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Kirchhoff’s Laws For Electrical EngineersKirchhoff’s Laws For Electrical Engineers– Beginners– Beginners

Kir chhoff’ s Laws For Electrical Engineers - Beginners (photo credit: Jesse Mason via Youtube)

Voltages and currents in electric circuitsVoltages and currents in electric circuits

Two extremely important principles in electric circuits were codified by Gustav RobertTwo extremely important principles in electric circuits were codified by Gustav Robert

Kirchhoff in the year 1847, known asKirchhoff in the year 1847, known as Kirchhoff’s LawsKirchhoff’s Laws. His two laws refer to. His two laws refer to

VoltagesVoltages  andandCurrents in electric circuitsCurrents in electric circuits, respectively., respectively.

Kirchhoff’s Voltage LawKirchhoff’s Voltage Law

Kirchhoff’s Voltage LawKirchhoff’s Voltage Law states that the algebraic sum of all voltages in a closed loopstates that the algebraic sum of all voltages in a closed loop

is equal to zero. Another way to state this law is to say that for every rise in potentialis equal to zero. Another way to state this law is to say that for every rise in potential

there must be an equal fall, if we begin at any point in a circuit and travel in a loopthere must be an equal fall, if we begin at any point in a circuit and travel in a loop

back to that same starting point.back to that same starting point.

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Hiking up a mountain // Hiking up a mountain // 

An analogy for visualizing Kirchhoff’s Voltage Law is hiking up a mountain.An analogy for visualizing Kirchhoff’s Voltage Law is hiking up a mountain.

Suppose we start at the base of a mountain and hike to an altitude ofSuppose we start at the base of a mountain and hike to an altitude of 5000 feet5000 feet to setto set

up camp for an overnight stay. Then, the next day we set off from camp and hikeup camp for an overnight stay. Then, the next day we set off from camp and hike

farther up anotherfarther up another 3500 feet3500 feet. Deciding we’ve climbed high enough for two days, we set. Deciding we’ve climbed high enough for two days, we set

up camp again and stay the night. The next day we hike downup camp again and stay the night. The next day we hike down 6200 feet6200 feet to a thirdto a third

location and camp once gain. On the fourth day we hike back to our original startinglocation and camp once gain. On the fourth day we hike back to our original starting

point at the base of the mountain.point at the base of the mountain.

We can summarize our hiking adventure as a series of rises and falls like this:We can summarize our hiking adventure as a series of rises and falls like this:

Hiking up a mountain analogy 

DayDay PathPath Altitude gain / LossAltitude gain / Loss

 Day 1Day 1 A to BA to B  +5000 feet+5000 feet

 Day 2Day 2 B to CB to C  +3500 feet+3500 feet

 Day 3Day 3 C to DC to D  -6200 feet-6200 feet

 Day 4Day 4 D to AD to A  -2300 feet-2300 feet

 TOTAL //TOTAL // ABCDAABCDA 0 feet0 feet

Of course, no one would tell their friends they spent four days hiking a total altitude ofOf course, no one would tell their friends they spent four days hiking a total altitude of

0 feet, so people generally speak in terms of the highest point reached: in this case0 feet, so people generally speak in terms of the highest point reached: in this case

8500 feet. However, if we track each day’s gain or loss in algebraic terms (maintaining8500 feet. However, if we track each day’s gain or loss in algebraic terms (maintaining

the mathematical sign, either positive or negative), we see that the end sum is zerothe mathematical sign, either positive or negative), we see that the end sum is zero(and indeed must always be zero) if we finish at our starting point.(and indeed must always be zero) if we finish at our starting point.

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If we view this scenarioIf we view this scenario from the perspective of potential energyfrom the perspective of potential energy  as we lift aas we lift a

constant mass from point to point, we would conclude that we were doing work on thatconstant mass from point to point, we would conclude that we were doing work on that

mass (i.e. investing energy in it by lifting it higher) onmass (i.e. investing energy in it by lifting it higher) on days 1days 1 andand 22, but letting the, but letting the

mass do work on us (i.e. releasing energy by lowering it) onmass do work on us (i.e. releasing energy by lowering it) on days 3days 3 andand 44. After the. After the

four-day hike, the net potential energy imparted to the mass is zero, because it endsfour-day hike, the net potential energy imparted to the mass is zero, because it ends

up at the exact same altitude it started at.up at the exact same altitude it started at.

