**Hey, you're in the
**__
power s____upply
zone__*--*YOUR HANDYMAN ZONE!

**
Power Supply 1-2-3: Direct Current (DC)**

# Chapter 10

# DC
NETWORK ANALYSIS

In many circuit applications, we
encounter components connected together in one of two ways to form a
three-terminal network: the “Delta,” or Δ (also known as the “Pi,”
or π) configuration, and the “Y” (also known as the “T”)
configuration.

It is possible to calculate the proper
values of resistors necessary to form one kind of network (Δ or Y)
that behaves identically to the other kind, as analyzed from the
terminal connections alone. That is, if we had two separate resistor
networks, one Δ and one Y, each with its resistors hidden from view,
with nothing but the three terminals (A, B, and C) exposed for
testing, the resistors could be sized for the two networks so that
there would be no way to electrically determine one network apart
from the other. In other words, equivalent Δ and Y networks behave
identically.

There are several equations used to
convert one network to the other:

Δ and Y networks are seen frequently
in 3-phase AC power systems (a topic covered in volume II of this
book series), but even then they're usually balanced networks (all
resistors equal in value) and conversion from one to the other need
not involve such complex calculations. When would the average
technician ever need to use these equations?

A prime application for Δ-Y conversion
is in the solution of unbalanced bridge circuits, such as the one
below:

Solution of this circuit with Branch
Current or Mesh Current analysis is fairly involved, and neither the
Millman nor Superposition Theorems are of any help, since there's
only one source of power. We could use Thevenin's or Norton's
Theorem, treating R_{3} as our load, but what fun would that
be?

If we were to treat resistors R_{1},
R_{2}, and R_{3} as being connected in a Δ
configuration (R_{ab}, R_{ac}, and R_{bc},
respectively) and generate an equivalent Y network to replace them,
we could turn this bridge circuit into a (simpler) series/parallel
combination circuit:

After the Δ-Y conversion . . .

If we perform our calculations
correctly, the voltages between points A, B, and C will be the same
in the converted circuit as in the original circuit, and we can
transfer those values back to the original bridge configuration.

Resistors R_{4} and R_{5},
of course, remain the same at 18 Ω and 12 Ω, respectively. Analyzing
the circuit now as a series/parallel combination, we arrive at the
following figures:

We must use the voltage drops figures
from the table above to determine the voltages between points A, B,
and C, seeing how the add up (or subtract, as is the case with
voltage between points B and C):

Now that we know these voltages, we
can transfer them to the same points A, B, and C in the original
bridge circuit:

Voltage drops across R_{4} and
R_{5}, of course, are exactly the same as they were in the
converted circuit.

At this point, we could take these
voltages and determine resistor currents through the repeated use of
Ohm's Law (I=E/R):

A quick simulation with SPICE will
serve to verify our work:[spi]

unbalanced bridge circuit

v1 1 0

r1 1 2 12

r2 1 3 18

r3 2 3 6

r4 2 0 18

r5 3 0 12

.dc v1 10 10 1

.print dc v(1,2) v(1,3) v(2,3) v(2,0) v(3,0)

.end

v1 v(1,2) v(1,3) v(2,3) v(2) v(3)

1.000E+01 4.706E+00 5.294E+00 5.882E-01 5.294E+00 4.706E+00

The voltage figures, as read from left
to right, represent voltage drops across the five respective
resistors, R_{1} through R_{5}. I could have shown
currents as well, but since that would have required insertion of
“dummy” voltage sources in the SPICE netlist, and since we're
primarily interested in validating the Δ-Y conversion equations and
not Ohm's Law, this will suffice.

**REVIEW:**
- “Delta” (Δ) networks are also known as
“Pi” (π) networks.
- “Y” networks are also known as “T”
networks.
- Δ and Y networks can be converted to their
equivalent counterparts with the proper resistance equations. By
“equivalent,” I mean that the two networks will be electrically
identical as measured from the three terminals (A, B, and C).
- A bridge circuit can be simplified to a
series/parallel circuit by converting half of it from a Δ to a Y
network. After voltage drops between the original three
connection points (A, B, and C) have been solved for, those
voltages can be transferred back to the original bridge circuit,
across those same equivalent points.