Name & Surname:
ID:
University of Bahçeşehir
Engineering Faculty
Date:
Electrical and Electronics Engineering Dept.
EEE 2101 Circuit Theory I - Laboratory 2
Wye-Delta Transformations, Wheatstone
Bridge, Nodal Analysis
Topics:
Delta to Wye Conversion
Wye to Delta Conversion
Resistance Measurement with using the Wheatstone Bridge
Nodal Analysis
Required Equipment and Components:
DMM (Digital Multi Meter)
Resistors
Breadboard
Potentiometer
DC power supply
Information:
Wye-Delta Transformations
Situations often arise in circuit analysis when the resistor are neither in parallel nor in series.
For example, consider the bridge circuit in fig 2.1. How do we combine resistors R1 through
R6 when the resistors are neither in series nor in parallel? Many circuits of the type shown in
figure 2.1 can be simplified by using three terminal equivalent networks.
Figure 2.1
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These are the wye (Y ) or tee (Τ) net shown in figure 2.2 and the delta (∆ ) or (Π ) pi network
shown in figure 2.3. These networks occur by themselves or as part of larger network.
Figure 2.2.b (T)
Figure 2.2.a (Y)
Figure 2.3.b
Figure 2.3.a
(Π )
(∆ )
Delta to Wye Conversion
Suppose it is more convenient to work with a wye network in a place where the circuit
contains a delta configuration. We superimpose a wye network on the existing delta network
and find the equivalent resistances in the wye network. To obtain the equivalent resistances
in the wye network, we compare the two networks and make sure that the resistance between
each pair of nodes in the ∆ network is the same as the resistance between the same pair of
nodes in the Y network. The following equations can be used for the transformation;
Figure 2.4
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R1 =
Rb Rc
Ra + Rb + Rc
(2.1)
R2 =
Rc R a
Ra + Rb + Rc
(2.2)
R3 =
Rb Ra
Ra + Rb + Rc
(2.3)
Wye to Delta Conversion
The following equations can be used for the transformation;
Figure 2.5
Ra =
R1 R2 + R2 R3 + R3 R1
R1
(2.4)
Rb =
R1 R2 + R2 R3 + R3 R1
R2
(2.5)
Rc =
R1 R2 + R2 R3 + R3 R1
R3
(2.6)
Resistance Measurement With Wheatstone Bridge
Although the ohmmeter method provides the simplest way to measure resistance, more
accurate measurement may be obtained using the Wheatstone bridge. Here we will use
Wheatstone bridge to measure an unknown resistance. The unknown resistance R x is
connected to the bridge as shown in figure 2.6.
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Figure 2.6 (The Wheatstone bridge;
R x is the
unknown resistance)
The variable resistance is adjusted until no current flows through the galvanometer (or
multimeter). Under this condition v1 = v 2 , and the bridge is said to be balanced. Since no
current flows through the galvanometer, R1 and R2 behave as though they were in series; so
do R3 and R x . The fact that no current flows through the galvanometer also implies that
v1 = v 2 . Applying the voltage division principle,
v1 =
Rx
R2
v = v2 =
v
R1 + R2
R x + R3
(2.7)
Hence, no current flows through the galvanometer (or multimeter) when;
Rx
R
R2
=
⇒ R2 R3 = R1 R x ⇒ R x = 3 R2
R1 + R2 R x + R3
R1
(2.8)
Nodal Analysis
Nodal analysis provides a general procedure for analyzing circuits using node voltages as the
circuit variables. Choosing node voltages instead of element voltages as circuit variables is
convenient and reduces the number of equations one must solve simultaneously.
In nodal Analysis, we are interested in finding the node voltages. Given a circuit with n
nodes without voltage sources, the nodal analysis of the circuit involves taking the following
three steps;
• Select a node as reference node. Assign voltages v1 , v 2 , v3 ....., v n −1 to the remaining n1 nodes. The voltages are referenced with respect to the reference node.
• Apply KCL to each of the n-1 non reference nodes. Use Ohm’s Law to express the
branch currents in terms of node voltages.
• Solve the resulting simultaneous equations to obtain the unknown node voltages.
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If the circuit has voltage sources, there are two possibilities;
• If a voltage source is connected between the reference node and a nonreference node,
we simply set the voltage at the nonreference node equal to the voltage of the voltage
source. In fig 2.7, for example;
v1 = 10V
(2.9)
Figure 2.7 (A circuit with supernode)
• If the voltage source (dependent or independent) is connected between two
nonreference nodes, the two nonreferenced nodes form a supernode, we apply both
KVL and KCL to determine node voltages. In Figure 2.7, nodes 2 and 3 form a
supernode.
We analyze a circuit with supernodes using the same three steps mentioned in the
previous section except that the supernodes treated differently. We apply the KCL rule to
the supernode only instead of applying it to two nodes separately. Hence, at the
supernode in figure 2.7;
i1 + i 4 = i 2 + i3
(2.10)
or
v1 − v 2 v1 − v3 v 2 − 0 v3 − 0
+
=
+
2
4
8
6
(2.11)
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Preliminary Work:
1) Convert the ∆ network (Fig 2.8) to an equivalent Υ network.
Figure 2.8
2) Calculate the node voltages in the circuit shown in figure 2.9.
Figure 2.9
Procedure:
1
1) Construct the circuit in figure 2.10 and measure the resistance between a and b ( Rab
).
Write the result on table 2.1 and then transform the 1.8K, 2.2K, 1.8K part. Use delta
to wye conversion, construct the new network with given resistances and measure the
2
resistance again ( Rab
). Then write the new value on table 2.1. Also draw the new
network on the empty part, Figure 2.11.
1
Rab
2
Rab
Table 2.1
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Figure 2.10
……………………………………………………..
Figure 2.11
1
2) Construct the circuit in figure 2.12 and measure the resistance between a and b ( Rab
).
Write the result on table 2.2 and then transform the 0.33K, 0.33K, 0.47K part. Use
wye to delta conversion, construct the new network with given resistances and
2
measure the resistance again ( Rab
). Then write the new value on table 2.2. Also draw
the new network on the empty part, Figure 2.13.
1
Rab
2
Rab
Table 2.2
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Figure 2.12
…………………………….
Figure 2.13
3) Construct the network in figure 2.14. Vary the potentiometer till the current value on
multimeter becomes zero. When the current in multimeter is zero you can use
equation (2.8) to find the value of unknown resistor. To do that firstly measure the
values of R1 and R2 , and then measure the value of potentiometer when current in
multimeter is zero. Use (2.8) to find the unknown resistor ( Runknown ) and write it to
table 2.3.
Runknown
Table 2.3
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Figure 2.14
4) Calculate the node voltages ( Vx and V y ) and the currents on ( R1 , R2 , R3 , R4 , R5 ) in the
network in figure 2.15. And then construct the network in figure 2.15, measure the
node voltages ( V x and V y ) and the currents on ( R1 , R2 , R3 , R4 , R5 ). Then write them
on table 2.4.
Figure 2.15
Calculated values
Measured values
Vx
Vy
I1
I2
I3
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I4
I5
Table 2.4
After Lab:
1) Construct circuit in Figure 2.15 in Multisim.
2) E-mail it to gorkem.serbes@bahcesehir.edu.tr.
EEE 2101 Circuit Theory I Page 10 of 10