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Lecture 2 Review • Methods of Analysis —Nodal analysis • Without Voltage Sources • With Voltage Sources —Mesh analysis • Without Current Sources • With Coltage Sources —Analysis by Inspection Quiz An open-loop gain: 2105 Input resistance: 2 M Output resistance: 50 v Find the closed-loop gain: o vs Lecture 3 DC Circuits Circuit Theorems: to simplify circuit analysis Lecture Objectives • Linear circuit • Superposition • Equivalent Circuits —Simple Circuits —Y- Transforms —Source transformation —Thevenin's theorem —Norton's theorem • Maximum power transfer Linear circuit v iR —Homogeneity: i1 ki0 , v1 ki0 R k (i0 R) kv0 —Additivity: i i1 i2 , v (i1 i2 ) R i1 R i2 R v1 v2 • Linearity Property Input(excitation),i Linear Circuit Output(response), v Example: Linearity 3A 6V 2A 5A 14V 2A 3V 1A 8V Assume I0 = 1 A,… the source current Is =? 5V 5A Since Is = 15 A = 3 × 5A, according to linearity (homogeneity), I0 = ? 3 A Superposition: several sources • Linear circuit: Additivity • Voltage or Current = algebraic sum of v or i due to each independent source acting alone, respectively. • Can we use superposition for power? No! Why? If there are N sources, you’ll need to do similar calculation N times. Example: Superposition Use the superposition theorem to find v in the circuit: Turn off v = 10 V Voltage source 0V Short circuit Current source 0A Open circuit How about dependent sources? Equivalence of circuits a b • • • • • What elements are in the box? What is the behaviour of the circuit? Voltage-Current relation Series-parallel combination Y- transformation Source Transformation Thevenin’s Theorem Norton’s Theorem Voltage-current relation at terminals a and b is identical to that of its equivalent circuit. Simple resistive circuits • Series circuits — Current: elements in series carry the same current — Resistance in series: the sum of the individual resistances • Parallel circuits — Voltage: elements in parallel have the same voltage — Conductance in parallel: the sum of the individual conductances L1 One path I L2 Equivalent L=L1+L2 V RT RT A1 two paths R L A L1 L2 L L 1 1 R1 R2 A A A L , A1 A2 1 1 A1 A2 1 A1 1 A2 1 1 RT L L L R1 R2 A2 A=A1+A2 GT G1 G2 - Y Transformation R1 Rb Rc Ra Rb Rc Rc Ra R2 Ra Rb Rc Ra Rb R3 Ra Rb Rc T Y Y- Transformation Example Obtain the equivalent resistance Rab for the circuit and use it to find current i Source transformation Replacing a voltage source vs in series with a resistor R by a current source is in parallel with a resistor R, or vice versa. A Voltage Source A CurrentSouce Check: Voltage Source Open Circuit Short Circuit vs vs R That’s how this part of circuit acts on the load as a ’source’. Current Source Turn-off source R is R R is R That’s what the external sources ‘feel’ this part of the circuit. Example Use source transformation to find vo in the circuit Thevenin's theorem Load – variable Other elements - fixed Vth : Open circuit voltage RTh : the input resistance at the terminals when the independent sources are turned off Example Find the Thevenin equivalent circuit of the circuit, to the left of the terminals a-b. Then find the current through RL = 6, 16, and 36 Norton's theorem Maximum power transfer • Minimizing power dissipated in the process of transmission and distribution • Maximize the power delivered to a load p i 2 RL i VTh RTh RL 2 VTh 2 RL p i RL RTh RL Apendix Op Amp - typical packages 1. 2. An active circuit element Perform mathematical operations Pin 1 is always to the left of the notch or dot. For package information: http://www.intersil.com/design/packages/ Looking from the top 741 General-purpose: Fairchild Semiconductor … Intel Feedback path - example Feedback: negative feedback An open-loop gain: 2105 Input resistance: 2 M Output resistance: 50 v o Find the closed-loop gain: vs Nodal analysis vo 1.9999699 vs Insensitive to A This is tedious. Ideal Op Amp Virtual open circuit i1 = i2 =0 Virtual close circuit vd = 0; v1 = v2 • Infinite open-loop gain, A • Infinite input resistance, Ri • Zero output resistance, Ro 0 Idea Op Amp - Example Rework it using the ideal op amp model Idea Op Amp: i1 i2 0 v1 i1 =0 v2 i2 =0 Negligibly small error results from assuming ideal op amp characteristics Virtual open circuit: i1 = i2 =0 Virtual close circuit: vd = 0; v1 = v2 Node 1: vs v1 v1 vo 10 103 20 103 Idea Op Amp: v1 v2 0 vs 0 0 vo 10 103 20 103 vo 2 vs Working with non-ideal: vo 1.9999699 vs Lecture 4 DC Circuits Capacitors and Inductors