Can someone explain what dependent and independent KCL equations mean to us, and how we can determine what is number of dependent/independent equations? I am reading about KCL and they said:

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I don't understand this part: This in turn shows that the four KCL equations are dependent? What that dependent means to us when we have 4 KCL equations?


3 Answers 3


If you have a network with n nodes, there will be \$n-1\$ independent KCL equations.

That means the KCL equations alone are not sufficient to produce a unique solution to the circuit. One more equation is required to produce a solvable set of equations.

Normally, the one additional equation is produced by designating one of the nodes as the "ground" node, and arbitrarily choosing that its potential will be considered as 0 V.

  • \$\begingroup\$ So mathematically we look at KCL equations as linear equations and applying dependent and independent equations rules to them like here youtube.com/watch?v=WSpF5uvApLA. \$\endgroup\$
    – Junior
    Commented Nov 23, 2015 at 22:43
  • 1
    \$\begingroup\$ Yes. If you have a linear circuit (containing only independent sources, linear resistors, and linear dependent sources, and linear capacitors and inductors in the phasor domain) then the KCL equations will be linear equations and all the rules of linear equations apply to solving them. \$\endgroup\$
    – The Photon
    Commented Nov 23, 2015 at 23:05

This may even become more clear when you start solving systems using matrices. The above system will produce an incidence matrix

and you chose to leave out one (usually the last ) row, which equates to choosing the node in the last row as a reference node(usually 0 V). The above system produces Nn-1 equations, however you need Nl equations in order to solve the system. The remaining equations arise by applying KVL to the circuit which yields Nl-Nn+1 equations. Notice when you add up everything Nn-1+Nl-Nn+1=Nl, you get an independent system of Nl linear equations.


A set of N linear equations are dependent if one equation can be written as the linear combination of the remaining equations.

Any set of values that satisfying the first N-1 equations will also satisfy their linear combination - the Nth equation. So there is no point in adding an extra redundant equation to the set when N-1 equations can do the same job.

Consider the set of equations given in the original post. Equation (2.8) can be represented as:

$$(2.8) = -(2.5)-(2.6)-(2.7)$$

So any set of values of currents satisfying equations (2.5),(2.6) and (2.7) will also satisfy their linear combination - and hence equation (2.8). So adding equation (2.8) is adding redundancy.

Dependency expressed in other words:
A set of N equations are dependent if there exist any linear combination of these N equation that gives a result 0=0:

$$\sum_i c_iE(i)\Rightarrow0=0$$

where \$c_i\ne 0\$. This definition is used in OP to say that the set of equations are dependent.


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