Chapter 2 Outline Resistors in Series Voltage Divider Conductances in Parallel Current Divider Node Voltage Analysis Mesh Current Analysis Superposition Th venin equivalent circuits Norton equivalent circuits Maximum Power Transfer EE40 Lecture 3 Prof Chang Hasnain 8 31 07 Reading Chap 2 EE40 Fall 2006 Slide 1 Prof Chang Hasnain EE40 Fall 2006 Slide 2 Resistors in Series Voltage Divider Consider a circuit with multiple resistors connected in series Find their equivalent resistance I R1 VSS R2 Prof Chang Hasnain KCL tells us that the same current I flows through every resistor I R1 VSS I VSS R1 R2 R3 R4 V1 R2 R3 KVL tells us V3 R4 R3 R4 Equivalent resistance of resistors in series is the sum EE40 Fall 2006 Slide 3 Prof Chang Hasnain EE40 Fall 2006 Slide 4 Prof Chang Hasnain Resistors in Parallel When can the Voltage Divider Formula be Used I I R1 R2 VSS R1 V2 VSS R3 V2 R2 R3 R4 R4 R5 ISS Consider a circuit with two resistors connected in parallel Find their equivalent resistance x KVL tells us that the same voltage is dropped I1 I2 across each resistor R1 R2 Vx I1 R1 I2 R2 KCL tells us V 2 R 2 V SS R R R R 1 2 3 4 V 2 Correct if nothing else is connected to nodes EE40 Fall 2006 R 2 V SS R R R R 1 2 3 4 Why What is V2 Slide 5 Prof Chang Hasnain EE40 Fall 2006 Slide 6 General Formula for Parallel Resistors Current Divider x What single resistance Req is equivalent to three resistors in parallel I I V R1 R2 R3 eq V Req Prof Chang Hasnain ISS I1 I2 R1 R2 Vx I1 R1 ISS Req Equivalent conductance of resistors in parallel is the sum EE40 Fall 2006 Slide 7 Prof Chang Hasnain EE40 Fall 2006 Slide 8 Prof Chang Hasnain Generalized Current Divider Formula Measuring Voltage Consider a current divider circuit with 2 resistors in parallel I V I3 I2 I1 R2 R1 R3 I3 V I 1 1 1 R1 R 2 R 3 To measure the voltage drop across an element in a real circuit insert a voltmeter digital multimeter in voltage mode in parallel with the element Voltmeters are characterized by their voltmeter input resistance Rin Ideally this should be very high typical value 10 M 1 R 3 V I R3 1 R 1 1 R 2 1 R 3 Ideal Voltmeter Rin EE40 Fall 2006 Slide 9 Prof Chang Hasnain EE40 Fall 2006 Slide 10 Prof Chang Hasnain Effect of Voltmeter circuit with voltmeter inserted undisturbed circuit R1 VSS EE40 Lecture 4 Prof Chang Hasnain R1 VSS V2 R2 R2 V2 Rin Compare to R2 R2 V2 VSS R1 R 2 9 5 07 Reading Chap 2 R 2 Rin V2 VSS R 2 Rin R1 Example VSS 10 V R 2 100 K R1 900 K V2 1V Rin 10 M V2 EE40 Fall 2006 Slide 11 Prof Chang Hasnain EE40 Fall 2006 Slide 12 Prof Chang Hasnain Measuring Current Effect of Ammeter To measure the current flowing through an element in a real circuit insert an ammeter digital multimeter in current mode in series with the element Ammeters are characterized by their ammeter input resistance Rin Ideally this should be very low typical value 1 Measurement error due to non zero input resistance undisturbed circuit circuit with ammeter inserted Imeas I V1 Rin V1 R2 R2 Ideal Ammeter I Rin V1 R1 R 2 Imeas V1 R1 R 2 Rin Example V1 1 V R1 R2 500 Rin 1 1V I 1mA I meas 500 500 EE40 Fall 2006 Slide 13 Prof Chang Hasnain Using Equivalent Resistances Simplify a circuit before applying KCL and or KVL EE40 Fall 2006 Slide 14 Prof Chang Hasnain 1 Choose a reference node ground Look for the one with the most connections I R1 Compare to R2 R2 Node Voltage Circuit Analysis Method Example Find I 7V ammeter R1 R1 R2 R3 R1 R2 3 k R3 6 k R6 R4 R5 5 k R6 10 k R4 R5 2 Define unknown node voltages those which are not fixed by voltage sources 3 Write KCL at each unknown node expressing current in terms of the node voltages using the I V relationships of branch elements Special cases floating voltage sources 4 Solve the set of independent equations N equations for N unknown node voltages EE40 Fall 2006 Slide 15 Prof Chang Hasnain EE40 Fall 2006 Slide 16 Prof Chang Hasnain Nodal Analysis Example 1 A floating voltage source is one for which neither side is connected to the reference node e g VLL in the circuit below R 3 R1 Nodal Analysis w Floating Voltage Source V1 R2 R4 VLL Va IS Vb I1 1 Choose a reference node 2 Define the node voltages except reference node and the one set by the voltage source R2 I2 R4 3 Apply KCL at the nodes with unknown voltage Problem We cannot write KCL at nodes a or b because there is no way to express the current through the voltage source in terms of Va Vb 4 Solve for unknown node voltages Solution Define a supernode that chunk of the circuit containing nodes a and b Express KCL for this supernode Incorporate voltage source constraint into KCL equation EE40 Fall 2006 Slide 17 Prof Chang Hasnain EE40 Fall 2006 Slide 18 Nodal Analysis Example 2 Nodal Analysis Example 3 supernode VLL Va R1 Va Vb R2 R3 V I1 Prof Chang Hasnain R4 1 R2 I1 R4 R5 V2 I2 Challenges Determine number of nodes needed Deal with different types of sources Eq n 1 KCL at supernode Substitute property of voltage source EE40 Fall 2006 Slide 19 Prof Chang Hasnain EE40 Fall 2006 Slide 20 Prof Chang Hasnain Formal Circuit Analysis Methods Mesh Analysis Example 1 MESH ANALYSIS Mesh Current Method 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 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 1 Select M mesh currents 2 Apply KVL to each mesh 3 Solve for mesh currents Simple method for planar circuits A mesh current is not necessarily identified with a branch current EE40 Fall 2006 Slide 21 Prof Chang Hasnain Mesh Analysis Example 2 ia EE40 Fall 2006 Slide 22 Prof Chang Hasnain 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 ib Eq n 1 KVL for supermesh Eq n 2 Constraint …
View Full Document
Unlocking...