EE 221 Review 2 Nodal and Mesh Analysis Superposition Source transformation Thevenin and Norton equivalent Operational Amplifier

Nodal Analysis - Approach 1.Redraw circuit to emphasize nodes. 2.Assign reference node and voltages. N nodes result in N-1 unknown voltages. 3.Use KCL to find N-1 equations. 4.Relate dependent sources to node voltages. 5.Form supernode to enclose voltage sources and apply KCL. Add voltage equations.

Nodal analysis - Example KCL requires that all currents flowing into the region must sum to zero, or we would pile up or run out of electrons. At node 1: (KCL) At the “supernode:” (KCL) At the “supernode:” (KVL)

Nodal analysis - Example Independent voltage source (supernode containing reference) v 1 = -12 (I) (II) (III) variables

Mesh analysis - approach 1.Redraw planar circuit to emphasize meshes. 2.Assign clockwise mesh currents. M meshes result in M unknown currents. 3.Apply KVL around each mesh. 4.Relate dependent sources to mesh currents. 5.Use supermesh for current source shared between two meshes. Add current equation.

Mesh analysis - Example Creating a “supermesh” from meshes 1 and 3: ( i 1 - i 2 ) + 3 ( i 3 - i 2 ) + 1 i 3 = 0 [1] Around mesh 2: 1 ( i 2 - i 1 ) + 2 i ( i 2 - i 3 ) = 0[2] Rearranging, i i i 3 = 7[1] -i i i 3 = 0[2] i 1 - i 3 = 7[3] Solving, i 1 = 9 A, i 2 = 2.5 A, and i 3 = 2 A. Finally, we relate the currents in meshes 1 and 3: i 1 - i 3 = 7[3]

Circuit analysis (a) A voltage source set to zero acts like a short circuit. (b) A current source set to zero acts like an open circuit.

Superposition (a) Linear circuits allow superposition. (b) Keep only one independent source at a time activate. (c) Always keep dependent sources.

Source transformation A general practical voltage source connected to a load resistor R L. A general practical current source connected to a load resistor R L. Convert between the two - Sources are related by: R S = R p, and Vs = R s I s = R p I s Useful when asked for: Maximum terminal voltage (v s ) and/or current (i s ) (Maximum) power transferred (P L = v L i L when R L = R s )

Thevenin and Norton "Dead" network to find equivalent source resistance R TH and R N Open loop voltage to determine V TH (any method) Short circuit current determines I N

Thenenin - Example Open loop voltage to determine V TH and short circuit current determines I N Find equivalent source resistance R TH and R N use "Dead" network use R TH = R N = V TH / I N (the only way in case of dependent sources) Source transformation is used here.

Operational Amplifier (a) Electrical symbol. (b) “Minimum" op amp. Ideal: (1) No input current. (2) No voltage difference between input terminals. Neglected: (1) Output voltage saturation. (2) Input/output resistance. (3) Limited open loop gain. (4) Input bias current. (5) Input offset voltage.

Operational Amplifier - Circuits Op amp connected as an Inverting amplifier. V out = - (R f / R 1 ) V in Output characteristics. 1st step: Determine voltage at input terminals 2nd step: Determine current i 3rd step: Find output voltage v out

Operational Amplifier - Circuits (c) (a) An op amp used to construct a noninverting amplifier circuit. (b) Circuit with currents and voltages labeled. (c) Output characteristics. V out = (1 + R f / R 1 ) V in

OpAmp - Example a b c d Your choice: Nodal analysis and/or superposition v out = -R b /R c v 1 + R d (R a +R b ) / (R a (R c +R d )) v 2 Difference amplifier