**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

You may have wondered where we got
that strange equation for the determination of “Millman Voltage”
across parallel branches of a circuit where each branch contains a
series resistance and voltage source:

Parts of this equation seem familiar
to equations we've seen before. For instance, the denominator of the
large fraction looks conspicuously like the denominator of our
parallel resistance equation. And, of course, the E/R terms in the
numerator of the large fraction should give figures for current,
Ohm's Law being what it is (I=E/R).

Now that we've covered Thevenin and
Norton source equivalencies, we have the tools necessary to
understand Millman's equation. What Millman's equation is actually
doing is treating each branch (with its series voltage source and
resistance) as a Thevenin equivalent circuit and then converting
each one into equivalent Norton circuits.

Thus, in the circuit above, battery B_{1}
and resistor R_{1} are seen as a Thevenin source to be
converted into a Norton source of 7 amps (28 volts / 4 Ω) in
parallel with a 4 Ω resistor. The rightmost branch will be converted
into a 7 amp current source (7 volts / 1 Ω) and 1 Ω resistor in
parallel. The center branch, containing no voltage source at all,
will be converted into a Norton source of 0 amps in parallel with a
2 Ω resistor:

Since current sources directly add
their respective currents in parallel, the total circuit current
will be 7 + 0 + 7, or 14 amps. This addition of Norton source
currents is what's being represented in the numerator of the Millman
equation:

All the Norton resistances are in
parallel with each other as well in the equivalent circuit, so they
diminish to create a total resistance. This diminishing of source
resistances is what's being represented in the denominator of the
Millman's equation:

In this case, the resistance total
will be equal to 571.43 milliohms (571.43 mΩ). We can re-draw our
equivalent circuit now as one with a single Norton current source
and Norton resistance:

Ohm's Law can tell us the voltage
across these two components now (E=IR):

Let's summarize what we know about the
circuit thus far. We know that the total current in this circuit is
given by the sum of all the branch voltages divided by their
respective currents. We also know that the total resistance is found
by taking the reciprocal of all the branch resistance reciprocals.
Furthermore, we should be well aware of the fact that total voltage
across all the branches can be found by multiplying total current by
total resistance (E=IR). All we need to do is put together the two
equations we had earlier for total circuit current and total
resistance, multiplying them to find total voltage:

The Millman's equation is nothing more
than a Thevenin-to-Norton conversion matched together with the
parallel resistance formula to find total voltage across all the
branches of the circuit. So, hopefully some of the mystery is gone
now!