Discussion #9 - Equivalent Circuits
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Schedule… Date Day 1 Oct Wed 2 Oct Thu 3 Oct Fri 4 Oct Sat 5 Oct Sun 6 Oct Mon 7 Oct Tue 8 Oct Wed ECEN 301 Class No. Title Chapters 9 Equivalent Circuits 3.6 Recitation 10 Energy Storage HW Due date Lab Due date Exam HW 4 3.7, 4.1 NO LAB NO LAB 11 Dynamic Circuits 4.2 – 4.4 Discussion #9 – Equivalent Circuits 1 Equivalence - Equality Mosiah 29: 38 38 Therefore they relinquished their desires for a king, and became exceedingly anxious that every man should have an equal chance throughout all the land; yea, and every man expressed a willingness to answer for his own sins. ECEN 301 Discussion #9 – Equivalent Circuits 2 Current Sources All current sources can be modeled as voltage sources (and vice-versa) Many sources are best modeled as voltage sources (batteries, electric outlets etc.) There are some things that are best modeled as current sources: • Van de Graaff generator Behaves as a current source because of its very high output voltage coupled with its very high output resistance and so it supplies the same few microamps at any output voltage up to hundreds of thousands of volts ECEN 301 Discussion #9 – Equivalent Circuits 3 Lecture 9 – Equivalent Circuits Thévenin Equivalent Norton Equivalent ECEN 301 Discussion #9 – Equivalent Circuits 4 Network Analysis Network Analysis Methods: Node voltage method Mesh current method Superposition Equivalent circuits Source transformation Thévenin equivalent Norton equivalent ECEN 301 Discussion #9 – Equivalent Circuits 5 Equivalent Circuits It is always possible to view a complicated circuit in terms of a much simpler equivalent source and equivalent load circuit. i i R2 + va + – v R4 + R1 R3 R5 Source Load – – ECEN 301 v Discussion #9 – Equivalent Circuits 6 Equivalent Circuits It is always possible to view a complicated circuit in terms of a much simpler equivalent source and equivalent load circuit. i R2 va + – R1 + R3 v i R4 + R5 Source – ECEN 301 Discussion #9 – Equivalent Circuits v Load – 7 Equivalent Circuits Equivalent circuits fall into one of two classes: Thévenin: voltage source vT and series resistor RT Norton: current source iN and parallel resistor RN RT vT + i i + + v – Load iN RN – v Load – Norton Equivalent Thévenin Equivalent NB: RT = RN ECEN 301 Discussion #9 – Equivalent Circuits 8 Equivalent Circuits Thévenin Theorem: when viewed from the load, any network comprised of independent sources and linear elements (resistors), may be represented by an equivalent circuit. Equivalent circuit consists of an ideal voltage source vT in series with an equivalent resistance RT RT i + vT + – v Load – A fancy way of saying: “The circuit that includes everything except for the load” “View from load” ECEN 301 Discussion #9 – Equivalent Circuits 9 Equivalent Circuits Norton Theorem: when viewed