An Illustrated Guide to Kirchhoff’s Laws

Understanding the fundamental principles of circuit analysis.

Kirchhoff’s Current Law (KCL)

The Principle: The algebraic sum of currents entering a node (or junction) is zero.

In simple terms: What goes in must come out. The total current flowing into any point in a circuit must equal the total current flowing out of that point.

Analogy: Think of it like water pipes connected at a junction. The total amount of water flowing into the junction per second must be the same as the total amount flowing out.

Formula: Σ I_in = Σ I_out

KCL Examples (15 Examples)

Example 1: Find I_x

In: 2A

Out: 3A, I_x

2 = 3 + I_x

I_x = -1A (meaning 1A flows in)

Example 2: Find I_x

_x 1A

In: 5A, 2A

Out: I_x, 1A

5 + 2 = I_x + 1

I_x = 6A

Example 3: Find I_a

_a 4A 3A

In: 10A, I_a, 4A

Out: 3A

10 + I_a + 4 = 3

I_a = -11A (11A flows out)

Example 4: Find I₁

₁ 7A 12A

In: 12A

Out: I₁, 7A

12 = I₁ + 7

I₁ = 5A

Example 5: Find I_y

In: 6A

Out: 2A, I_y

6 = 2 + I_y

I_y = 4A

Example 6: Find I_z

In: 15A

Out: 8A, I_z

15 = 8 + I_z

I_z = 7A

Example 7: All currents leaving

In: 0

Out: 2A, 5A, I_x

0 = 2 + 5 + I_x

I_x = -7A (7A flows in)

Example 8: All currents entering

In: 3A, 4A, I_x

Out: 0

3 + 4 + I_x = 0

I_x = -7A (7A flows out)

Example 9: Find I_b

_b 5A

In: 2A

Out: 1A, I_b, 5A

2 = 1 + I_b + 5

I_b = -4A (4A flows in)

Example 10: Find I_c

_c 8A

In: 8A, 8A

Out: I_c

8 + 8 = I_c

I_c = 16A

Example 11: Find I_d

_d 3A 9A

In: I_d, 9A

Out: 3A

I_d + 9 = 3

I_d = -6A (6A flows out)

Example 12: Find I_e

_e 12A

In: 12A

Out: I_e, 12A

12 = I_e + 12

I_e = 0A

Example 13: Find I_f

In: 2A, 3A

Out: 4A, I_f

2 + 3 = 4 + I_f

I_f = 1A

Example 14: Find I_g

_g 5A 5A

In: I_g

Out: 5A, 5A

I_g = 5 + 5

I_g = 10A

Example 15: Find I_h

In: 6A, 2A

Out: I_h

6 + 2 = I_h

I_h = 8A

Kirchhoff’s Voltage Law (KVL)

The Principle: The algebraic sum of all voltages around any closed loop in a circuit is equal to zero.

In simple terms: The voltage drops equal the voltage rises in any closed path. If you start at one point in a loop and measure the voltage changes as you go around, you’ll end up with the same voltage you started with.

Analogy: Imagine a roller coaster. You start at the bottom, the lift hill (voltage source) takes you up (voltage rise). As you go through the loops and turns (resistors), you lose that height (voltage drops), and you end up back at the bottom where you started.

Formula: Σ V = 0 or Σ V_rises = Σ V_drops

KVL Examples (15 Examples)

Example 16: Find V_R2

_R1=5V R2V_R2=?

Loop direction: Clockwise from source.

+12V – V_R1 – V_R2 = 0

+12V – 5V – V_R2 = 0

V_R2 = 7V

Example 17: Find V_S

_S=? R1=3V R2=6V R3=2V

Loop direction: Clockwise.

+V_S – 3V – 6V – 2V = 0

V_S – 11V = 0

V_S = 11V

Example 18: Find V_R3

Loop direction: Clockwise.

+24V – 10V – 8V – V_R3 = 0

6V – V_R3 = 0

V_R3 = 6V

Example 19: Opposing Source

Loop direction: Clockwise.

