2. Nodal Analysis
• The variables selected in the circuit are node
voltages
• Node voltages are defined w.r.t to a common
point in the circuit
• All voltages are defined w.r.t a reference node• All voltages are defined w.r.t a reference node
• Quite often largest number of branches are
connected to this node and is called ground
– it is said to be at ground-zero potential
• In most of the cases, the selected variables
(voltages) will be positive w.r.t to the reference
node
4. • In the figure all voltages are defined w.r.t to the bottom
reference (ground) node
• Reference node is shown by the ground symbol
• Vs, Va, Vb, Vc are measured w.r.t bottom node
• The voltage across an element between two nodes is equal to
the difference between the voltages at those two nodes
– e.g; V3 = Va – Vb = 3
– e.g; V3 = Va – Vb =
2
5. Or we can also use KVL →
The voltage at reference
(ground) node is zero
6. I = ∆V/R
• Thus, as a general rule if we know the node
voltages in a circuit, we can calculate the current
throught any resistive element using Ohm’s law
as:
Nm vv
i
NR
i
7. • In nodal analysis, KCL is employed such that
– Variables in the equations are the unknown
voltages
• In an N-node circuit
– One node is selected as reference node
– Voltages at non-reference nodes are measured
w.r.t this reference node.
– N-1 linearly independent KCL equations are
required to determine the N-1 unknown node
voltages.
– It is always important to specify a reference
8. • Voltage of 12 V at node-A means nothing unless we specify 12 V w.r.t
some reference node
• Voltage at node-1 w.r.t reference node-3 is V1 = 4V
• Voltage at node-2 w.r.t reference node-3 is V2 = -2V
– Voltage at node-1 w.r.t node-2 = ?
– Voltage at node-2 w.r.t node-1 = ?
• Current flows from node of higher potential to node of lower potential
Top to bottom current flow Bottom to top current flow
Left to right current flow
9. Circuits Containing Only independent
Current Sources
• The network shown below contains 3 nodes
– Hence we need N – 1 = 3 – 1 = 2 linearly
independent KCL equations to determine N – 1 = 2
unknown node voltagesunknown node voltages
• Selecting the bottom node as reference node
the voltages at non-reference nodes are
labeled as v1 and v2
10. • Assuming the currents flowing away from the
non-reference nodes to be positive
– Applying KCL at node 1
• Ohm’s law → i = v/R = Gv
• And reference node is at zero potential
– Applying KCL at node 2
11. • Hence, the two equations for the unknown node
voltages are,
• The result is two simultaneous equations in
unknowns v1 and v2
• Any circuit analysis technique can be used for
solving these equationssolving these equations
– Gaussian Elimination
– Matrices
– MATLAB
• Note that nodal analysis employs KCL in
conjunction with Ohm’s law
12. Example 3.1, pg: 105
• Find all node voltages if
– IA = 1 mA, IB = 4 mA, R1 = 12 kΩ, R2 = 6 kΩ, R3 = 6 kΩ
13. Solving in MATLAB
• The results of manual calculations can often be
checked through MATLAB, e.g; for a given case the
following three commands can be used
• R = [2 3, 4 6]; //declares resistance matrix
• I = [ 0.001; 0.005]; //declares current matrix
• G = inv (R); //calculates the inverse of R matrix
• V = G*I or V = inv(R)*I //calculates the node voltages
15. • The analyses produces 3 simultaneous
equationsequations
– The equations can be written in matrix form as:
16. Another approach is to assume all
currents to be flowing away from the
node unless the direction is already
specified by a current source. Again
we consider node-2 in the same
figure shown below. After applying
KCL at node-2, we shall arrive at the
same equation as the one to the left.
Two approaches can be followed for
current directions at a node. One is shown
below, where once a direction is assumed
for a current, we follow that same current
direction for every other node. As an
example we consider node-2 in the
following figure.
Identical
Equations
17. Example 3.2, pg: 109
• Find the node voltages in the following network
if
– R1 = R2 = 2 kΩ, R3 = R4 = 4 kΩ, R5 = 1 kΩ, iA = 4 mA
and iB = 2 mAand iB = 2 mA
18. Circuits Containing Dependent Current
Sources
• The presence of a dependent source may
destroy the symmetry of the nodal equations
• What are the nodal voltages in the network
shown below ifshown below if
19. Example 3.4, pg: 112
• Find the node voltages if
– R1 = 1 kΩ, R2 = R3 = 2 kΩ, R4 = 4 kΩ, iA = 2 mA,
iB = 4 mA and α = 2
20. Circuits Containing Independent
Voltage Sources
• In case of independent voltage
sources, the equations will be
symmetric
• In some cases, through careful
observation, one or more nodal
voltages might already be givenvoltages might already be given
– This will reduce the number of nodal
equations required for finding the
unknown voltages
– In the figure shown to the right, node
voltages V1 and V3 are already known
as both have a direct connection to
the reference ground node
• Thus only one equation is required to
solve for the unknown node voltage V2
21. Example 3.5, pg: 113
• Find V1, V2 and V3 in the following network
using nodal analyses
