# KIRCHHOFF'S LAWS Click or Tap the Example circuits below to invoke TINACloud and select the Interactive DC mode to Analyze them Online.
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Many circuits are too complex to be solved using the rules for series or parallel circuits or the techniques for conversion to simpler circuits described in previous chapters. For these circuits we need more general solution methods. The most general method is given by KirchhoffŌĆÖs laws, which permit the calculation of all circuit voltages and currents of circuits by a solution of a system of linear equations.

There are two Kirchhoff laws, the voltage law and the current law. These two laws can be used to determine all voltages and currents of circuits.

KirchhoffŌĆÖs voltage law (KVL) states that the algebraic sum of the voltage rises and voltage drops around a loop must be zero.

A loop in the above definition means a closed path in the circuit; that is, a path that leaves a node in one direction and returns to that same node from another direction.

In our examples, we will use clockwise direction for loops; however, the same results will be obtained if the counterclockwise direction is used.

In order to apply KVL without error, we have to define the so called reference direction. The reference direction of the unknown voltages points from the + to the ŌĆō sign of the assumed voltages. Imagine using a voltmeter. You would place the voltmeter positive probe (usually red) at the componentŌĆÖs reference + terminal. If the real voltage is positive, it is in the same direction as we assumed, and both our solution and the voltmeter will show a positive value.

When deriving the algebraic sum of the voltages, we must assign a plus sign to those voltages where the reference direction agrees with the direction of the loop, and negative signs in the opposite case.

Another way to state KirchhoffŌĆÖs voltage law is: the applied voltage of a series circuit equals the sum of the voltage drops across the series elements.

The following short example shows the use of KirchhoffŌĆÖs voltage law.

Find the voltage across resistor R2, given that the source voltage, VS = 100 V and that the voltage across resistor R1 is V1 = 40 V.

The figure below can be created with TINA Pro Version 6 and above, in which drawing tools are available in the schematic editor. The solution using KirchhoffŌĆÖs voltage law: -VS + V1 + V2 =0, or VS = V1 + V2

hence: V2 = VS ŌĆō V1 = 100-40 = 60V

Note that normally we donŌĆÖt know the voltages of the resistors (unless we measure them), and we need to use both KirchhoffŌĆÖs laws for the solution.

KirchhoffŌĆÖs current law (KCL) states that the algebraic sum of all the currents entering and leaving any node in a circuit is zero.

In the following, we give a + sign to currents leaving a node and a - sign to currents entering a node.

HereŌĆÖs a basic example demonstrating KirchhoffŌĆÖs current law. Find the current I2 if the source current IS = 12 A, and I1 = 8 A. Using KirchhoffŌĆÖs current law at the circled node: -IS + I1 + I2 = 0 , hence: I2= IS ŌĆō I1 = 12 ŌĆō 8 = 4 A, as you can check using TINA (next figure).

In the next example, we will use both KirchhoffŌĆÖs laws plus OhmŌĆÖs law to calculate the current and the voltage across the resistors.

In the figure below, you will note the Voltage Arrow above resistors. This is a new component available in Version 6 of TINA and works like a voltmeter. If you connect it across a component, the arrow determines the reference direction (to compare to a voltmeter, imagine placing the red probe at the tail of the arrow and the black probe at the tip). When you run DC analysis, the actual voltage on the component will be displayed on the arrow. Click/tap the circuit above to analyze on-line or click this link to Save under Windows

To begin using KirchhoffŌĆÖs current law, we see that the currents through all the components are the same, so letŌĆÖs denote that current by I.

According to KirchhoffŌĆÖs voltage law: VS = V1+V2+V3

Now using OhmŌĆÖs law: VS=I*R1+ I*R2+I *R3

And from here the current of the circuit:

I=VS /(R1+R2+R3)= 120/(10+20+30) = 2 A

Finally the voltages of the resistors:

V1=I*R1 = 2*10 = 20 V; V2 = I*R2 = 2*20 = 40 V; V3 = I*R3 =2*30 = 60 V

The same results will be seen on the Voltage Arrows by simply running TINAŌĆÖs interactive DC analysis. In this next, more complex circuit, we also use both KirchhoffŌĆÖs laws and OhmŌĆÖs law, but we find that we most solve a linear system of equations.

