Download EE 221 Review 1

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)
v1  v2 v1  v3
83 

3
4
At the “supernode:” (KCL)
v2  v1 v3  v1 v3 v2
3  25 

 
3
4
5 1
At the “supernode:” (KVL)
22  v3  v2
Nodal analysis - Example
(I)
(II)
(III)
variables
Independent voltage source (supernode containing reference)
v1 = -12
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:
-7 + 1 ( i1 - i2 ) + 3 ( i3 - i2 ) + 1 i3 = 0
[1]
Around mesh 2:
1 ( i2 - i1 ) + 2 i2 + 3 ( i2 - i3 ) = 0
[2]
Finally, we relate the currents in meshes 1 and 3:
i1 - i3 = 7
[3]
Rearranging,
i1 - 4 i2 + 4 i3 = 7
[1]
-i1 + 6 i2 - 3 i3 = 0
[2]
i1
[3]
- i3 = 7
Solving,
i1 = 9 A, i2 = 2.5 A, and i3 = 2 A.
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
connected
to 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
Source
transformation
is used here.
• 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)
Operational Amplifier
(a) Electrical
symbol.
Neglected:
(1) Output voltage saturation.
(b) “Minimum"
op amp.
Ideal:
(1) No input current.
(2) No voltage difference
between input terminals.
(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.
Output characteristics.
• 1st step: Determine voltage at input terminals
• 2nd step: Determine current i
Vout = - (Rf / R1) Vin
• 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
b
Difference amplifier
a
c
d
• Your choice: Nodal analysis and/or superposition
• vout = -Rb/Rc v1 + Rd(Ra+Rb) / (Ra(Rc+Rd)) v2