46. ELECTRONICS
Craig G. Shaefer
46.4. KIRCHHOFF’S
46.4. Kirchhoff ’s Laws
This Kirchhoff’s Laws Section 46.4[49] should be cited as “Craig G. Shaefer, Math for the
Motivated: Electronics: Kirchhoff’s Laws, 2012 ed., 5687–5694, (self-published), Rapid City,
SD, 2012.”
As we point out in Figure 46.6.4 on page 5708 of the “Resistance: Series and Parallel”
Example, there are some circuit networks that cannot be simplified using the formulae for
series and parallel topologies of resistors. For these we require two further rules that are
known as the Kirchhoff’s Laws.
These principles we will find most useful for analyzing the voltages and currents in electronic
circuits. These two principles are really nothing more than applications of the Conservation
of Energy and Conservation of Charge Laws from physics.
Before we state these two principles, let’s recall how an electric motor operates. An electric
motor is nothing more than a coil of wire carrying a current. The magnet field created by
the current interacts with the magnetic fields of permanent magnets in such a way as to
generate circular motion. If the current is switched off, the rotor stops spinning. Thus the
electric motor requires energy in order to produce rotary motion.
We also know that whenever two points are at different voltage potentials and there is a
conductor allowing for current to flow between these two points, then current will flow from
the higher potential point to the lower potential. We will employ this concept below.
Now for the two rules. The first principle states that the voltage drops around any closed
loop in a circuit must sum to zero. This stems directly from Conservation of Energy, because
if the voltage drops did not sum to zero then after transversing the loop we would have some
nonzero voltage difference remaining. This voltage difference would mean that a current
would flow, and this current would not have any energy source. Essentially it would be a
perpetually flowing current, and, just like the electric motor, work could be derived from the
current flow. This would constitute a perpetual motion machine; free energy in other words.
But the Conservation of Energy Law eliminates this possibility and hence there cannot be
any remaining voltage differences after circumnavigating a loop. In other words, the voltage
drops around any loop must sum to zero.
This is how electronics translates the property that the circulation of the electrostatic field
is zero (Equation (45.3.82) on page 5423):
!
0 = E · t̂ds
Circulation Vanishes
C
into a statement about a circuit.
The second principle stems from the Conservation of Charge Law. This rule says that we
cannot destroy or create electric charges. When applied to electronic circuits, this means
that the current, which is nothing more than flowing electric charges, cannot be created or
destroyed by passive devices. Currents can be split or added, but the total current must
remain the same. Thus when we look at three wires connected at a common point, the
current flowing into that point must equal the current flowing out of that point, otherwise
the Conservation of Charge Law would be violated.
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46.4. KIRCHHOFF’S
Craig G. Shaefer
46. ELECTRONICS
The Continuity Equation (45.3.43) (page 5384) states
∂ρ
∂t
0 = ∇·J +
Continuity Equation
which, for a circuit in equilibrium having a steady current means that ∂ρ/∂t = 0 and thus
0 = ∇·J
.
Steady State
This means that the sum of the currents into and out of a point must be zero.
When these two principles are valid, then the current through each element of a circuit must
obey Ohm’s Law, that is, the current through the element equals the voltage across the
element divided by the element’s resistance.
These two principles are summarized in the following two laws named for the German physicist Gustav Robert Kirchhoff (1824-1887).
Theorem 46.4.1 (Kirchhoff’s Laws "T:Kirchhoff#).
Kirchhoff’s Current Law (KCL)
10
At any point in a circuit, the sum of all currents entering and leaving must equal zero.
The Kirchhoff Current Law is nothing more than a Conservation of Charge rule, it states
that the sum of all currents into a node (place where two or more wires connect so that the
current can split between alternative paths) must equal the sum of all currents out of the
node, thus:
(46.4.1)
0=
any
node
"
Ik
,
KCL
k
where Ik denotes the k-th current flowing into the node.
This is a statement of Conservation of Charge, it says that electrical charges cannot be
created nor destroyed, the net charge must remain the same.
Sometimes the KCL is called the Kirchhoff’s Node Law.
Note: Sometimes a node refers simply to any point along a conductor that a person wishes
to designate. In this case, the node has only two conductors connected to it. And in this
case, when three or more conductors connect at a point, the node is known as a principal
node.
Kirchhoff’s Voltage Law (KVL)
For any loop in an electronic circuit, the sum of all voltage drops and voltage sources is zero.
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Craig G. Shaefer
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46. ELECTRONICS
Craig G. Shaefer
46.4. KIRCHHOFF’S
The Kirchhoff Voltage Law is really just the expression of Conservation of Energy, it states
that around any closed loop in a circuit, the sum of all voltages across all of the circuit
elements in the loop must be zero:
(46.4.2)
0=
any
loop
"
Vk
,
KVL
k
where Vk denotes the voltage across the terminals of the k-th element in the closed loop.
This is a statement of Conservation of Energy, the net electrical energy must be constant.
Sometimes the KVL is called the Kirchhoff’s Loop Law.
Together these two laws are often called the Kirchhoff’s Circuit Laws
Proof :
Proof of Theorem 46.4.1.