The total number of independent applications of KirchhoffŌĆÖs laws in a circuit is the number of circuit branches, while the total number of unknowns (the current and voltage of each branch) is twice that . However, by also using OhmŌĆÖs law at each resistor and the simple equations defining the applied voltages and currents, we get a system of equation where the number of unknowns is the same as the number of equations.

Find the branch currents I1, I2, I3 in the circuit below. Click/tap the circuit above to analyze on-line or click this link to Save under Windows

The set of equations follows:

The nodal equation for the circled node:

- I1 - I2 - I3 = 0

or multiplying by -1

I1 + I2 + I3 = 0

The loop equations (using the clockwise direction) for the loop L1, containing V1, R1 and R3

-V1+I1*R1-I3*R3 = 0

and for the loop L2, containing V2, R2 and R3

I3*R3 ŌĆō I2*R2 +V2 = 0

Substituting the component values:

I1+ I2+ I3 = 0 -8+40*I1 ŌĆō 40*I3 = 0 40*I3 ŌĆō20*I2 +16 = 0

Express I1 using the nodal equation: I1 = -I2 ŌĆō I3

then substitute it into the second equation:

-V1 ŌĆō (I2 + I3)*R1 ŌĆōI3*R3 = 0 or ŌĆō8- (I2 + I3)*40 ŌĆō I3*40 = 0

Express I2 and substitute it into the third equation, from which you can already calculate I3:

I2 = - (V1 + I3*(R1+R3))/R1 or I2 = -(8 + I3*80)/40

I3*R3 + R2*(V1 + I3*(R1+R3))/R1 +V2 = 0 or I3*40 + 20*(8 + I3*80)/40 + 16 = 0

And: I3 = - (V2 + V1*R2/R1)/(R3+(R1+R3)*R2/R1) or I3 = -(16+8*20/40)/(40 + 80*20/40)

Therefore I3 = - 0.25 A; I2 = -(8-0.25*80)/40 = 0.3 A and I1 = - (0.3-0.25) = - 0.05 A

Or: I1 = -50 mA; I2 = 300 mA; I3 = -250 mA.

Now letŌĆÖs solve the same equations with TINAŌĆÖs interpreter:

 {Solution by TINA's Interpreter} Sys I1,I2,I3 I1+I2+I3=0 -V1+I1*R1-I3*R3=0 I3*R3-I2*R2+V2=0 end; I1=[-50m] I2=[300m] I3=[-250m]

Finally letŌĆÖs check the results using TINA: Next, letŌĆÖs analyze the following even more complex circuit and determine its branch currents and voltages.

Click/tap the circuit above to analyze on-line or click this link to Save under Windows

LetŌĆÖs denote the unknown voltages and currents by adding voltage and current arrows to components, and also show the loops (L1,L2, L3) and the nodes (N1,N2) where we will use the KirchhoffŌĆÖs equations.

Here is the set of Kirchhoff equations for the loops (using the clockwise direction) and the nodes.

-IL + IR1 - Is = 0 (for N1)

- IR1 + IR2 + Is3 = 0 (for N2)

-Vs1 - VR3 + VIs + VL = 0 (for L1)

-VIs + Vs2 +VR2 +VR1 = 0 (for L2)

-VR2 - Vs2 + Vs3 = 0 (for L3)

Applying OhmŌĆÖs law:

VL = IL*RL

VR1 =IR1*R1

VR2 = IR2*R2

VR3 = - IL*R3

This is 9 unknowns and 9 equations. The easiest way to solve this is to use TINAŌĆÖs

interpreter. However, if we are pressed to use hand calculations, we note that this set of equations can be easily reduced to a system of 5 unknowns by substituting the last 4 equations into the L1, L2, L3 loop equations. Also, by adding equations (L1) and (L2), we can eliminate VIs , reducing the problem to a system of 4 equations for 4 unknowns (IL, IR1 IR2, Is3). When we have found these currents, we can easily determine VL , VR1, VR2, and VR3 using the last four equations (OhmŌĆÖs law).