from the load, any network comprised of independent sources and linear elements (resistors), may be represented by an equivalent circuit. Equivalent circuit consists of an ideal current source iN in parallel with an equivalent resistance RN i + iN RN v Load – A fancy way of saying: “The circuit that includes everything except for the load” “View from load” ECEN 301 Discussion #9 – Equivalent Circuits 10 Thévenin and Norton Resistances Computation of Thévenin and Norton Resistances: 1. Remove the load (open circuit at load terminal) 2. Zero all independent sources Voltage sources Current sources short circuit (v = 0) open circuit (i = 0) 3. Compute equivalent resistance (with load removed) ECEN 301 Discussion #9 – Equivalent Circuits 11 Thévenin and Norton Resistances Example1: find the equivalent resistance as seen by the load RL is =0.5A, vs = 10V, R1 = 4Ω, R2 = 6Ω , R3 = 10Ω , R4 = 2Ω , R5 = 2Ω , R6 = 3Ω, R7 = 5Ω R4 R2 vs is R1 ECEN 301 + – R7 R5 R6 RL R3 Discussion #9 – Equivalent Circuits 12 Thévenin and Norton Resistances Example1: find the equivalent resistance as seen by the load RL is =0.5A, vs = 10V, R1 = 4Ω, R2 = 6Ω , R3 = 10Ω , R4 = 2Ω , R5 = 2Ω , R6 = 3Ω, R7 = 5Ω R2 vs is R1 ECEN 301 1. R4 + – Remove the load R7 R5 R6 R3 Discussion #9 – Equivalent Circuits 13 Thévenin and Norton Resistances Example1: find the equivalent resistance as seen by the load RL is =0.5A, vs = 10V, R1 = 4Ω, R2 = 6Ω , R3 = 10Ω , R4 = 2Ω , R5 = 2Ω , R6 = 3Ω, R7 = 5Ω 2. R4 R2 R7 R5 R1 ECEN 301 Zero all independent sources • short circuit voltage sources • Open circuit current sources R6 R3 Discussion #9 – Equivalent Circuits 14 Thévenin and Norton Resistances Example1: find the equivalent resistance as seen by the load RL is =0.5A, vs = 10V, R1 = 4Ω, R2 = 6Ω , R3 = 10Ω , R4 = 2Ω , R5 = 2Ω , R6 = 3Ω, R7 = 5Ω 3. R4 R2 R7 R5 R1 ECEN 301 REQ1 R6 R3 Compute equivalent resistance R1 R2 4 6 10 Discussion #9 – Equivalent Circuits REQ2 R4 R5 R4 R5 (2)(2) (2) (2) 4 4 1 15 Thévenin and Norton Resistances Example1: find the equivalent resistance as seen by the load RL is =0.5A, vs = 10V, R1 = 4Ω, R2 = 6Ω , R3 = 10Ω , R4 = 2Ω , R5 = 2Ω , R6 = 3Ω, R7 = 5Ω 3. Compute equivalent resistance R7 REQ2 REQ1 REQ3 R6 R3 REQ1 10 REQ1 R3 REQ1 R3 (10 )(10 ) (10 ) (10 ) 100 20 5 REQ2 1 ECEN 301 Discussion #9 – Equivalent Circuits 16 Thévenin and Norton Resistances Example1: find the equivalent resistance as seen by the load RL is =0.5A, vs = 10V, R1 = 4Ω, R2 = 6Ω , R3 = 10Ω , R4 = 2Ω , R5 = 2Ω , R6 = 3Ω, R7 = 5Ω R7 3. Compute equivalent resistance REQ2 REQ4 R6 1 REQ3 5 ECEN 301 REQ2 5 1 6 REQ3 REQ2 REQ3 Discussion #9 – Equivalent Circuits 17 Thévenin and Norton Resistances Example1: find the equivalent resistance as seen by the load RL is =0.5A, vs = 10V, R1 = 4Ω, R2 = 6Ω , R3 = 10Ω , R4 = 2Ω , R5 = 2Ω , R6 = 3Ω, R7 = 5Ω R7 3. Compute equivalent resistance REQ5 REQ4 REQ4 ECEN 