#04 Kirchhoff's voltage and current laws — worked examples

Below are all the lines written in the notebook together with the full narration transcript.

1. 1. KVL rule

   

   Closed loop: ∑v_k = 0

   − to + across a source → rise (+)

   With current through R → drop (−)

   Voltage rises = voltage drops

   Narration transcript

   Kirchhoff's Voltage Law, or KVL, states that the sum of all voltages around any closed loop equals zero. In other words, if you start at any point in a circuit and travel around a complete loop back to the same point, the voltage rises must equal the voltage drops. We write this as: the summation of V around a loop equals zero. Convention: when we travel through a source from minus to plus, that's a voltage rise, so we write it as positive. When we travel through a resistor in the direction of current, that's a voltage drop, so we write it as negative.

2. 2. KVL worked loop

   

   $24 - 3I - 5I - 4I = 0$

   $24 - 12I = 0$

   $I = 2 A$

   $V_{1} = 6 V, V_{2} = 10 V, V_{3} = 8 V$

   $6 + 10 + 8 = 24 V$

   Narration transcript

   Let's apply KVL to a simple series circuit. We have a twenty four volt source connected to three resistors: three ohms, five ohms, and four ohms, all in series. Since they're in series, the same current I flows through all of them. Applying KVL around the loop: twenty four minus three I minus five I minus four I equals zero. Combining: twenty four minus twelve I equals zero. So I equals twenty four over twelve, which is two amperes. Now we can find each voltage: V one equals I times three, which is six volts. V two equals I times five, which is ten volts. V three equals I times four, which is eight volts. Let's verify: six plus ten plus eight equals twenty four. KVL checks out!

3. 3. KCL rule

   

   At a node: ∑i_k = 0

   Entering currents (+)

   Leaving currents (−)

   Total entering = total leaving

   Basis: conservation of charge

   Narration transcript

   Kirchhoff's Current Law, or KCL, states that the sum of all currents entering a node equals the sum of all currents leaving that node. Equivalently, the algebraic sum of all currents at a node is zero. We write this as: the summation of I at a node equals zero. Convention: currents entering the node are positive, currents leaving the node are negative. Or you can use the opposite convention, as long as you're consistent. KCL is based on conservation of charge: charge cannot accumulate at a node, so whatever flows in must flow out.

4. 4. KCL worked node

   

   Node a: i₁, i₂ enter

   i₃, i₄ leave

   $i_{1} + i_{2} - i_{3} - i_{4} = 0$

   $i_{1} + i_{2} = i_{3} + i_{4}$

   Check: 2 − 4 + 2 = 0

   Narration transcript

   Let's apply KCL at node a. Four currents meet at this node. Using our convention: entering is positive, leaving is negative. Currents i one and i two enter the node. Currents i three and i four leave the node. Writing KCL at node a: plus i one plus i two minus i three minus i four equals zero. We can rearrange this as: i three plus i four equals i one plus i two. In words: total current out equals total current in. You can write KCL at every node in any circuit. For example, at another node with known values: two amperes minus four amperes plus two amperes equals zero. The sum is indeed zero, confirming KCL.

5. 5. Conservation summary

   KVL → voltage balance

   KCL → current balance

   Ohm's law → element relation

   Ohm + KCL + KVL → circuit analysis

   Narration transcript

   Let's recap Kirchhoff's two laws. KVL says the sum of voltages around any closed loop is zero. Voltage rises equal voltage drops. KCL says the sum of currents at any node is zero. Currents entering equal currents leaving. Together with Ohm's law, these two laws give us everything we need to analyze any linear circuit. In the next video, we'll explore voltage and current dividers in more detail.

Source video: Circuit Theory #04 — Kirchhoff's Laws (KVL & KCL) with Solved Examples (3:52)