Kirchhoff's Lawsm0015Kirchhoff's Laws2.92000/07/112009/10/19 14:51:29.459 GMT-5DonH.JohnsonDon Johnsondhj@rice.eduDonH.JohnsonDon Johnsondhj@rice.eduRoyHaRoy Harha@alumni.rice.eduDonH.JohnsonDon Johnsondhj@rice.educircuitKCLKirchhoffKVLScience and TechnologyA brief description of Kirchhoff's Laws (current and voltage)enKirchhoff's Current Law
At every node, the sum of all currents entering a node must equal zero. What
this law means physically is that charge cannot accumulate in a node; what
goes in must come out. In the example, ,
below we have a three-node circuit and thus have three KCL equations.
ii10i1i20ii20
Note that the current entering a node is the negative of the current leaving
the node.
Given any two of these KCL equations, we can find the other by adding or subtracting
them. Thus, one of them is redundant and, in mathematical terms, we can discard any one
of them. The convention is to discard the equation for the (unlabeled) node at the bottom of the
circuit.
In writing KCL equations, you will find that in an
n-node circuit, exactly one of them is always
redundant. Can you sketch a proof of why this might be true? Hint: It has to
do with the fact that charge won't accumulate in one place on its own.
KCL says that the sum of currents entering or leaving a node must be zero. If
we consider two nodes together as a "supernode", KCL applies as well to currents
entering the combination. Since no currents enter an entire circuit, the sum of
currents must be zero. If we had a two-node circuit, the KCL equation of one
must be the negative of the other, We can combine all but one
node in a circuit into a supernode; KCL for the supernode must be the negative of
the remaining node's KCL equation. Consequently, specifying
n1
KCL equations always specifies the remaining one.
Kirchhoff's Voltage Law (KVL)
The voltage law says that the sum of voltages around every closed loop in the circuit
must equal zero. A closed loop has the obvious definition: Starting at a node, trace
a path through the circuit that returns you to the origin node. KVL expresses the fact
that electric fields are conservative: The total work performed in moving a test charge
around a closed path is zero. The KVL equation for our circuit is
v1v2v0
In writing KVL equations, we follow the convention that an element's voltage enters
with a plus sign if traversing the closed path, we go from the positive to the
negative of the voltage's definition.