Let’s apply this principle to a real circuit, where total current and all voltage drops haveLet’s apply this principle to a real circuit, where total current and all voltage drops have

already been calculated for us:already been calculated for us:

Arrow shows current in the direction of conventional flow notation 

If we trace a path ABCDEA, we see that theIf we trace a path ABCDEA, we see that the algebraic voltage sum in this loop isalgebraic voltage sum in this loop is

zerozero::

PathPath Voltage gain / LossVoltage gain / Loss

 A to BA to B  -4 volts-4 volts

 B to CB to C  -6 volts-6 volts

 C to DC to D  +5 volts+5 volts

 D to ED to E -2 volts-2 volts

 E to AE to A +7 volts+7 volts

 ABCDEAABCDEA 0 volts0 volts

We can even trace a path that does not follow the circuit conductors or include allWe can even trace a path that does not follow the circuit conductors or include all

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components, such as EDCBE, and we will see thatcomponents, such as EDCBE, and we will see that the algebraic sum of all voltagesthe algebraic sum of all voltages

is still zerois still zero::

PathPath Voltage gain / LossVoltage gain / Loss

 A to BA to B  +2 volts+2 volts

 B to CB to C  -5 volts-5 volts

 C to DC to D  +6 volts+6 volts

 D to ED to E -2 volts-2 volts

 E to AE to A -3 volts-3 volts

 ABCDEAABCDEA 0 volts0 volts

Kirchhoff’s Voltage Law is oftenKirchhoff’s Voltage Law is often a difficult subject for students a difficult subject for students ,,

precisely because voltage itself is a difficult concept to grasp.precisely because voltage itself is a difficult concept to grasp.

Remember that there is no such thing as voltage at a single point;Remember that there is no such thing as voltage at a single point;  rather, voltagerather, voltage

exists only as a differential quantity. To intelligently speak of voltage, we must refer toexists only as a differential quantity. To intelligently speak of voltage, we must refer to

either a loss or gain of potential between two points.either a loss or gain of potential between two points.

Our analogy of altitude on a mountain is particularly apt. We cannot intelligently speakOur analogy of altitude on a mountain is particularly apt. We cannot intelligently speak

of some point on the mountain as having a specific altitude unless we assume a pointof some point on the mountain as having a specific altitude unless we assume a point

of reference to measure from. If we say the mountain summit is 9200 feet high,of reference to measure from. If we say the mountain summit is 9200 feet high, wewe

usually mean 9200 feet higher than sea levelusually mean 9200 feet higher than sea level, with the level of the sea being our, with the level of the sea being our

common reference point. However, our hiking adventure where we climbed 8500 feet incommon reference point. However, our hiking adventure where we climbed 8500 feet in

two days did not imply that we climbed to an absolute altitude of 8500 feet above seatwo days did not imply that we climbed to an absolute altitude of 8500 feet above sea

level. Since I never specified the sea-level altitude at the base of the mountain, it islevel. Since I never specified the sea-level altitude at the base of the mountain, it isimpossible to calculate our absolute altitude at the end of day 2.impossible to calculate our absolute altitude at the end of day 2.

All you can tell from the data given is that we climbed 8500 feet above the mountainAll you can tell from the data given is that we climbed 8500 feet above the mountain

base, wherever that happens to be with reference to sea level.base, wherever that happens to be with reference to sea level.

So it is with electrical voltage as well:So it is with electrical voltage as well:  most circuits have a point labeledmost circuits have a point labeled

as ground where all other voltages are referenced.as ground where all other voltages are referenced.

