Title: Kirchoffs Current Law
1Kirchoffs Current Law
The circuit below is the simplest possible
electrical circuit. Current flows out of the
batterys positive terminal, into the upper
terminal of the resistor and out the lower one,
and returns to the battery through the negative
terminal. The current leaving the battery has
nowhere to go except to flow through the wire
connecting the postive terminal of the battery to
the resistor, through the resistor, and through
the other wire into the batterys negative
terminal
The number of electrons leaving the battery in a
given time (e.g., 1 second) must equal the number
returning to the battery in the same time
interval. If more electrons enter than leave, or
vice versa, the battery as a whole is no longer
electrically neutral.
I
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2Kirchoffs Current Law
The battery as a whole must remain electrically
neutral. Charging a battery means storing
energy in it through an electrochemical reaction,
NOT storing a surplus of electrical charge (i.e.,
a quantity of electrons either greater than or
less than the number of protons in the atoms the
battery is made of). The Positive terminal is
positive, but the negative terminal is equally
negative, so the two cancel and the overall
battery remains neutral. Discharging it means
extracting energy by means of an
Electrochemical reaction. Likewise, current into
the resistor must equal current out of the
resistor, because it must also remain
electrically neutral.
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3Kirchoffs Current Law
We can pick a point somewhere within the wire
connecting the positive battery terminal to the
resistor, and call this point a simple node,
designate a. In another sense, the entire wire
is a node, but for now think of the dot in the
schematic below as the node (simple node) where
two lengths of wire are connected together the
length of wire to the left of the dot, and the
length of wire to the right of the dot. The
current flowing into the node is desgnated I1,
and the current leaving the node is I2.
I1 must equal I2. If I1 were greater than I2, a
surplus of electrons would quickly build up at
the node, which would make a small portion of the
wire negatively charged. If I2 were greater, an
electron deficit would develop, making that
portion of the wire positively charged.
a
I1
I2
E
R
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4Kirchoffs Current Law
If the wire is a conductor, no portion of it may
be positive or negative compared to any other
portion. The wire must remain neutral, so I1 and
I2 must be equal. Electron traffic jams are not
allowed! Kirchoffs Current Law (KCL) says that
the total current entering a node must equal the
total current leaving the node.
a
I1
I2
E
R
-
b
I4
I3
5Kirchoffs Current Law
We can also designate another node (node b) in
the wire connecting the resistor to the negative
battery terminal, with current I3 flowing into it
and I4 flowing out of node b. I3 must equal I4 ,
for the same reason that I1 must equal I1.
Because the current entering the resistor must
equal the current leaving it, we observe that I2
I3. By the same token, the current entering
the battery must equal the current leaving the
battery, so I4 I1.
a
Therefore, all four currents are equal. The same
current flows everywhere in this simple circuit.
I1
I2
E
R
-
b
I4
I3
6Kirchoffs Current Law
Now lets add another resistor. Like the first
resistor, the new one is connected between nodes
a and b. We redesignate the original resistor
R1, and call the new one R2. We call the current
through the new resistor I5. I1 is still the
current entering node a, but now the current
leaving node a is the sum of I2 and I5.
Kirchoffs Current Law still applies the total
current entering a node must equal the total
current leaving the node. The total current
entering node a is I1, and the
a
If we know what I1 and I5 are, we can find I2
I1
I2
E
R2
R1
I5
-
Or if we know I1 and I2
b
I4
I3
7Kirchoffs Current Law
What do we really mean by a node? Is it a point
on a schematic diagram? How many nodes in the
circuit below? A node isnt a point on a diagram.
Its an electrical point where two or more
circuit elements (resistors, voltage sources,
etc.) are connected together. Everything
connected to a particular node has the same
potential.
Two points connected by an ideal conductor (or a
conductor that approximates and ideal conductor)
arent electrically distinct points theyre part
of the same node. This means that the positive
battery terminal, the upper resistor terminals,
and the wires connecting them are all part of
node a.
a
I1
I2
E
R2
R1
I5
-
b
I4
I3
8Kirchoffs Current Law
The area enclosed in the dashed line is actually
node a. A similar dashed line could be drawn
around node b. a and b are what Herrick calls a
voltage node. Theyre really just nodes, and
thats what well usually call them. The current
leaving the battery is still I1, and the currents
leaving it are still I2 and I5, so nothing
actually changes in this circuit. But now lets
add a third resistor
a
The current leaving the battery is still I1, and
the currents leaving it are still I2 and I5, so
nothing actually changes in this circuit. But
now lets add a third resistor
I1
I2
E
R2
R1
I5
-
b
I4
I3
9Kirchoffs Current Law
Since we no know that two points connected by an
ideal conductor are part of the same node, we see
that the upper two dots are not two nodes,
theyre both part of node a. The current
entering node a is I1, and the total current
leaving is I2 I5 I6.
a
I1
I2
R2
R3
E
R1
I5
I6
-
b
I4
I3
10Bridge Circuit
This circuit is called a bridge circuit, and its
widely used in measurement and instrumentation
circuits, power supplies, and motor controls.
The bridge has four nodes. On the left, it is
drawn as it appears in the text. On the right is
an equivalent way of drawing it.
a
a
Isupply
IR3
Isupply
IR3
R1
E
IR1
R3
IR1
R3
R1
-
b
c
E
Rload
b
-
c
Rload
Iload
Iload
R2
R4
IR2
I4
IR2
R4
R2
d
I4
Isupply
d
I3
11Bridge Circuit
We can make several observations based on KCL.
First, at node a
Similarly, at node d
At node b
a
Isupply
IR3
And at node c
R1
E
IR1
R3
-
IRload may either be positive (its actual
direction the same as the reference arrow on the
diagram), negative (opposite the reference
arrow), or zero (the balanced bridge condition.