Substituting VL ,VR1,VR2 ,VR3 :

-IL + IR1 - Is = 0 (for N1)

- IR1 + IR2 + Is3 = 0 (for N2)

-Vs1 + IL*R3 + VIs + IL*RL = 0 (for L1)

-VIs + Vs2 + IR2*R2 + IR1*R1 = 0 (for L2)

- IR2*R2 - Vs2 + Vs3 = 0 (for L3)

Adding (L1) and (L2) we get

-IL + IR1 - Is = 0 (for N1)

- IR1 + IR2 + Is3 = 0 (for N2)

-Vs1 + IL*R3 + IL*RL + Vs2 + IR2*R2 + IR1*R1 = 0 (L1) + (L2)

- IR2*R2 - Vs2 + Vs3 = 0 (for L3)

After substituting the component values, the solution to these equations comes readily.

-IL+IR1 ŌĆō 2 = 0 (for N1)

-IR1 + IR2 + IS3 = 0 (for N2)

-120 - + IL*90 + IL*20 + 60 + IR2*40 + IR1*30 = 0 (L1) + (L2)

-IR2*40 ŌĆō 60 + 270 = 0 (for L3)

from L3 IR2 = 210/40 = 5.25 A (I)

from N2 IS3 ŌĆō IR1 = - 5.25 (II)

from L1+L2 110 IL + 30 IR1 = -150 (III)

and for N1 IR1 ŌĆō IL = 2 (IV)

Multiply (IV) by ŌĆō30 and add to (III) 140 IL = -210 hence IL = - 1.5 A

Substitute IL into (IV) IR1 = 2 + (-1.5) = 0.5 A

and IR1 into (II) IS3 = -5.25 + IR1 = -4,75 A

And the voltages: VR1 = IR1*R1 = 15 V; VR2 = IR2*R2 = 210 V;

VR3 = - IL*R3= 135 V; VL = IL*RL = - 30 V; VIs = VS1+VR3-VL = 285 V

 {Solution of the original equations by TINA's Interpreter} Sys IL,IR1,IR2,Is3,VIs,VL,VR1,VR3,VR2 -IL-Is+IR1=0 -IR1+IR2+Is3=0 -Vs1+VR3+Vis-VL=0 -Vis+VR1+VR2+Vs2=0 -Vs3+VR2+Vs2=0 VR1=IR1*R1 VR2=IR2*R2 VR3=-IL*R3 VL=IL*RL end; IL=[-1.5] IR1=[500m] IR2=[5.25] Is3=[-4.75] VIs= VL=[-30] VR1= VR2= VR3=

Solution of the reduced set of equations using the interpreter:

 {Solution of the reduced set of equations by TINA's Interpreter} Sys Il,Ir1,Ir2,Is3 -Il+Ir1-2=0 -Ir1+Ir2+Is3=0 -120+110*Il+60+40*Ir2+30*Ir1=0 -40*Ir2+210=0 end; Il=[-1.5] Ir1=[500m] Ir2=[5.25] Is3=[-4.75]

We can also enter expressions for the voltages and have TINAŌĆÖs Interpreter calculate them:

 Il:=-1.5; Ir1:=0.5; Ir2:=5.25; Is3:=-4.75; Vl:=Il*RL; Vr1:=Ir1*R1 Vr2:=Ir2*R2; Vr3:=-Il*R3; VIs:=Vs1-Vl+Vr3; Vl=[-30] Vr1= Vr2= Vr3= VIs= We can check the result with TINA by simply turning on TINAŌĆÖs DC interactive mode or using Analysis / DC Analysis / Nodal Voltages SUPERPOSITION THEOREM   NODE POTENTIAL METHOD