301 R6 REQ4 R6 REQ4 R6 (6)(3) (6) (3) 18 9 2 6 Discussion #9 – Equivalent Circuits 18 Thévenin and Norton Resistances Example1: find the equivalent resistance as seen by the load RL is =0.5A, vs = 10V, R1 = 4Ω, R2 = 6Ω , R3 = 10Ω , R4 = 2Ω , R5 = 2Ω , R6 = 3Ω, R7 = 5Ω R7 REQ5 3. REQ Compute equivalent resistance REQ5 R7 2 5 7 REQ5 ECEN 301 2 Discussion #9 – Equivalent Circuits 19 Thévenin and Norton Resistances Example1: find the equivalent resistance as seen by the load RL is =0.5A, vs = 10V, R1 = 4Ω, R2 = 6Ω , R3 = 10Ω , R4 = 2Ω , R5 = 2Ω , R6 = 3Ω, R7 = 5Ω 3. REQ ECEN 301 REQ Compute equivalent resistance 7 Discussion #9 – Equivalent Circuits 20 Thévenin Voltage Thévenin equivalent voltage: equal to the open-circuit voltage (voc) present at the load terminals (load removed) RT i RT + vT + – v vT Load + – – + i=0 voc = vT – Remove load ECEN 301 Discussion #9 – Equivalent Circuits 21 Thévenin Voltage Computing Thévenin voltage: 1. Remove the load (open circuit at load terminals) 2. Define the open-circuit voltage (voc) across the load terminals 3. Chose a network analysis method to find voc node, mesh, superposition, etc. 4. Thévenin voltage vT = voc ECEN 301 Discussion #9 – Equivalent Circuits 22 Thévenin Voltage Example2: find the Thévenin voltage vs = 10V, R1 = 4Ω, R2 = 6Ω , R3 = 10Ω R1 vs + – ECEN 301 R3 R2 iL RL Discussion #9 – Equivalent Circuits 23 Thévenin Voltage Example2: find the Thévenin voltage vs = 10V, R1 = 4Ω, R2 = 6Ω , R3 = 10Ω +R1– vs + – ECEN 301 + R3 – + R2 – + 1. 2. Remove the load Define voc voc – Discussion #9 – Equivalent Circuits 24 Thévenin Voltage Example2: find the Thévenin voltage vs = 10V, R1 = 4Ω, R2 = 6Ω , R3 = 10Ω i3 = 0 +R1– vs + – i + R3 – + R2 – 3. + Choose a network analysis method • Voltage divider voc – voc ECEN 301 Discussion #9 – Equivalent Circuits R2 R1 R2 vs 25 Thévenin Voltage Example2: find the Thévenin voltage vs = 10V, R1 = 4Ω, R2 = 6Ω , R3 = 10Ω i3 = 0 +R1– vs + – i ECEN 301 + R3 – + R2 – 4. vT = voc + voc – vT R2 R1 R2 Discussion #9 – Equivalent Circuits vs 26 Thévenin Equivalent Circuit Computing Thévenin Equivalent Circuit: 1. Compute the Thévenin resistance RT 2. Compute the Thévenin voltage vT RT i + vT + – v Load – ECEN 301 Discussion #9 – Equivalent Circuits 27 Thévenin Equivalent Circuit Example3: find iL by finding the Thévenin equivalent circuit vs = 10V, R1 = 4Ω, R2 = 6Ω, R3 = 10Ω, RL = 10Ω R1 vs + – ECEN 301 R3 R2 iL RL Discussion #9 – Equivalent Circuits 28 Thévenin Equivalent Circuit Example3: find iL by finding the Thévenin equivalent circuit vs = 10V, R1 = 4Ω, R2 = 6Ω, R3 = 10Ω, RL = 10Ω +R1– vs + – ECEN 301 + R3 – 1. Compute RT • Remove RL + R2 – Discussion #9 – Equivalent Circuits 29 Thévenin Equivalent Circuit Example3: find iL by finding the Thévenin equivalent circuit vs = 10V, R1 = 4Ω, R2 = 6Ω, R3 = 10Ω, RL = 10Ω +R1– + R3 – + R2 – ECEN 301 1. Compute RT • Remove RL • Zero sources Discussion #9 – Equivalent Circuits 30 Thévenin Equivalent Circuit Example3: find iL by finding the Thévenin equivalent circuit vs = 10V, R1 = 4Ω, R2 = 6Ω, R3 = 10Ω, RL = 10Ω 1. REQ RT ECEN 301 Compute RT • Remove RL • Zero sources • Compute RT = REQ R3 R1 || R2 Discussion #9 – Equivalent Circuits 31 Thévenin Equivalent Circuit Example3: find iL by finding the Thévenin equivalent circuit vs = 10V, R1 = 4Ω, R2 = 6Ω, R3 = 10Ω, RL = 10Ω R1 vs + – R3 R2 iL 1. 2. Compute RT Compute vT • (previously computed) RL vT ECEN 301 Discussion #9 – Equivalent Circuits R2 R1 R2 vs 32 Thévenin Equivalent Circuit Example3: find iL by finding the Thévenin equivalent circuit vs = 10V, R1 = 4Ω, R2 = 6Ω, R3 = 10Ω, RL = 10Ω RT R1 R3 R3 R1 || R2 RT iL iL vs + – R2 RL vT + RL – vT R2 R1 R2 vs NB: iL is the same in both circuits ECEN 301 Discussion #9 – Equivalent Circuits 33 Thévenin Equivalent Circuit Example3: find iL by finding the Thévenin equivalent circuit vs = 10V, R1 = 4Ω, R2 = 6Ω, R3 = 10Ω, RL = 10Ω RT iL R3 R1 || R2 RT – ECEN 301 RT RL R2 vT + vT vT R2 R1 R2 vs vs iL R1 R2 RL (6)(10 ) ( 4) ( 6) R3 1 R1 || R2 RL 1 (10 ) (4) || (6) (10 ) 60 1 10 20 2.4 6 22 .4 0.27 A Discussion #9 – Equivalent Circuits 34 Norton Current Current equivalent current: equal to the short-circuit current (isc) present at the load terminals (load replaced with short circuit) i iN = isc + iN RN v Load iN RN – Short circuit load ECEN 301 Discussion #9 – Equivalent Circuits 35 Norton Current Computing Norton current: 1. Replace the load with a short circuit 2. Define the short-circuit current (isc) across the load terminals 3. Chose a network analysis method to find isc node, mesh, superposition, etc. 4. Norton current iN = isc ECEN 301 Discussion #9 – Equivalent Circuits 36 Norton Current Example4: find the Norton equivalent current iN vs = 6V, is = 2A, R1 = 6Ω, R2 = 3Ω, R3 = 2Ω, RL = 10Ω vs – + R3 is R1 ECEN 301 R2 RL Discussion #9 – Equivalent Circuits 37 Norton Current Example4: find the Norton equivalent current iN vs = 6V, is = 2A, R1 = 6Ω, R2 = 3Ω, R3 = 2Ω, RL = 10Ω vs 1. 2. – + R3 is R1 ECEN 301 R2 Short-circuit the load Define isc isc Discussion #9 – Equivalent Circuits 38 Norton Current Example4: find the Norton equivalent current iN vs = 6V, is = 2A, R1 = 6Ω, R2 = 3Ω, R3 = 2Ω, RL = 10Ω vs Node a Node b – + is + R1 – iv i1 i3 3. Choose a network analysis method • Node voltage 1. 2. va is independent vb is dependent (actually va and vb are dependent on each other but choose vb) vb = va + vs + R3 – + R2 – i2 isc Node d ECEN 301 Discussion #9 – Equivalent Circuits 39 Norton Current Example4: find the Norton equivalent current iN vs = 6V, is = 2A, R1 = 6Ω, R2 = 3Ω, R3 = 2Ω, RL = 10Ω vs Node a Node b – + is + R1 – iv i1 3. i3 KCL at node a : + R3 – + R2 – i2 Choose a network analysis method • Node voltage isc is i1 iv 0 va R1 is iv Node d ECEN 