3. R1
R2
R3
c a
b
c a
b
Rc
Ra
Rb
Delta or ∆ Wye or Y
R
R R
R R R
R
R R
R R R
R
R R
R R R
a
b
c
=
+ +
=
+ +
=
+ +
1 2
1 2 3
2 3
1 2 3
1 3
1 2 3
Wye-Delta Transformation
R
R R R R R R
R
R
R R R R R R
R
R
R R R R R R
R
a b b c a c
b
a b b c a c
c
a b b c a c
a
1
2
3
=
+ +
=
+ +
=
+ +
Y to Δ Equations Δ to Y
4. Kirchhoff’s Voltage Law, KVL
3 V
R1
2.2 k
R2
3.7 k
R4
5.4 k
R3
1.0 k
1 2 3
0
+ VR1
- - VR2
+
-
VR4
+
-
VR3
+
Loop (0, 1, 2, 0)
V0,1 + V1,2 + V2,0 = 0 V
-3 V + VR1 - VR3 = 0 V
Loop (0, 3, 2, 0)
V0,3 + V3,2 + V2,0 = 0 V
VR4 + VR2 - VR3 = 0 V
For example:
V1,2 = - V2,1 = VR1
5. Nodal Analysis 1
entering
leaving
A current is entering the node if the current is leaving the device.
A current is leaving the node if the current is entering the device.
6. Nodal Analysis 2
Apply Ohm’s Law (I = V/R) for R1:
current entering node 2 = (V1 - V2) / R1
current leaving node 2 = (V2 - V1) / R1
R1
1.5 k1 2
7. Kirchhoff’s Current Law, KCL
KCL at node 2:
Σ ΣI I
I I I
V V V V V V
entering leaving
R R R
=
+ =
−
+
−
=
−
1 3 2
1 2 0 2 2 3
2 2 1 3. k .0 k .7 k
Shade indicates a node
At node 1, V1=3 v. due to source
At node 0, V0=0 v. due to ground
KCL at node 3:
V V V V2 3 3 0
3 7
−
=
−
. k 5.4 k
3 V
R1
2.2 k
R2
3.7 k
R4
5.4 k
R3
1.0 k
1 2 3
0
IR1
IR2
IR3
IR2
IR4
8. Nodal Analysis
Shade indicates a node
3 V
R1
2.2 k
R2
3.7 k
R4
5.4 k
R3
1.0 k
1 2 3
0
IR1
IR2
IR3
IR2
IR4
Substitute the known voltages into the equations that were obtained at nodes 2 & 3
Solve the two equations for V2 and V3
You can put the two equations in matrix form and solve using Matlab or any
mathematics program (or your calculator).
