Difficulty: Intermediate | Prerequisites: Ohm's Law, basic series and parallel resistor combinations, understanding of voltage and current.
This lecture pulls together everything you have learnt about resistors, voltage sources, and Ohm's Law into a single systematic framework for analysing real circuits. Up to now you could handle one-loop circuits by combining resistors and applying V = IR. That stops working the moment a circuit has multiple loops or multiple batteries. Kirchhoff's two laws, and the five-step method covered here, are how you solve those more complex circuits. You will use this approach for virtually every DC circuit problem from here on, and it reappears in AC analysis later in the course.
Kirchhoff's Voltage Law (KVL) says the voltage changes around any closed loop sum to zero. Kirchhoff's Current Law (KCL) says the total current entering a junction equals the total current leaving it. Together with a systematic labelling method, these two rules let you write enough equations to solve for every unknown current and voltage in any DC circuit, including circuits with real (non-ideal) batteries.
Kirchhoff's Voltage Law (KVL)
The sum of all voltage changes (rises and drops) around any closed loop in a circuit equals zero: ΣΔV = 0. In simple terms, this means that if you walk around a loop and add up every voltage gain and loss, you end up back where you started, so the total must be zero.
Kirchhoff's Current Law (KCL)
The total current flowing into any junction equals the total current flowing out: ΣI_in = ΣI_out. Think of it as conservation of charge at a junction: current cannot pile up or vanish at a node.
Junction (node)
A point in a circuit where three or more wires meet. This is where you apply KCL.
Loop
Any closed path through a circuit. You apply KVL around each loop.
Ideal battery
A voltage source with zero internal resistance. The terminal voltage equals the EMF regardless of the current drawn: V_L = V₀.
Real battery (non-ideal battery)
A voltage source modelled as an ideal EMF in series with a small internal resistance r. The terminal voltage drops below the EMF when current flows: V_L = V₀ – Ir. In simple terms, this is why a battery "sags" under heavy load.
Internal resistance (r)
The small resistance inside a real battery that causes its terminal voltage to decrease as current increases. This is what limits the current a real battery can deliver, even into a short circuit.
Voltage sagging
The drop in terminal voltage that occurs when a real battery supplies large currents. The battery "can't supply too much current with load without voltage sagging."
The current through each resistor is the same: I₁ = I₂
Total resistance adds: R_total = R₁ + R₂ + ...
Voltage divides across the resistors in proportion to their resistances
The voltage across each resistor is the same: V₁ = V₂
Reciprocal rule for total resistance: 1/R_total = 1/R₁ + 1/R₂ + ...
Current divides between branches, with more current through the smaller resistor
For any closed loop: ΣΔV = 0
Every element in the loop either raises or drops the voltage
A battery traversed from – to + is a voltage rise (+V₀); from + to – is a drop (–V₀)
A resistor traversed in the direction of current is a voltage drop (–IR); against the current is a rise (+IR)
You can write one KVL equation per independent loop
At any junction: ΣI_in = ΣI_out
Example from the lecture: at a node where three branches meet, I₁ = I₂ + I₃
You get one KCL equation per junction (minus one, since the last junction's equation is redundant)
This is the systematic procedure from the lecture. Follow it exactly and you will always generate enough equations.
Label all currents. Assign a variable (I₁, I₂, I₃, ...) and draw an arrow for each branch. The direction you choose is a guess; if the answer comes out negative, the actual current flows the other way.
Label +/– signs on every element.
For resistors: current enters the + side and exits the – side (voltage drops in the direction of current)
For batteries: the long side of the symbol is +
Choose loops and traversal directions. Pick enough independent loops to cover every branch. Start somewhere on the loop and walk around in one direction (clockwise or anticlockwise, your choice).
Write voltage drops around each loop. As you traverse the loop, record the first sign you hit on each element. If you hit + first, that element contributes a positive term; if you hit – first, it contributes a negative term.
Write the node equations. Apply KCL at each junction. Together with your loop equations, you should now have as many equations as unknowns.
Ideal battery:
Terminal voltage equals EMF: V_L = V₀
Current through the load: I = V₀ / R
As load resistance R → 0, current I → ∞ (physically unrealistic, which is why real batteries matter)
Real battery:
Terminal voltage is less than EMF: V_L < V₀
Current through the load: I = V₀ / (R + r)
As load resistance R → 0, current remains finite: I_max = V₀ / r
The voltage the load actually sees drops as current increases, which is the "sagging" effect
This is why a battery that reads 1.5 V on a voltmeter (no load) may deliver noticeably less voltage when powering a device
The lecture works through a circuit with three branches, multiple batteries (E₁, E₃), and multiple resistors (R₁, R₄, R₅). Applying the method above:
KVL around one loop gives: –E₁ + I₁R₁ + E₃ – I₄R₄ + I₅R₅ = 0
KCL at the junction gives: I₂ + I₃ = I₁
A second example with three loops (R₁/V₁, R₂/V₂, R₃/V₃) shows how to generate a full system of equations:
Loop 1: 0 = –V₂ – I₂R₂ – I₁R₁ + V₁
Loop 2: 0 = –V₃ + I₃R₃ + I₂R₂ + V₂
Loop 3: 0 = –V₁ + I₁R₁ – I₃R₃ + V₃ = 0
Junction: I₂ = I₃ + I₁
This gives four equations and three unknown currents, which is one more than needed (the extra loop equation is a linear combination of the other two). You solve the system with standard algebra or matrix methods.