+20V – 8V + 5V – V_R2 = 0

17V – V_R2 = 0

V_R2 = 17V

Example 20: Find V_R1

Loop direction: Clockwise.

+9V – V_R1 – 4V = 0

5V – V_R1 = 0

V_R1 = 5V

Example 21: Aiding Sources

Loop direction: Clockwise.

+10V – V_R1 – 5V – 12V = 0

-7V – V_R1 = 0

V_R1 = -7V (Polarity is reversed)

Example 22: Find V_S2

_S2=? R2=15V

Loop direction: Clockwise.

+50V – 20V + V_S2 – 15V = 0

15V + V_S2 = 0

V_S2 = -15V

Example 23: Find V_R2

Loop direction: Clockwise.

+18V – V_R2 – 18V = 0

-V_R2 = 0

V_R2 = 0V

Example 24: Find V_S (Counter-Clockwise)

_S=? R1=7V R2=3V

Loop direction: Counter-Clockwise.

+3V + 7V – V_S = 0

10V – V_S = 0

V_S = 10V

Example 25: Find V_x

_x R3=15V

Loop direction: Clockwise.

+40V – 15V – V_x – 15V = 0

10V – V_x = 0

V_x = 10V

Example 26: Unknown Source Polarity

_S R1=5V R2=5V

Assume V_S is a rise (CW). Current flows from R2 to R1.

+V_S – 5V – 5V = 0

V_S = 10V. Positive result means assumed polarity was correct (+ on top).

Example 27: Find V_R3

Loop direction: Clockwise.

+100V – 25V – 50V – V_R3 = 0

25V – V_R3 = 0

V_R3 = 25V

Example 28: Find V_S1

_S1=? R1=12V –+ 3V R2=9V

Loop direction: Clockwise.

+V_S1 – 12V + 3V – 9V = 0

V_S1 – 18V = 0

V_S1 = 18V

Example 29: Find V_R1

Loop direction: Clockwise.

+15V – V_R1 – 5V – 5V = 0

5V – V_R1 = 0

V_R1 = 5V

Example 30: Find V_R2

Loop direction: Clockwise.

-6V – 9V – V_R2 = 0

-15V – V_R2 = 0

V_R2 = -15V (Polarity is reversed)

Kirchhoff’s Current Law (KCL)

KCL Examples (15 Examples)

Example 1: Find Ix

Example 2: Find Ix

Example 3: Find Ia

Example 4: Find I1

Example 5: Find Iy

Example 6: Find Iz

Example 7: All currents leaving

Example 8: All currents entering

Example 9: Find Ib

Example 10: Find Ic

Example 11: Find Id

Example 12: Find Ie

Example 13: Find If

Example 14: Find Ig

Example 15: Find Ih

Kirchhoff’s Voltage Law (KVL)

KVL Examples (15 Examples)

Example 16: Find VR2

Example 17: Find VS

Example 18: Find VR3

Example 19: Opposing Source

Example 20: Find VR1

Example 21: Aiding Sources

Example 22: Find VS2

Example 23: Find VR2

Example 24: Find VS (Counter-Clockwise)

Example 25: Find Vx

Example 26: Unknown Source Polarity

Example 27: Find VR3

Example 28: Find VS1

Example 29: Find VR1

Example 30: Find VR2

Example 1: Find I_x

Example 2: Find I_x

Example 3: Find I_a

Example 4: Find I₁

Example 5: Find I_y

Example 6: Find I_z

Example 9: Find I_b

Example 10: Find I_c

Example 11: Find I_d

Example 12: Find I_e

Example 13: Find I_f

Example 14: Find I_g

Example 15: Find I_h

Example 16: Find V_R2

Example 17: Find V_S

Example 18: Find V_R3

Example 20: Find V_R1

Example 22: Find V_S2

Example 23: Find V_R2

Example 24: Find V_S (Counter-Clockwise)

Example 25: Find V_x

Example 27: Find V_R3

Example 28: Find V_S1

Example 29: Find V_R1

Example 30: Find V_R2