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 - ExampleKCL requires that all currents flowing into the region must sum to zero, or we would pile up or run out of electrons.

4

3 38 3121 vvvv

At node 1: (KCL)

154

3 253 231312 vvvvvv

At the “supernode:” (KCL)

23 22 vv

At the “supernode:” (KVL)

Nodal analysis - Example

Independent voltage source (supernode containing reference)

v1 = -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 - ExampleCreating a “supermesh” from meshes 1 and 3:

-7 + 1 ( i1 - i2 ) + 3 ( i3 - i2 ) + 1 i3 = 0 [1]

Around mesh 2:

1 ( i2 - i1 ) + 2 i2 + 3 ( i2 - i3 ) = 0 [2]

Rearranging,

i1 - 4 i2 + 4 i3 = 7 [1]

-i1 + 6 i2 - 3 i3 = 0 [2]

i1 - i3 = 7 [3]

Solving,

i1 = 9 A, i2 = 2.5 A, and i3 = 2 A.

Finally, we relate the currents in meshes 1 and 3:

i1 - i3 = 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 RL.

A general practical current source connectedto a load resistor RL.

• Convert between the two - Sources are related by:• RS = Rp, and• Vs = Rs Is = Rp Is

• Useful when asked for:• Maximum terminal voltage (vs) and/or current (is)• (Maximum) power transferred (PL = vL iL when RL = Rs)

Thevenin and Norton

• "Dead" network to find equivalent source resistance RTH and RN

• Open loop voltage to determine VTH (any method)

• Short circuit current determines IN

Thenenin - Example

• Open loop voltage to determine VTH and short circuit current determines IN

• Find equivalent source resistance RTH and RN

• use "Dead" network

• use RTH = RN = VTH / IN (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.

Vout = - (Rf / R1) Vin

Output characteristics.

• 1st step: Determine voltage at input terminals

• 2nd step: Determine current i

• 3rd step: Find output voltage vout

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. Vout = (1 + Rf / R1) Vin

OpAmp - Example

a

b

c

d

• Your choice: Nodal analysis and/or superposition

• vout = -Rb/Rc v1 + Rd(Ra+Rb) / (Ra(Rc+Rd)) v2

Difference amplifier