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Kevin D. Donohue, University of Kentucky1 Nodal and Loop Analysis Systematic methods for labeling circuits and finding a solvable set of equations, Operational Amplifiers

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Kevin D. Donohue, University of Kentucky2 Example Simple circuits with single loops or node-pairs can result in one equation with one unknown, when properly labeled. R R0 R1 i1 R2 R3 R4 V

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Kevin D. Donohue, University of Kentucky3 Example of Nodal Analysis For more complex circuits a set of labels and equations in terms of node voltages can be developed. + Vx - 2Vx

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Kevin D. Donohue, University of Kentucky4 Example of Mesh Analysis For more complex circuits a set of labels and equations only in terms of loop currents can be developed. I2I2 3I 2

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Kevin D. Donohue, University of Kentucky5 Nodal Analysis Identify and label all nodes in the system. Select one node as a reference node (V=0) Perform KCL at each non-reference node expressing each branch current in terms of the node voltages If any branch contains a voltage source One way: Make reference node the negative end of the voltage source and set node values on the positive end equal to the source values (reduces number of equations and unknowns by one) Another way: Create an equation where the difference between the node voltages on either end to source is equal to the source value and then use a surface around both nodes for KCL (provides an extra equation lost from the unknown current in voltage source)

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Kevin D. Donohue, University of Kentucky6 Examples Perform nodal analysis on circuits with current sources and resistors. Perform nodal analysis on circuits with voltage sources and and resistors.

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Kevin D. Donohue, University of Kentucky7 Loop Analysis Create loop current labels that include every circuit branch where each loop contains a branch included by no other loop and no loops cross each other. Perform KVL around each loop expressing all voltages in terms of loop currents. If any branch contains a current source, One way: Let only only one loop current pass through source so loop current then equals the source value (reduces number of equations and unknowns by one) Another way: Let more than one loop pass through source and set combination of loop current equal to source value (provides an extra equation lost from the unknown voltage drop on current source)

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Kevin D. Donohue, University of Kentucky8 Examples Perform loop analysis on circuits with voltage sources and resistors. Perform loop analysis on circuits with current sources and and resistors.

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Kevin D. Donohue, University of Kentucky9 Example Operational amplifiers (op amps) were originally developed to amplify DC voltage levels in analog computers. Today, their applications are many. Apply the model for the ideal op amp to find the voltage gain (Vo/Vi) in the given circuit: + Vo -

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Kevin D. Donohue, University of Kentucky10 Op Amp Model The actual op amp is composed of many transistors, but can be approximated with a simpler circuit model: V+V+ V-V- Ri Ro A(V + - V - ) + Vo - V+V+ V-V- + Vo -

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Kevin D. Donohue, University of Kentucky11 Ideal Approximation Typical values for Ri = 2 M , A=10 6, and Ro=50 . For circuit in the first example, use op amp model with dependent source to justify the ideal approximations made in the first example. Ideal op amp approximation: -Vd0+-Vd0+ I+0I+0 I-0I-0

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Kevin D. Donohue, University of Kentucky12 Analyzing Ideal Op Amp Circuits The simplifications for the op amp model suggest that nodal analysis will often be the best method of analyzing op amp circuits. Do examples of circuits with op amps, independent sources, and resistors.

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Kevin D. Donohue, University of Kentucky13 Op Amp Web Pages 741.htm (tutorial) 741.htm (tutorial – WARNING: crude language and humor used at this site. Not recommended for more sensitive or unstable students!) df (data sheet) df

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Kevin D. Donohue, University of Kentucky14 Design Example 2 Microphones with sensitivities of 3 mV/dB and 6 mV/dB and are denoted as independent voltage sources V 1 and V 2, respectively. Determine resistor values so that Vo is the difference between the microphone sound pressure levels (SPL) such that a 1 dB change in SPL corresponds to a 30 mV change in Vo. V1V1 V2V2 + Vo - R1R1 R3R3 R2R2 R4R4 R5R5 R6R6

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Kevin D. Donohue, University of Kentucky15 Design Formula It can be shown that for the subcircuit: V1V1 V2V2 + Va - R1R1 R3R3 R2R2 R4R4

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Kevin D. Donohue, University of Kentucky16 Design Formula It can be shown that for subcircuit: + Vo - R6R6 R5R5 + Va -

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Kevin D. Donohue, University of Kentucky17 Design Formula It can be shown that for the entire circuit: V1V1 V2V2 + Vo - R1R1 R3R3 R2R2 R4R4 R5R5 R6R6

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Kevin D. Donohue, University of Kentucky18 Design Strategy The structure of the formula suggests breaking the problem into 2 parts: Part 1: Let the part of the formula associated with the first subcircuit take care of the scaling the microphone sensitivities (to make both sensitivities equal) and then subtracting them. Part 2: Let the part of formula associated with second subcircuit take care of the scaling the difference signal to the 30 mV/dB specification.

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Kevin D. Donohue, University of Kentucky19 Specification Equations Scale the 3 mV/dB microphone circuit 2 times the amount as the other: Scale the 3 mV/dB microphone through the first and second circuit to achieve a 30 mV/dB scale:

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Kevin D. Donohue, University of Kentucky20 Design Decisions Note in the last example there are many solutions and several ways to set up the design equations and solve. Can you determine if one way is better than the other? What additional criteria may be added to this problem?

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