin’s Theorems + - E + - E2 R9 R8 2 2 I3 1Aac TRAN = 0 I4 R6 R7 4 6 1Aac TRAN = 1 0 31 Using PSpice to verify Norton’s and Thevenin’s Theorems + - E RTh = 6/1 = 6 + - 1.333V E2 4.000V R9 R8 2 2 6.000V I3 1Aac TRAN = 0 I4 R6 R7 4 6 1Aac TRAN = 1 0 32 Using PSpice to verify Norton’s and Thevenin’s Theorems + - E VTh = 20V + - 6.667V E2 20.00V R9 R8 2 2 20.00V I3 1Aac TRAN = 5 I4 R6 R7 4 6 1Aac TRAN = 0 0 33