Millman's Theorem.

Fundamental Series FS06 - June, 2006

Millman's Theorem

Millman's Theorem states: The node voltage of a network of parallel branches, each consisting of a voltage source and a conductance in series (Fig 1), can be calculated by the formula:

V0 = (G1V1+ G2V2 + ... + GNVN)
      -------------------------------------
        (G1 + G2 + ... + GN)

Notes:

Biography and Background

Jacob Millman (1911 - 1991)

Millman's Theorem is named after Jacob Millman. Dr. Millman was born in Russia in 1911 and came to the US in 1913. Dr. Millman received a Ph.D. from MIT in 1935.

From 1936 to 1952, except for three years during World War II, when he was a scientist with the Radiation Laboratory at M.I.T, Dr. Millman was a professor of engineering at City College of New York

He taught at Columbia University From 1952 until he retired in 1976. He was named chairman of the department of electrical engineering in 1965.

At his retirement, Dr. Millman became the Charles Batchelor Professor Emeritus of Electrical Engineering. Dr. Millman, an expert on radar, electronic circuits and pulse-circuit techniques was perhaps best known as an author between 1941 and 1987 of several textbooks on electronics.

Today, in recognition of Dr. Millman's contributions to Electrical Engineering text book writing, the IEEE Education Society offers the McGraw-Hill/Jacob Millman Award which "recognizes an author of an exceptional textbook relating to the field of Electrical Engineering, as evaluated on excellence in writing, impact of the textbook on electrical engineering education, and innovative pedagogical features of the textbook."

Jacob Millman (1911 - )

Millman's Theorem

In certain cases, the topology of a circuit lends itself to easier analysis if the circuit is analyzed in terms of Conductances instead of Resistances.

By definition conductance G, is the reciprocal of resistance:

G = 1/R

Conductance is usually measured in units called mhos, i.e. the reciprocal of ohms. Ohm's Law can be restated in terms of conductance as follows:

I = GV         eq.1

Conductance is particularly useful when analyzing parallel elements such as resistors. For example, the total resistance of a string of parallel resistors is calculated as follows:

1/RT = 1/R1 + 1/R2 + 1/R3 + ... + 1/Rn

But 1/R by definition is Conductance, so using Conductance, the total Conductance of same circuit is

CT = C1 + C2 + C3+ ... Cn

A much simpler problem to solve.

If an entire circuit can be redrawn as a network of parallel branches of voltages and conductances, Millman's theorem can be used to simplify analysis.

Millman's Theorem states: The node voltage of a network of parallel branches, each branch consisting of a voltage source and a conductance in series (Fig 1) can be calculated by the formula:

V0 =   (G1V1 + G2V2 + ... + GNVN)
        -------------------------------------------         eq.2
            (G1 + G2 + ... + GN)

To illustrate the use of Millman's Theorem, let's take a look at Figure 2, from the article on Thevenin's Theorem.

In that article, we needed to calculate the Thevenin voltage of the circuit looking back into the Thevenin equivalent circuit. Figure 3 shows the Thevenin equivalent circuit re-drawn.

Taking Figure 3 one step more, we redraw the circuit so each branch looks the same - we show a single Conductance and a voltage source (even if it is 0) for each branch (Fig. 4).

From Kirchoff's Law, we know that the Algebraic Sum of all the currents entering into the top node is:

i1 + i2 + i3 + i4 + i5 = 0         eq.3

Looking at branch 1 of fig.4, if the voltage drop across G1 is VG1, then

VTH = V1 - VG1

and

VG1 = V1 - VTH

Multiplying through by the conductance of R1 which is G1, we get

G1VG1 = G1V1 - G1VTH         eq.4

From eq.1 we can write

i1 = G1V1 - G1VTH         eq.5

Similarly, for all branch circuits, we write:

i2 = G2V2 - G2VTH         eq.6
i3 = G3V3 - G3VTH         eq.7
i4 = G4V4 - G4VTH         eq.8
i5 = G5V5 - G5VTH         eq.9

Substituting eqs.4 - 8 into eq.3:

(G1V1 + ... + G5V5) - (G1 + ... + G5)VTH = 0

Solving for VTH we get

VTH =   (G1V1 + ... + G5V5)
        ----------------------------         eq.10
            (G1 + ... + G5)

Referring to Figure 3, plugging in the values, we get

VTH =   (0.046x15 + 0.125x0 + 0.067x24 + 0.125x0 + 0.111x12)
        --------------------------------------------------------------------
            (0.046 + 0.125 + 0.067 + 0.125 + 0 .111)

= 7.658V

Which, as you may recall is lot easier than solving for 4 mesh currents as we did before.

This relationship can be generalized for any circuit of N parallel branches as shown in Fig. 1. The node Voltage V0 can be calculated by the equation:

V0 =   (G1V1 + G2V2 + ... + GNVN)
        -------------------------------------
            (G1 + G2 + ... + GN)

Millman's Theorem and Current Sources.

We organized each of our parallel branches of the Millman equivalent circuit as a voltage source and a series resistance. But what if we are working with current sources, does the theorem still apply? The answer is yes.

By now you should recognize that each of the branches is a Thevenin equivalent. If you want to use current sources instead of voltage sources, convert each branch to its Norton equivalent. The Norton current Ib of any branch just happens to be VbGb, the Norton conductance is the same as the Thevenin conductance. Our Norton equivalent circuit now looks like:

Now apply Norton's Theorem to the complete new circuit and we find that:

Ieq = i1 + i3 + i5

and

Geq = G1 + G2 + G3 + G4 + G5

and

V0 x Geq = Ieq

so, substituting back the sub terms, we get

V0 =   (I1 + I2+ ... + I5)
        -------------------------------------
         (G1 + G2 + ... + G5)

Referring to Figure 5, plugging in the values, we get

VTH =   (0.69 + 0 + 1.61 + 0 + 1.33)
        --------------------------------------------------------------------
            (0.046 + 0.125 + 0.067 + 0.125 + 0.111)

= 7.658V

In general, for any circuit of N parallel branches, the Millman equation can be stated as

V0 =   (I1 + I2+ ... + In)
        -------------------------------------
         (G1 + G2 + ... + Gn)

Or, the output voltage is the sum of the currents of parallel branches divided by the sum of the conductances of the branches - Which is really cool!

In conclusion

Millman's Theorem is another tool we can used to analyze complex circuits IF we can reconstruct the circuit into a circuit of parallel branches consisting of a voltage source and a series conductance -or- a current source and a parallel conductance.

Millman's Theorem turns out to be especially useful in transistor modeling and amplifier design.