Announcements New topics: Mesh (loop) method of circuit analysis Superposition method of circuit analysis Equivalent circuit idea (Thevenin, Norton) Maximum power transfer from a circuit to a load To stop blowing fuses in the lab, note how the breadboards are wired … EECS 42, Spring 2005 Week 3a 1

Top view of board EECS 42, Spring 2005 Week 3a 2

Top view of board EECS 42, Spring 2005 Week 3a 3

Bottom view of board – note which way the wires go EECS 42, Spring 2005 Week 3a 4

Primary Formal Circuit Analysis Methods MESH ANALYSIS (“Mesh-Current Method”) 1) Select M independent mesh currents such that at least one mesh current passes through each branch* M = #branches - #nodes + 1 2) Apply KVL to each mesh, expressing voltages in terms of mesh currents => M equations for M unknown mesh currents 3) Solve for mesh currents => determine node voltages NODAL ANALYSIS (“Node-Voltage Method”) 0) Choose a reference node 1) Define unknown node voltages 2) Apply KCL to each unknown node, expressing current in terms of the node voltages => N equations for N unknown node voltages 3) Solve for node voltages => determine branch currents *Simple method for planar circuits A mesh current is not necessarily identified with a branch current. EECS 42, Spring 2005 Week 3a 5

Mesh Analysis: Example #1 Select M mesh currents. Apply KVL to each mesh. Solve for mesh currents. EECS 42, Spring 2005 Week 3a 6

Mesh Analysis with a Current Source ia ib Problem: We cannot write KVL for meshes a and b because there is no way to express the voltage drop across the current source in terms of the mesh currents. Solution: Define a “supermesh” – a mesh which avoids the branch containing the current source. Apply KVL for this supermesh. EECS 42, Spring 2005 Week 3a 7

Mesh Analysis: Example #2 ia ib Eq’n 1: KVL for supermesh Eq’n 2: Constraint due to current source: EECS 42, Spring 2005 Week 3a 8

Mesh Analysis with Dependent Sources Exactly analogous to Node Analysis Dependent Voltage Source: (1) Formulate and write KVL mesh eqns. (2) Include and express dependency constraint in terms of mesh currents Dependent Current Source: (1) Use supermesh. (2) Include and express dependency constraint in terms of mesh currents EECS 42, Spring 2005 Week 3a 9

Superposition Method (Linear Circuits Only) A linear circuit is constructed only of linear elements (linear resistors, linear dependent sources) and independent sources. 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. Procedure: Determine contribution due to an independent source Set all other sources to zero (voltage source  short circuit; current source  open circuit) Repeat for each independent source Sum individual contributions to obtain desired voltage or current EECS 42, Spring 2005 Week 3a 10

Superposition Example Find Vo 4 V 2  + – + Vo – 24 V + – 4 A 4  EECS 42, Spring 2005 Week 3a 11

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Source Combinations ≡ ≡ Voltage sources in series can be replaced by an equivalent voltage source: Current sources in parallel can be replaced by an equivalent current source: v1 + – v1+v2 + – ≡ + – v2 ≡ i1+i2 i1 i2 EECS 42, Spring 2005 Week 3a 14

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 a network of sources and resistors + vL – iL + vL – iL ≡ + – RL VTh RL b b “load” resistor EECS 42, Spring 2005 Week 3a 15

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 the across the resistor (iRTh) i RTh a v = VTh+ iR i v + vab – + – VTh b I-V characteristic of resistor: v = iR I-V characteristic of voltage source: v = VTh EECS 42, Spring 2005 Week 3a 16

Finding VTh and RTh Only two points are needed to define a line. Choose two convenient points: 1. Open circuit across terminals a,b i = 0, vab ≡ voc 2. Short circuit across terminals a,b vab = 0, i ≡ -isc = -VTh/RTh i i voc R vab Th i -isc i sc + V Th – v = VTh+ iR EECS 42, Spring 2005 Week 3a 17

Calculating a Thévenin Equivalent Calculate the open-circuit voltage, voc 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 + voc – b a network of sources and resistors isc EECS 42, Spring 2005 Week 3a b 18

Thévenin Equivalent Example Find the Thevenin equivalent with respect to the terminals a,b: EECS 42, Spring 2005 Week 3a 19

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Alternative Method of Calculating RTh For a network containing only independent sources and linear resistors: Set all independent sources to zero voltage source  short circuit current source  open circuit Find equivalent resistance Req between the terminals by inspection Or, set all independent sources to zero Apply a test voltage source VTEST Calculate ITEST network of independent sources and resistors, with each source set to zero Req ITEST network of independent sources and resistors, with each source set to zero + – VTEST EECS 42, Spring 2005 Week 3a 21

RTh Calculation Example #1 Set all independent sources to zero: EECS 42, Spring 2005 Week 3a 22

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: 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). EECS 42, Spring 2005 Week 3a 23

RTh Calculation Example #2 Find the Thevenin equivalent with respect to the terminals a,b: EECS 42, Spring 2005 Week 3a 24

Networks Containing Time-Varying Sources Care must be taken in summing time-varying sources! Example: 10 sin (100t) 1 k – + + – 20 cos (100t) 1 k EECS 42, Spring 2005 Week 3a 25

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 a RL iL + vL – iN b RN network of sources and resistors + vL – iL ≡ RL b EECS 42, Spring 2005 Week 3a 26

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: i a i + vab – i = -IN+ Gv v iN RN b I-V characteristic of resistor: i=Gv I-V characteristic of current source: i = -IN EECS 42, Spring 2005 Week 3a 27

Finding IN and RN = RTh Analogous to calculation of Thevenin Eq. Ckt: 1) Find open-circuit voltage and short-circuit current IN ≡ isc = VTh/RTh 2) Or, find short-circuit current and Norton (Thevenin) resistance EECS 42, Spring 2005 Week 3a 28

Finding IN and RN We can derive the Norton equivalent circuit from a Thévenin equivalent circuit simply by making a source transformation: RTh a a + vL – iL + vL – iL vTh + – iN RL RN RL b b EECS 42, Spring 2005 Week 3a 29

Maximum Power Transfer Theorem Thévenin equivalent circuit Power absorbed by load resistor: RTh + vL – iL + – VTh RL To find the value of RL for which p is maximum, set to 0: A resistive load receives maximum power from a circuit if the load resistance equals the Thévenin resistance of the circuit. EECS 42, Spring 2005 Week 3a 30

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