301 Discussion #9 – Equivalent Circuits 40 Norton Current Example4: find the Norton equivalent current iN vs = 6V, is = 2A, R1 = 6Ω, R2 = 3Ω, R3 = 2Ω, RL = 10Ω vs Node a Node b – + is + R1 – iv i1 3. i3 Choose a network analysis method • Node voltage KCL at node b : + R3 – + R2 – i2 iv i2 i3 isc iv Node d ( va iv vb R2 vb R3 vs ) ( va vs ) R2 iv va ECEN 301 Discussion #9 – Equivalent Circuits R3 1 R2 1 R3 0 0 0 vs 1 R2 1 R3 41 Norton Current Example4: find the Norton equivalent current iN vs = 6V, is = 2A, R1 = 6Ω, R2 = 3Ω, R3 = 2Ω, RL = 10Ω vs Node a Node b – + is + R1 – iv i1 + R2 – i2 Choose a network analysis method • Node voltage 6iv 5va + R3 – Node d ECEN 301 3. i3 isc 6iv va 30 12 iv 2.5 A va 3V Discussion #9 – Equivalent Circuits 42 Norton Current Example4: find the Norton equivalent current iN vs = 6V, is = 2A, R1 = 6Ω, R2 = 3Ω, R3 = 2Ω, RL = 10Ω vs Node a Node b – + is + R1 – iv i1 3. i3 Choose a network analysis method • Node voltage + R3 – + R2 – i2 isc isc (vs vb ) 2 3 A 2 Node d ECEN 301 vb R3 Discussion #9 – Equivalent Circuits 43 Norton Current Example4: find the Norton equivalent current iN vs = 6V, is = 2A, R1 = 6Ω, R2 = 3Ω, R3 = 2Ω, RL = 10Ω vs – + is + R1 – ECEN 301 iv i1 4. i3 iN = isc + R3 – + R2 – i2 isc iN isc 1.5 A Discussion #9 – Equivalent Circuits 44 Norton Equivalent Circuit Computing Norton Equivalent Circuit: 1. Compute the Norton resistance RN 2. Compute the Norton current iN i + iN RN v Load – ECEN 301 Discussion #9 – Equivalent Circuits 45 Norton Equivalent Circuit Example5: find the Norton equivalent circuit vs = 6V, is = 2A, R1 = 6Ω, R2 = 3Ω, R3 = 2Ω, RL = 10Ω vs – + R3 is R1 ECEN 301 R2 RL Discussion #9 – Equivalent Circuits 46 Norton Equivalent Circuit Example5: find the Norton equivalent circuit vs = 6V, is = 2A, R1 = 6Ω, R2 = 3Ω, R3 = 2Ω, RL = 10Ω vs 1. – + R3 is R1 ECEN 301 Compute RN • Remove RL R2 Discussion #9 – Equivalent Circuits 47 Norton Equivalent Circuit Example5: find the Norton equivalent circuit vs = 6V, is = 2A, R1 = 6Ω, R2 = 3Ω, R3 = 2Ω, RL = 10Ω 1. R3 R1 ECEN 301 Compute RN • Remove RL • Zero sources R2 Discussion #9 – Equivalent Circuits 48 Norton Equivalent Circuit Example5: find the Norton equivalent circuit vs = 6V, is = 2A, R1 = 6Ω, R2 = 3Ω, R3 = 2Ω, RL = 10Ω 1. REQ RN ECEN 301 Compute RN • Remove RL • Zero sources • Compute RN = REQ R3 Discussion #9 – Equivalent Circuits R1 || R2 49 Norton Equivalent Circuit Example5: find the Norton equivalent circuit vs = 6V, is = 2A, R1 = 6Ω, R2 = 3Ω, R3 = 2Ω, RL = 10Ω vs 1. 2. – + R3 is R1 R2 RL Compute RN Compute iN • (previously computed) iN isc 1.5 A RN ECEN 301 Discussion #9 – Equivalent Circuits R3 R1 || R2 4 50 Norton Equivalent Circuit Example5: find the Norton equivalent circuit vs = 6V, is = 2A, R1 = 6Ω, R2 = 3Ω, R3 = 2Ω, RL = 10Ω vs – + R3 is R1 R2 RL iN iN ECEN 301 Discussion #9 – Equivalent Circuits RL RN 1.5 A RN 4 51
