Transform your understanding of Basic Circuit connections and laws with this indispensable set of notes, crafted specifically for electrical and computer engineering students. Dive into key concepts like circuit analysis, Ohm’s Law, Kirchhoff’s Laws, Thevenin’s and Norton’s theorems, and AC...
2.1 Circuit Terminology
Node: junction of two or more circuit elements.
Essential node: junction of three or more circuit elements.
Path: set of one or more adjoining circuit elements that may be traversed in
succession without passing through the same node more than once. A path
generally has an initial node at its beginning and a final node at its end.
If the initial and final nodes are the same, the path is closed and becomes a loop.
Mesh: a loop that does not enclose any other loop.
Branch: path that connects two nodes.
Essential branch: branch that connects two essential nodes without passing
through an essential node.
In Figure 2.1.1 the nodes 1 Rsrc a
labeled a, b, c, and d, are L1
essential nodes, whereas
nodes 1, 2, and 4 are not C2 R1
2
essential nodes. Nodes 3 R5
vSRC +
and b are one and the c b
– 3
same, as are nodes 3’, d,
4 C3
and 5 because no circuit R4
element is connected 3'
L4
between them, only a R3
connection of zero 5 d
resistance that is used for Figure 2.1.1
convenience of illustration.
Rsrc, vSRC, L1, R1, L4, R4, C3, and R3 taken individually, are branches. The
combinations Rsrc-vSRC, L1-R1, L4-R4, and the individual branches C3 and R3 are
essential branches. The closed paths d-5-1-a-b-d and d-5-1-a-c-b-d are loops. The
loops d-5-1-a-c-d, a-b-c-a, and d-c-b-d (going through C3) are meshes. C3 and R3
can also be considered to form a mesh.
2-1/16
,2.2 Kirchhoff’s Laws
Kirchhoff’s Current Law
Statement At any instant of time, the sum of currents entering a node is equal
to the sum of currents leaving the node.
At node N in Figure 2.2.1 KCL gives: iA + iB = iC + iD.
KCL is a direct expression of conservation of current, which iC
follows from conservation of charge, just as conservation of power iA iB
follows from conservation of energy.
N
Alternative statement of KCL: at any instant of time, the algebraic iD
sum of all the currents at any node is zero.
Figure 2.2.1
iA + iB −i C −i D =0 , where opposite signs are assigned to currents
flowing towards the node and to currents flowing away from the node.
KCL may be applied S S
not just to a node i
i=0
but also to
A B A B
interconnected
circuits or parts of a i
(a) (b)
circuit.
Figure 2.2.2
Figure 2.2.2a:
equal and opposite currents flow in the two connections. In Figure 2.2.2b, i = 0.
Kirchhoff’s Voltage Law
Statement At any instant of time, the sum of voltage rises around any loop is
equal to the sum of voltage drops – v3 +
around the loop. + +
v2 3 4
v1 + v2 + v3 = v4 + v5 in Figure 2.2.3. v4
–
Equivalent statement of KVL: at any instant of –
2 +q 5
time, the algebraic sum of the voltages around
any loop is zero, since voltage drops and + +
v1 1 v5
voltage rises have opposite signs: – –
v1 + v2 + v3 – v4 – v5 = 0.
Concept KCL and KVL together are an Figure 2.2.3
2-2/16
, expression of conservation of energy.
In taking a charge +q clockwise around the loop, the work done on the charge is q(v1
+ v2 + v3). But the charge can do more work, q(v4 + v5), if (v1 + v2 + v3) < (v4 + v5),
which violates conservation of energy. Taking the charge +q around a loop, which
necessarily involves passing through nodes or essentials nodes between the circuit
elements, implies that the charge +q is conserved at these nodes, that is, KCL
applies.
2.3 Voltage Division and Series Connection of Resistors
Definition In a series connection of elements, the same current flows
through all the elements.
In Figure 2.3.1a: + v1 – + v2 – i + vn –
a
v SRC =v 1 + v 2 +. ..+ v n
R1 R2 Rn
(2.3.1) vSRC +
Applying Ohm’s law to –
i
the individual resistors:
b (a) i
v SRC =R 1 i+ R 2 i+. . .+ R n i a
(2.3.2)
vSRC + Reqs
If Rm denotes one of the
–
resistors R1 to Rn and vm b i
is the voltage drop
Figure 2.3.1 (b)
across Rm, so that vm =
Rmi, then dividing this relation by Equation 2.3.2:
vm Rm
=
v SRC R 1 + R2 +. . .+ R n , m = 1, 2, …., n (2.3.3)
vSRC divides across the string of resistors in the ratio of each individual resistance to
the total resistance. If vSRC = 6 V, R1 = 1 , R2 = 2 , and R3 = 3 , the voltages
across the resistors are v1 = 1 V, v2 = 2 V, and v3 = 3 V.
It also follows that the voltages across any two resistors are in the ratio of the
corresponding resistances and in the inverse ratio of the conductances:
v1 R1 Gm
= =
vm Rm G1 (2.3.4)
2-3/16
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