Transcript nodal analysis

Basic Electrical
Engineering
Lecture # 10
Course Instructor:
Engr. Sana Ziafat
Agenda
• Planner vs Non planner circuits
• Node analysis
Branches and Nodes
Branch:
elements connected end-to-end,
nothing coming off in between (in series)
Node:
place where elements are joined—includes entire wire
Nodal Analysis
 Kirchhoff’s current law is used to develop the
method referred to as nodal analysis
 A node is defined as a junction of two or more
branches
 Application of nodal analysis
1. Determine the number of nodes within the
network.
2. Pick a reference node, and label each remaining
node with a subscript value of voltage: V1, V2, and so
on.
Nodal Analysis
3. Apply Kirchhoff’s current law at each node
except the reference. Assume that all
unknown currents leave the node for each
application of Kirchhoff’s current law. In
other words, for each node, don’t be
influenced by the direction that an unknown
current for another node may have had.
Each node is to be treated as a separate
entity, independent of the application of
Kirchhoff’s current law to the other nodes.
4. Solve the resulting equation for the nodal
voltages.
Nodal Analysis
 On occasion there will be independent voltage sources in
the network to which nodal analysis is to be applied. If so,
convert the voltage source to a current source (if a series
resistor is present) and proceed as before or use the
supernode approach:
1. Assign a nodal voltage to each independent node of
the network.
2. Mentally replace independent voltage sources with
short-circuits.
3. Apply KCL to the defined nodes of the network.
4. Relate the defined nodes to the independent voltage
source of the network, and solve for the nodal
voltages.
Nodal Analysis
1.
Choose a reference node and assign a
subscripted voltage label to the (N – 1) remaining
nodes of the network.
2. The number of equations required for a
complete solution is equal to the number of
subscripted voltages (N – 1). Column 1 of each
equation is formed by summing the
conductances tied to the node of interest and
multiplying the result by that subscripted nodal
voltage.
Nodal Analysis
(Format Approach)
3.
We must now consider the mutual terms that
are always subtracted form the first column. It
is possible to have more than one mutual term if
the nodal voltage of current interest has an
element in common with more than one nodal
voltage. Each mutual term is the product of the
mutual conductance and the other nodal voltage
tied to that conductance.
Nodal Analysis
(Format Approach)
4.
The column to the right of the equality sign is
the algebraic sum of the current sources tied to
the node of interest. A current source is assigned
a positive sign if it supplies current to a node and
a negative sign if it draws current from the node.
5. Solve the resulting simultaneous equations for
the desired voltages.
Node Voltages
The voltage drop from node X to a reference node (ground)
is called the node voltage Vx.
Example:
a
+
Va
_
b
+
+
_
Vb
_
ground
Nodal Analysis Method
1. Choose a reference node ( ground, node 0)
(look for the one with the most connections,
or at the bottom of the circuit diagram)
2. Define unknown node voltages (those not connected to ground by voltage
sources).
3. Write KCL equation at each unknown node.
▫ How? Each current involved in the KCL equation will either come from
a current source (giving you the current value) or through a device like
a resistor.
▫ If the current comes through a device, relate the current to the node
voltages using I-V relationship (like Ohm’s law).
4. Solve the set of equations (N linear KCL equations for N unknown node voltages).
Example
node voltage set
R1

+
-
V1
Va
R
3
R2
What if we
used different
ref node?
Vb
R4
IS
 reference node
•
•
•
Choose a reference node.
Define the node voltages (except reference node and the one set by the
voltage source).
Apply KCL at the nodes with unknown voltage.
Va  V1 Va Va  Vb


0
R1
R2
R3
•
Vb  Va Vb

 IS
R3
R4
Solve for Va and Vb in terms of circuit parameters.
Example
R1
Va
R3
V
1
R2
I1
R4
R5
V2
Va   V1 Va Va  V2


 I1
R1
R4
R5
Q&A