3. DUALITY
• Several analogous situations will have been noted in the
preceding discussions, while analysing the networks. The
statements of two Kirchhoff’s laws are almost similar
word of word with ‘voltage’ substituted for ‘current’,
‘independent loop’ for ‘independent node pair’, etc. Like
wise the integral-differential equations that resulted from
the applications of two Kirchhoff's laws are similar in
appearance. This repeated similarity is only part of larger
pattern of identical behaviour patterns in the roles played
by voltage and current in network analysis.
4.
5. • L di/dt + Ri + 1/C∫idt = v(t) (1)
• C’ dv’/dt + G’v’ + 1/L’∫v’dt = I’(t) (2)
• The role of current and voltage in the two networks have
been interchanged.
• The solution of one equation is also the solution of the
other.
• This similarity is termed the principle of duality.
6. • An inspection of the terms of equations (1) and (2) shows that the following are
the analogous quantities(with primes omitted)
• Ri and Gv
• L di/dt and C dv/dt
• 1/C∫ idt and 1/L∫ vdt
Evidently, the following pairs are dual quantities.
• R and G
• L and C
• Loop current, I and node voltage v
• Q or Ψ or
• ∫ idt ∫ vdt
• Loop and node-pair
• Short circuit and open circuit
• Switch open and switch closed
• Current source and voltage source
7. The following procedure may be followed in constructing the dual of a
given network:
1. Inside each loop place a node and give it a number for all loops for
convenience. Place an extra node, the datum node, external to the network
say 0 node. Place an extra the same numbered nodes on a separate space
on the paper for construction of the dual.
2. Draw a line between two nodes of the original network traversing only one
elements at a time. Thus draw lines from node to node through the
elements in the original network. For each elements traversed in the
original network, connect the dual elements from the listing just given – on
dual network being constructed.
3. Continue this process until the number of possible paths through single
elements is exhausted. If we go through a connecting leads assumed to be
a short, the dual elements is an open circuit.
4. The network so constructed is a dual network. This construction may be
verified by witting the mesh current equations for the one network and
nodal equations for the other network.
12. Network conventional for
determining source orientation
• In a linear bilateral network containing a single energy
source , the magnitude of the branch voltage or current is
fix and does not depend upon the orientation of the
source.
• However , if the network contains two or more energy
source then the branch voltage or current depends on the
orientation of the source .
• We are already aware that the resistances , inductances ,
and capacitances in the original network appear in dual
network as conductance , capacitances and inductances
respectively with the same numerical values .
13. • The voltage source in the original network are replace by the current
sources in the dual network and vice – versa.
• The most important task is to decide the orientation of these sources in
the dual network it is determine as follows:
• First of all , we assume a clockwise direction for all mesh currents is
original network N1. each mesh current enclosing a dot(node) of the
dual network say N2.
• It is important to note that the orientation of the voltage source is
identified with the direction of the mesh current whereas the
orientation of the current source is identified with the node which is
enclosed by mesh current.
14. • If the direction of the voltage source and clockwise mesh current is
same in the network N1, then orientation of the current source in the
dual network N2 is towards the node which is enclosed by the said
mesh current in N1 .
• Conversely if the directions of the voltage source and the clockwise
mesh current are opposite then the current of the current source in N2
is directed away from the node which is enclosed by the said mesh
current in N1 .
• The principle is valid for the original network N1 as well as the dual
network N2.