As we have already discussed, Conservation of Energy via the zero circulation of the electrostatic field requires that the sum of all voltage drops around any closed circuit or loop
vanishes. And the Conservation of Charge, via the Continuity Equation means that at any
node in a circuit the current into the node must equal the current out of the node.
!
Notice that in Figure 46.4.1 there are three currents flowing into the node, I1 (t), I2 (t), and
I5 (t). There are two currents flowing out of the node, I3 (t) and I4 (t). [In this context, the
current values Ik (t) are the magnitudes of the currents, i.e., Ik (t) really denotes |Ik (t)|.] In
the steady state, the sum of the inflowing currents must equal the sum of the outflowing
currents, otherwise charge would be building up at the node. Thus we have
I1 (t) + I2 (t) + I5 (t) = I3 (t) + I4 (t)
.
By KCL
This equation may be rearranged to
0 = I1 (t) + I2 (t) + I5 (t) + (−I3 (t)) + (−I4 (t))
,
Rearrange
which states that the sum of all currents into and out of a node must equal zero, where the
sign of the current specifies whether it is flowing into (+) or out of (−) the node.
This rule generalizes to any number of conductors, thus we may rewrite it as
"
"
(46.4.3)
Ij (t) ,
Alternative KCL
Ik (t) =
k∈{in}
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j∈{out}
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46.4. KIRCHHOFF’S
Craig G. Shaefer
46. ELECTRONICS
I1(t)
I4(t)
node
I2(t)
I5(t)
I3(t)
Figure 46.4.1. KirchhoffNews: This schematic diagram shows a portion
of an electronic circuit consisting only of five conductors meeting at a point,
called a node. Arrows indicate the directions of current flow, with those
pointing towards the node indicating that current is flowing into the node
while those pointing away from the node means that current is flowing out
of the node. The sum of the currents into the node must equal the sum of
the currents out of the node.
where k ∈ {in} indicates that Ik belongs to the set of all inflowing currents while j ∈ {out}
means that Ij belongs to the set of all outflowing currents. This equation is just an alternative
expression of the regular Kirchhoff’s Current Law, Equation (46.4.1). Sometimes it is easier
to think in terms of inflowing and outflowing currents than using the sign of the current to
indicate whether it is inflowing or outflowing.
L
V2(t)
I(t)
V3(t)
V1(t)
R
C
Figure 46.4.2. KirchhoffNews: This schematic diagram shows a closed
loop from an electronic circuit consisting of three voltage sources whose
polarities are indicated by arrows and a resistor, inductor, and capacitor.
The direction of the current is simply picked, and if the current ends up
being negative then its actual direction is opposite from that assumed.
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46. ELECTRONICS
Craig G. Shaefer
46.4. KIRCHHOFF’S
Notice that in Figure 46.4.2 there are three voltage sources, two are in the direction of the
assumed current. The two in the same direction as the current, i.e., V1 (t) and V2 (t), are
given positive voltages while the one in the opposite direction is given a negative voltage.
In the steady state, the voltages are sources to be summed since they produce a rise in the
potential difference across their terminals. The passive elements in this circuit, the R, L,
and C, produce voltage drops across their terminals. In order for energy to be conserved,
the sum of the voltage rises must equal the sum of the voltage drops around the circuit.
Thus we have
#
dI(t)
1
V1 (t) + V2 (t) − V3 (t) = I(t)R + L
+
I(t)dt .
By KVL
dt
C
This equation may be rearranged to
0 = V1 (t) + V2 (t) − V3 (t)
$
%
#
1
dI(t)
− I(t)R + L
+
I(t)dt
dt
C
Rearrange
,
which states that the sum of all voltage rises and drops around a closed loop must equal
zero.
This rule generalizes to any number of voltage rises and drops, thus we may rewrite it as
"
"
(46.4.4)
Ij (t) ,
Alternative KVL
Vk (t) =
k∈{rises}
j∈{drops}
where k ∈ {rises} indicates that Vk is a volage source oriented so that it produces a volate
rise in the direction of the current I(t) while j ∈ {drop} means that Vj belongs to the set of
all passive elements that yield voltage drops. This equation is just an alternative expression
of the regular Kirchhoff’s Voltage Law, Equation (46.4.2). If the sum of the potential rises
equals the sum of the potential drops, then the total sum of all voltages around the closed
circuit must be zero. Sometimes it is easier to think in terms of potential rises and drops
when considering closed loops in an electronic circuit.
Note: In this Chapter we employ the convention that when two lines on a schematic diagram
cross it signifies a junction between the wires. If crossing wires are not meant to denote a
conductive connection, then one of the crossing lines is drawn so that it appears to “pass
over” the other line. Another convention is also employed for schematics, and that convention is that crossing lines do not denote a conductive junction between wires, rather a filled
dot or small circle at a crossing point is necessary to denote a junction. This second convention is very useful for very dense schematic diagrams with numerous crossing wires that
do not represent junctions. However, in this Chapter we do not plan or drawing this type
of schematics so our schematics will have very few crossings that do not represent junctions.
