Transcript Lecture 6
Lecture #6 OUTLINE • Complete Mesh Analysis Example(s) • Superposition • Thévenin and Norton equivalent circuits • Maximum Power Transfer Reading Chapter 2 EECS40, Fall 2004 Lecture 6, Slide 1 Prof. White Superposition A linear circuit is one constructed only of linear elements (linear resistors, and linear capacitors and inductors, linear dependent sources) and independent sources. Linear means I-V charcteristic of elements/sources are straight lines when plotted Principle of Superposition: • In any linear circuit containing multiple independent sources, the current or voltage at any point in the network may be calculated as the algebraic sum of the individual contributions of each source acting alone. EECS40, Fall 2004 Lecture 6, Slide 2 Prof. White Superposition Procedure: 1. Determine contribution due to one independent source • Set all other sources to 0: Replace independent voltage source by short circuit, independent current source by open circuit 2. Repeat for each independent source 3. Sum individual contributions to obtain desired voltage or current EECS40, Fall 2004 Lecture 6, Slide 3 Prof. White Superposition Example • Find Vo – + 24 V 2W 4V +– 4A + 4 W Vo – EECS40, Fall 2004 Lecture 6, Slide 4 Prof. White Equivalent Circuit Concept • A network of voltage sources, current sources, and resistors can be replaced by an equivalent circuit which has identical terminal properties (I-V characteristics) without affecting the operation of the rest of the circuit. iA network A of sources and resistors iB + vA _ ≡ network B of sources and resistors + vB _ iA(vA) = iB(vB) EECS40, Fall 2004 Lecture 6, Slide 5 Prof. White Source Combinations • Voltage sources in series can be replaced by an equivalent voltage source: ≡ v1+v2 – + – + v2 – + v1 • Current sources in parallel can be replaced by an equivalent current source: i1 EECS40, Fall 2004 i2 ≡ i1+i2 Lecture 6, Slide 6 Prof. White Thévenin Equivalent Circuit • Any* linear 2-terminal (1-port) network of indep. voltage sources, indep. current sources, and linear resistors can be replaced by an equivalent circuit consisting of an independent voltage source in series with a resistor without affecting the operation of the rest of the circuit. Thévenin equivalent circuit RTh a + iL vL RL ≡ VTh – – + network of sources and resistors a b + iL vL RL – b “load” resistor EECS40, Fall 2004 Lecture 6, Slide 7 Prof. White I-V Characteristic of Thévenin Equivalent • The I-V characteristic for the series combination of elements is obtained by adding their voltage drops: For a given current i, the voltage drop vab is equal to the sum of the voltages dropped across the source (VTh) and across the resistor (iRTh) RTh a v i + – + VTh i vab = VTh+ iR vab – b I-V characteristic of resistor: v = iR I-V characteristic of voltage source: v = VTh EECS40, Fall 2004 Lecture 6, Slide 8 Prof. White Finding VTh and RTh Only two points are needed to define a line. Choose two convenient points: RTh 1. Open circuit across terminals a,b i i = 0, vab ≡ voc – + VTh 2. Short circuit across terminals a,b vab = 0, i ≡ -isc = -VTh/RTh + voc = VTh – i voc RTh vab i EECS40, Fall 2004 – + VTh isc -isc v = VTh+ iR Lecture 6, Slide 9 Prof. White Calculating a Thévenin Equivalent 1. Calculate the open-circuit voltage, voc a network of sources and resistors + voc – b 2. Calculate the short-circuit current, isc • Note that isc is in the direction of the open-circuit voltage drop across the terminals a,b ! a network of sources and resistors isc RTh b EECS40, Fall 2004 Lecture 6, Slide 10 VTh voc voc isc Prof. White Thévenin Equivalent Example Find the Thevenin equivalent with respect to the terminals a,b: EECS40, Fall 2004 Lecture 6, Slide 11 Prof. White Alternative Method of Calculating RTh For a network containing only independent sources network of and linear resistors: independent sources and resistors, with each source set to zero 1. Set all independent sources to zero voltage source short circuit current source open circuit Req 2. Find equivalent resistance Req between the terminals by inspection RTh Req Or, set all independent sources to zero RTh EECS40, Fall 2004 VTEST I TEST Lecture 6, Slide 12 network of independent sources and resistors, with each source set to zero VTEST – + 1. Apply a test voltage source VTEST 2. Calculate ITEST ITEST Prof. White RTh Calculation Example #1 Set all independent sources to 0: EECS40, Fall 2004 Lecture 6, Slide 13 Prof. White Comments on Dependent Sources A dependent source establishes a voltage or current whose value depends on the value of a voltage or current at a specified location in the circuit. (device model, used to model behavior of transistors & amplifiers) To specify a dependent source, we must identify: 1. 2. 3. the controlling voltage or current (must be calculated, in general) the relationship between the controlling voltage or current and the supplied voltage or current the reference direction for the supplied voltage or current The relationship between the dependent source and its reference cannot be broken! – Dependent sources cannot be turned off for various purposes (e.g. to find the Thévenin resistance, or in analysis using Superposition). EECS40, Fall 2004 Lecture 6, Slide 14 Prof. White RTh Calculation Example #2 Find the Thevenin equivalent with respect to the terminals a,b: EECS40, Fall 2004 Lecture 6, Slide 15 Prof. White Networks Containing Time-Varying Sources Care must be taken in summing time-varying sources! Example: 10 sin (100t) 1 kW – + + 20 cos (100t) 1 kW – EECS40, Fall 2004 Lecture 6, Slide 16 Prof. White EECS40, Fall 2004 Lecture 6, Slide 17 Prof. White Norton Equivalent Circuit • Any* linear 2-terminal (1-port) network of indep. voltage sources, indep. current sources, and linear resistors can be replaced by an equivalent circuit consisting of an independent current source in parallel with a resistor without affecting the operation of the rest of the circuit. Norton equivalent circuit a network of sources and resistors + iL vL RL ≡ iN RN – + iL vL RL – b EECS40, Fall 2004 a b Lecture 6, Slide 18 Prof. White I-V Characteristic of Norton Equivalent • The I-V characteristic for the parallel combination of elements is obtained by adding their currents: For a given voltage vab, the current i is equal to the sum of the currents in each of the two branches: a i + iN RN i i = -IN+ Gv v vab – b I-V characteristic of resistor: i=Gv I-V characteristic of current source: i = -IN EECS40, Fall 2004 Lecture 6, Slide 19 Prof. White Finding IN and RN = RTh Analogous to calculation of Thevenin Eq. Ckt: 1) Find o.c voltage and s.c. current IN ≡ isc = VTh/RTh 2) Or, find s.c. current and Norton (Thev) resistance EECS40, Fall 2004 Lecture 6, Slide 20 Prof. White Finding IN and RN • We can derive the Norton equivalent circuit from a Thévenin equivalent circuit simply by making a source transformation: RTh – + vTh a a + iL vL RL iN RN – + iL vL RL – b b voc vTh RN RTh ; iN isc isc RTh EECS40, Fall 2004 Lecture 6, Slide 21 Prof. White Maximum Power Transfer Theorem Thévenin equivalent circuit Power absorbed by load resistor: RTh 2 – + VTh + iL vL RL VTh RL p i RL RTh RL 2 L – dp To find the value of RL for which p is maximum, set to 0: dRL 2 dp R R RL 2RTh RL 2 Th L VTh 0 4 dRL RTh RL RTh RL RL 2RTh RL 0 2 RTh RL EECS40, Fall 2004 A resistive load receives maximum power from a circuit if the load resistance equals the Thévenin resistance of the circuit. Lecture 6, Slide 22 Prof. White