What Have We Learned In This Lecture?
Circuit theory axioms: KVL, KCL, lumped circuit axiom
Modelling circuits using directed graphs:
Half of the required linearly independent equations come from KVL, KCL.
Another half come from element equations:
2-terminal elements: resistors, capacitors, inductors, memristors
Obtaining element equations for 3-terminal or 2-port elements
3-terminal or 2-port elements: OPAMP, transistor
DC-operating points
Linearization
Small-signal analysis
What Is Next?
Methods for solving circuits:
Node analysis
Mesh analysis
Additivity and multiplicativity
Norton and Thevenin Theorems
Dynamic circuits:
Obtaining and solving state equations of dynamic circuits
Solving Circuit Equations
We are going to apply some methods to solve linear time-invariant
resistive circuits!
What does it mean to solve? ............................
Which equations can we use? ................................
Why not just solve them all? ................................
Are the methods applicable to more general circuits? ............................
Then why linear resistive circuits? ..............................
General resistive circuit: A circuit that contains only resistors,
linear or nonlinear, time-varying or time-invariant, active or passive,
such as .................................................................................................................
.................................................................................................................................
.................................................................................................................................
•  A resistive circuit is said to be time-invariant if all elements of
the circuit are time-invariant.
•  A resistive circuit is said to be linear if all elements of the circuit
except the independent sources are linear n-ports.
Node Analysis
A method for solving time-invariant, linear circuits.
A method for finding node voltages.
Is this enough? ..............
v = AT e element
node voltages
voltages
i = G ⋅ v + is
What is the requirement? ...................
Can we generalize? ..............
element
currents
Method:
Step 1: Write KCL equations for nd − 1 nodes: A ⋅i = 0
Step 2: Substitute voltage-controlled element equations: A ⋅ [G ⋅ v + is ] = 0
Step 3: Write element voltages in terms of node voltages (KVL):
A ⋅ [G ⋅ AT ⋅ e + is ] = 0
T
Step 4: Solve the node voltages: A ⋅ G ⋅ A ⋅ e = − A ⋅ is
(
)
T −1
e = − A⋅G ⋅ A
⋅ A ⋅ is
An example
Find the node voltages of the following circuit:
Generalized Node Analysis
Can we generalize? YES
Group the elements into two classes:
1. class: voltage-controlled elements
2. class: not voltage-controlled elements
Method:
Step 1: Write KCL equations for nd − 1 nodes: Ai = 0
⎡ i1 ⎤
[ A1 A2 ]⎢ ⎥ = 0
⎣i2 ⎦
Step 2: Substitute voltage controlled element equations for
the elements in the 1. class and for the 2. class write EE’s:
[ A1G1
⎡v1 ⎤
A2 ]⎢ ⎥ = − A1is
⎣ i2 ⎦
⎡v2 ⎤
[ M N ]⎢ ⎥ = w
⎣ i2 ⎦
Step 3: Write element voltages in terms of node voltages (KVL):
v1 = A1T e
v2 = A2T e
⎡ A1G1 A1T
⎢
T
MA
2
⎣
A2 ⎤ ⎡ e ⎤ ⎡− A1is ⎤
⎥ ⎢ ⎥ = ⎢
⎥
N ⎦ ⎣i2 ⎦ ⎣ w ⎦
Step 4: Solve the node voltages and the currents of the 2. class elements.
An example
Solve the circuit
using the
generalized node
analysis method!
Mesh Analysis
A method for solving time-invariant, linear, planar circuits.
Consider the mesh currents?
A method for finding mesh currents.
T
Is this enough? ..............
i = Bm im
mesh currents
element
currents
v = R ⋅ i + vs
What is the requirement? ...................
Can we generalize? ..............
element
voltages
Method:
Step 1: Write KVL equations for ne − nd + 1 meshes: Bm ⋅ v = 0
Step 2: Substitute current-controlled element equations:Bm ⋅ [ R ⋅ i + vs ] = 0
Step 3: Write element currents in terms of mesh currents (KCL):
T
Bm ⋅ [ R ⋅ Bm ⋅ im + vs ] = 0
Step 4: Solve the mesh currents:
T
Bm ⋅ R ⋅ Bm ⋅ im = − Bm ⋅ vk
(
im = − Bm ⋅ R ⋅ Bm
)
T −1
Bm ⋅ vk
An example
Find the mesh
currents of the
circuit shown in
the figure!
Generalized Mesh Analysis
Can we generalize? YES
Group the elements into two classes:
1. class: current-controlled elements
2. class: not current-controlled elements
Method:
Step 1: Write KVL equations for ne − nd + 1 meshes: Bm ⋅ v = 0
⎡ v1 ⎤
[ B1 B2 ]⎢ ⎥ = 0
⎣v2 ⎦
Step 2: Substitute current-controlled element equations for
the elements in the 1. class and for the 2. class write EE’s:
[ B1R 1
⎡v2 ⎤
[ M N ]⎢ ⎥ = w
⎣ i2 ⎦
⎡ i1 ⎤
B2 ]⎢ ⎥ = − B1vs
⎣v2 ⎦
Step 3: Write element currents in terms of mesh currents:
i1 = B1T im
i2 = B2T im
⎡ B1 R1 B1T
⎢
T
NB
2
⎣
B2 ⎤ ⎡im ⎤ ⎡− B1vs ⎤
⎥ ⎢ ⎥ = ⎢
⎥
M ⎦ ⎣v2 ⎦ ⎣ w ⎦
Step 4: Solve the mesh currents and the voltages of the 2. class elements.
An example
Solve the circuit
using the
generalized mesh
analysis method!