Rload
b
c
Iload
IR2
R4
R2
I4
Isupply
d
I3
12Super Nodes
A super node is a region of the circuit,
including everything within that region. In our
bridge circuit, we may define a super node by
drawing a dashed line around R1 and R2, the super
node is everything within the dashed line. KCL
applies to super nodes as well as ordinary nodes.
For this super node, the current flowing in
is Isupply IR4 and the current leaving the
super node is Iload IR3 Isupply. Now, based
on KCL, we may write
Isupply
IR3
IR1
E
R1
R3
-
Rload
Which simplifies to
Iload
Which we found previously, without defining a
super node.
R4
R2
IR2
IR4
Isupply
I3
13A BJT Circuit
BJT stands for Bipolar Junction Transistor,
almost the first type of transistor (the first
was actually a closely-related type called the
point-contact transistor). The BJT is still
widely used. There are two basic varieties
The PNP transistor, and the NPN transistor. The
schematic symbol for an NPN type is shown below
The BJT has three terminals, called the collector
(C on the symbol shown here), the base (B) and
the emitter (E). As the arrow which is part of
the symbol suggests, the emitter current IE flows
out of the transistor. It can only be positive
or zero (ideally), so it must actually flow in
the indicated direction. The collector and base
currents also flow in the indicated directions if
they are nonzero. All three directions would be
reversed if this were a PNP transistor.
C
IC
IB
B
IE
E
I3
14A BJT Circuit
Theres no reason we cant treat the BJT as a
super node, with IB and IC entering the super
node and IE leaving. We now observe that
The BJT is useful because it has forward current
gain Increasing IB causes a larger increase in
IC
In which b is a constant called the forward
current gain parameter (sometimes written hfe
instead of b). Combining these two relationships
gives us this one
C
IC
IB
B
IE
or
E
I3
15A BJT Circuit
The gang at Bell Labs who discovered the BJT
thought (correctly) that it was pretty cool. It
isnt magic, though. A small increase in IB can
result (in a properly designed circuit) in a much
larger increase in IE and IC, but this increased
current doesnt come out of thin air, and isnt
conjured out of parts of amphibians and flying
mammals. It has to be supplied by an external
power source connected to either the collector or
the emitter, or nothing
happens. Think of the BJT (or other transistor
types) as a control valve. A small turn can
cause a big change in current flow, but like the
valve for a garden hose, something has to be
available to flow through it. In England,
transistors are sometimes referred to as valves.
C
IC
IB
B
IE
or
E
I3
16A BJT Circuit
The gang at Bell Labs who discovered the BJT
thought (correctly) that it was pretty cool. It
isnt magic, though. A small increase in IB can
result (in a properly designed circuit) in a much
larger increase in IE and IC, but this increased
current doesnt come out
Let the LED current be 10 mA. to light the LED.
Obviousley, ILED IC.
ILED
LED
IC
RB
Q b 100
IB
EIN
IE
RE
So the LED current can be turned on or off by a
mere 100 mA change in base current.
17A BJT Circuit
The gang at Bell Labs who discovered the BJT
thought (correctly) that it was pretty cool. It
isnt magic, though. A small increase in IB can
result (in a properly designed circuit) in a much
larger increase in IE and IC, but this increased
current doesnt come out
Let the LED current be 10 mA. to light the LED.
Obviousley, ILED IC.
ILED
LED
IC
RB
Q b 100
IB
EIN
IE
RE
So the LED current can be turned on or off by a
mere 100 mA change in base current.
18A BJT Circuit
Recall that
ILED
LED
IC
RB
Q b 100
IB
EIN
IE
RE
19An Op-Amp Circuit
Op-amp is short for operational amplifer, so
named because the first operational amplifiers
were used to perform mathematical operations in
analog computers. The op-amp has two inputs an
inverting input, and a noninverting input. An
increase in voltage applied to the noninverting
input causes an increase in output voltage, but
an
Increase in the voltage applied to the
noninverting input causes a decrease in output
voltage. Analyzing op-amp circuits can often be
simplified by using the ideal op-amp
approximation. One of the assumptions implied in
this approximation is that the input resistance
of both the inverting and noninverting inputs are
so great that Ini and Iinv are both zero.
Esupply
Ini
Iout
-
Iinv
-Esupply
A stone-age device discovered in caves near Palo
Alto, CA. It appeared to have been carved out
using flint knives, and radiocarbon tests showed
it to date to the 1940s.
20An Op-Amp Circuit
The op-amp also has two power supply terminals,
for a positive supply and a negative supply.
These are sometimes omitted from simplified
schematics, since its obvious that power supply
connections are needed. In the circuit below,
lets assume that weve caused a current IR1 to
flow toward the inverting input through R1. The
point at which the
inverting input, R1 and Rf are connected is a
node, so we apply KCL. The current entering the
node is IR1 IRf, and the current leaving is
Iinv. Weve decided to assume that Iinv is zero
(remember?), so
Ini
R1
IR1
IRf
Rf
21An Op-Amp Circuit
Now lets add a resistor, R2. Make the current
flowing toward the inverting input throught R2
IR2. Now, the point at which the three resistors
and the inverting input are connected is a node.
The current entering is the sum of IR1, IR1 and
IRf. The current leaving is still Iinv, still
zero, so
IRf is the additive inverse of the sum of IR1 and
IR2, so this circuit performs the mathematical
operation of addition. A third current, or a
fourth, could be summed if additional resistors
were added to the circuit.
Ini
R1
IR1
R2
IRf
IR2
Rf
22Alternate Form of KCL
As weve been using it so far, KCL can be written
this way
That is, the sum of all currents entering a node
is equal to the sum of all currents leaving that
node. Another way of writing KCL is
The two forms are equivalent.