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In DC-powered circuits, this ground point is often the negative pole of theIn DC-powered circuits, this ground point is often the negative pole of the DC powerDC power

sourcesource. Voltage is fundamentally a quantity relative between two points: a measure of. Voltage is fundamentally a quantity relative between two points: a measure of

how much potential has increased or decreased moving from one point to another.how much potential has increased or decreased moving from one point to another.

Go back to LawsGo back to Laws !!!!

Kirchhoff’s Current LawKirchhoff’s Current Law

Kirchhoff’s Current LawKirchhoff’s Current Law is a much easier concept to grasp. This law states that theis a much easier concept to grasp. This law states that the

algebraic sum of all currents at a junction point (called a node) is equal to zero.algebraic sum of all currents at a junction point (called a node) is equal to zero.

Another way to state this law is to say that for every electron entering a node, oneAnother way to state this law is to say that for every electron entering a node, one

must exit somewhere.must exit somewhere.

An analogy for visualizing Kirchhoff’s Current Law isAn analogy for visualizing Kirchhoff’s Current Law is

water flowing into and out of awater flowing into and out of a

“tee” fitting“tee” fitting::

Visualizing Kirchhoff’s Current Law is water flowing 

So long as there are no leaks in this piping system, every drop of water entering theSo long as there are no leaks in this piping system, every drop of water entering the

tee must be balanced by a drop exiting the tee. For there to be a continuous mismatchtee must be balanced by a drop exiting the tee. For there to be a continuous mismatch

between flow rates would implybetween flow rates would imply a violation of the Law of Mass Conservationa violation of the Law of Mass Conservation..

Let’s apply this principle to a real circuit, where all currents have been calculated forLet’s apply this principle to a real circuit, where all currents have been calculated for

us:us:

Arrow shows current in the direction of conventional flow notation 

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At nodesAt nodes

where justwhere just

two wirestwo wires

connectconnect

(such as(such as

points A, B,points A, B,

and C), theand C), the

amount ofamount of

currentcurrent

going in togoing in to

the nodethe node

exactlyexactly

equals theequals the

amount ofamount ofcurrent going out (4 mA, in each case). At nodes wherecurrent going out (4 mA, in each case). At nodes where three wires jointhree wires join (such as(such as

points D and E), we see one large current and two smaller currents (one 4 mA currentpoints D and E), we see one large current and two smaller currents (one 4 mA current

versus two 2 mA currents), with the directions such that the sum of the two smallerversus two 2 mA currents), with the directions such that the sum of the two smaller

currents form the larger current.currents form the larger current.

Just as the balance of water flow rates into and out of a piping “tee” Just as the balance of water flow rates into and out of a piping “tee” 

is a consequence of the Law of Mass Conservation,is a consequence of the Law of Mass Conservation, the balance of the balance of electric currents flowing into and out of a circuit junction electric currents flowing into and out of a circuit junction  is a is a 

consequence of the Law of Charge Conservation, another consequence of the Law of Charge Conservation, another 

fundamental conservation law in physics.fundamental conservation law in physics.

Go back to LawsGo back to Laws !!!!

Resistors in Series and Parallel // Kirchhoff’s RulesResistors in Series and Parallel // Kirchhoff’s Rules

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Share with engineers // Share with engineers //  % & ' (

!"#$%&$ ( )*$%$+,-$ %. /*-%*$ 0.1 !0-022*2 34 ,5 67 8%-&"",559$ ):2*$

Go back to LawsGo back to Laws !!!!

Reference // Reference //  Lessons In Industrial Instrumentation – Tony R. Kuphaldt Lessons In Industrial Instrumentation – Tony R. Kuphaldt 

Recommended EE articles // Recommended EE articles // 

About Author // About Author // 

Edvard CsanyiEdvard Csanyi

EdvardEdvard  -- Electrical engineer, programmer and founder ofElectrical engineer, programmer and founder of EEPEEP. Highly specialized for. Highly specialized for

design of LV high power busbar trunking (<6300A) in power substations, buildings anddesign of LV high power busbar trunking (<6300A) in power substations, buildings and

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industry fascilities. Designing of LV/MV switchgears.Professionalindustry fascilities. Designing of LV/MV switchgears.Professional

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