Basic Electrical Engineering Series
Definition
Millmans Theorem
Millman’s Theorem (due to J E Millman, also called the parallel generator theorem) is useful for circuits were the attention is centered on a single node. It also leads naturally into Rosen’s theorem – the general Star – Mesh transform and then onto star-delta and delta-star transforms which are just instances of Rosen’s Theorem.
“ If any number of linear impedance’s meet at a common node point, and the potentials from another point to the free ends of these impedance’s are known, then the potential at the node is the ratio of the sum of the products of potentials and admittance’s, and the sum of admittance’s”…
Therefore consider a set of impedance’s meeting at a node point:
The potential at the node (0') with respect to (0) is given by:
Example
So lets look at single node, two mesh problem, where we have to find I3
1) E1=10, E2=12, R1=120, R2=120, R3=100 Find the currents indicated
Vn = 6.87 volts, therefore
I3 = Vn/100 = 68.7 mA
and hence
I1 = (10 – Vn)/120 = 26 mA
I2 = (12 – Vn)/120 = 42 mA
This is a much simpler solution than solving the mesh equations using Kirchoffs Laws. Millmans is therefore very usefull in many problems
Practical Example
In the above circuit, the zener diode provides a reference input for the operational amplifier which is assumed to have a ‘virtual earth’ at the summing point. By replacing the Zener with an equivalent circuit consisting of a battery with an EMF equal to the zener voltage in series with a resistor equal to the zener slope resistance, calculate the following utilising Millmans theorem:
The output current IL
The zener current Iz
The current from the supply Is
The % stabalisation if the supply increases to 12
volts
Hint : first draw out the equivalent circuit without the details of the op amp – ie just the load presented to the zener circuit – you will find that it’s a simple two mesh circuit like the one above with two batteries – one for the supply and one for the zener.