Therefore we employ the first convention where any crossing is a junction unless it is drawn
to appear as if one wire passes over the top of the other wire.
46.4.1. Examples: Electronic Circuits.
Here are a number of examples of electronic circuits.
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46.4. KIRCHHOFF’S
Craig G. Shaefer
46. ELECTRONICS
To cement your understanding of this new nomenclature, I highly recommend that you work
your way through each and every one of these samples. You might rework each example on
scrap paper.
Example 4.
Resistance: 2 Resistors in Series and Parallel
&
4
We have now examined Ohm’s Law in greater detail to understand how resistance arises.
We may now employ this knowledge to understand how resistances are added together in
various configurations.
As we are already familiar, electronic components are given special symbols that allow us to draw circuits diagrams to specify the interconnections between various components. We employ these so-called schematic
diagrams for the RC circuit for the “Charging a Capacitor: Derivation”
Example on page 5750. The electronic schematic symbol for a resistor
is drawn at the left. It is often labelled with an R that both labels the
component as well as it often represents the resistor’s value in Ωs.
R
Let’s next think about what happens if we connect two resistors together. Now there are two alternative topologies for
doing this, one labelled series and the other called parallel.
In the series topology, the resistors are connected end-to-end,
with one lead of the first resistor as input and the second lead
of the second resistor as output. All of the electric current
must flow through both of the resistors. The schematic diagram at the right best explains this configuration.
R1
R2
R1
R2
In the parallel topology, both ends of one resistor are connected
to both ends of the other resistor. Current flowing through a
parallel configuration thus splits, with a portion of the current
flowing through one resistor and the remainder flowing through
the other resistor. The schematic diagram at the left best explains this configuration. Both the series and parallel topologies
generalize to any number of resistors, say N . When N = 2 the
total resistances are simple expressions for both the series and
parallel configurations. Derive these expressions.
Solution:
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46. ELECTRONICS
Craig G. Shaefer
46.4. KIRCHHOFF’S
First of all, we note that Equation (46.3.10) tells us that the resistance is proportional to the
length and inversely proportional to the area. Since two resistors in series essentially add their
lengths, we would surmise that for resistors in series we should add their resistances. Also, if the
two resistors are the same, then two identical resistors in parallel would present twice the area to
the electric current and thus should half the net resistance. In fact, since the resistance depends
upon the reciprocal of the area, then resistors in parallel should depend in some fashion on the
sum of the reciprocals. Let’s derive the actual expressions using Ohm’s Law,
V = IR
.
Ohm’s Law
Let say that we have 2 resistors having resistances of R1 and R2 , respectively, in series and we
apply a voltage V across them. Now in this topology the current must flow through each resistor
in turn, thus it is the same current, say I, that flows through each resistor since the net charge
must be conserved. Thus the voltage across resistor R1 is V1 = IR1 while the voltage across the
second resistor is V2 = IR2 . Since the voltage across both resistors equals V (by conservation
of energy, which we will see in a moment), then we have
V = IR1 + IR2 = I(R1 + R2 ) = IRseries
,
Total V
where Rseries denotes the single resistance equivalent to both resistors. Thus we see that
(46.4.5)
Rseries = R1 + R2
,
Rseries
that is, a single resistor whose resistance is equal to the sum of the resistances of both resistors in
series is equivalent to the series resistors, just as we expected from the above informal argument.
Let’s think about what happens when the resistors are in a parallel topology. Now the voltage
across both of the resistors is the same as the supply voltage, V . But each resistor has its own
amount of current passing through it depending upon its own resistance value. Thus we have
V = I1 R1 and V = I2 R2 . Since charges cannot be created nor destroyed, the total current must
equal the sum of all of the individual currents, I = I1 + I2 . If we denote the total equivalent
resistance of the 2 parallel resistors as Rparallel , then
V = IRparallel = (I1 + I2 ) Rparallel
,
but we also have Ik = V /Rk for k = {1, 2} and thus
$
%
$
%
V
V
1
1
V =
+
Rparallel = V
+
Rparallel
R1
R2
R1
R2
V
.
'
V
Sub Ik =
Rk
(
This is possible if and only if
$
%
1
1
1=
+
Rparallel
R1
R2
or
(46.4.6)
Rparallel =
1
R1
1
+
1
R2
.
Rparallel
Notice that we may rewrite this last expression as
(46.4.7)
Rparallel =
=
1
R1
1
+
1
R2
1
R1 +R2
R1 R2
=
=
R2
R1 R2
1
+
R1 R2
R1 + R2
R1
R1 R2
Rparallel
,
an expression affectionately known in electronics as “product over sum”, for obvious reasons.
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46.4. KIRCHHOFF’S
Craig G. Shaefer
46. ELECTRONICS
We have thus derived the resistances, given in Equations (46.4.5) and (46.4.6), equivalent to 2
resistors in both a series and a parallel configuration.
Note: An analogous argument to that above will be given in the “Resistance: N Resistors in
Series and Parallel” Example (page 5706) generalizing these results for two resistors to cases
having N resistors in series and parallel.
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