Formula | Meaning |
|---|---|
ΣΔV = 0 | KVL: voltage changes around a closed loop sum to zero |
ΣI_in = ΣI_out | KCL: current into a node equals current out |
R_series = R₁ + R₂ + ... | Total resistance in series |
1/R_parallel = 1/R₁ + 1/R₂ + ... | Total resistance in parallel |
I = V₀ / R | Current from an ideal battery |
I = V₀ / (R + r) | Current from a real battery with internal resistance r |
V_L = V₀ – Ir | Terminal voltage of a real battery under load |
Every circuit you encounter in practice, from phone chargers to car electrical systems, involves multiple loops and real (non-ideal) components. The internal resistance model explains why car batteries struggle in cold weather (internal resistance rises) and why you should not short-circuit a battery (the current is limited only by r, which can still be dangerously large). Voltage sagging is the reason USB chargers have voltage regulation circuitry: without it, plugging in a device would cause the output voltage to drop.
"The direction I pick for current must be correct." It does not need to be. If you guess wrong, the algebra simply gives a negative value, which tells you the current flows opposite to your arrow. The method works regardless of your initial guess.
"KVL only works for simple single-loop circuits." KVL works for any closed loop in any circuit, no matter how many loops or branches exist. That is precisely what makes it powerful.
"An ideal battery can supply infinite current in practice." The ideal battery is a model. Real batteries always have internal resistance, which limits the maximum current to V₀ / r. The ideal model is useful for simplifying calculations when the load resistance is much larger than r.
"Voltage sagging means the battery is dead." Sagging is normal behaviour for any real battery under load. A battery can sag noticeably and still have plenty of charge remaining; the sag is caused by internal resistance, not by depletion.
⚠️ The five-step method is the standard framework for multi-loop circuit problems. Expect at least one problem on any exam covering this material.
⚠️ Sign errors in KVL loops are the most common source of mistakes. Be methodical: always record the first sign you encounter on each element as you traverse the loop.
⚠️ Know the difference between ideal and real battery equations. A problem that gives you internal resistance is asking you to use I = V₀ / (R + r), not I = V₀ / R.
⚠️ KCL junction equations are easy marks. Write them first; they are simpler than loop equations and constrain your unknowns immediately.
True or False: If you assume a current flows clockwise and get a negative answer, you made an error and must redo the problem.
False. A negative current simply means the actual direction is opposite to your assumption.
Fill in the blank: The terminal voltage of a real battery under load is V_L = ______.
V₀ – Ir
True or False: KVL states that the sum of all currents at a junction is zero.
False. That is KCL. KVL states that the sum of voltage changes around a closed loop is zero.
Fill in the blank: For a real battery with EMF V₀ and internal resistance r, the maximum possible current (when R = 0) is ______.
V₀ / r
True or False: You need exactly one KVL equation per branch in a circuit.
False. You need one KVL equation per independent loop, and one KCL equation per independent junction, such that the total number of equations matches the number of unknown currents.
Q: A 12 V battery with internal resistance 0.5 Ω is connected to a 5.5 Ω resistor. What is the current through the circuit, and what is the terminal voltage of the battery?
A: I = 12 / (5.5 + 0.5) = 2.0 A. Terminal voltage V_L = 12 – (2.0)(0.5) = 11.0 V.
Q: At a junction, three wires carry currents I₁ = 3 A (into the junction), I₂ = 1 A (out of the junction), and I₃ (unknown direction). What is I₃, and which way does it flow?
A: By KCL, 3 = 1 + I₃, so I₃ = 2 A flowing out of the junction.
Q: You traverse a loop and encounter a 9 V battery from + to –, then a 4 Ω resistor carrying 2 A in the same direction you are walking. Write the KVL terms for these two elements.
A: The battery contributes –9 V (you hit + first, then –, so it is a drop). The resistor contributes –(2)(4) = –8 V (traversed in the direction of current, so it is a drop). Together: –9 – 8 = –17 V from these two elements.
Q: Why does a real battery's terminal voltage decrease as you draw more current from it?
A: The internal resistance r causes a voltage drop Ir inside the battery itself. As I increases, so does Ir, leaving less voltage available at the terminals: V_L = V₀ – Ir.
Q: A circuit has three unknown currents. How many independent equations (loop + junction) do you need to solve for all of them?
A: Three independent equations. Typically this is two KVL loop equations and one KCL junction equation, though the exact split depends on the circuit topology.
This material connects directly to RC circuits (coming up next), where you add capacitors to Kirchhoff loops and the equations become differential equations instead of algebraic ones. The five-step method also extends to AC circuits later in the course, where impedances replace resistances but the loop and node structure is identical. If you are comfortable with the systematic approach here, AC analysis will feel like a natural extension rather than a new topic.
Kirchhoff's voltage law, KVL, Kirchhoff's current law, KCL, junction rule, loop rule, node analysis, mesh analysis, multi-loop circuits, real battery, ideal battery, internal resistance, terminal voltage, voltage sag, voltage sagging, series resistors, parallel resistors, DC circuit analysis, University Physics Electricity and Magnetism, PHYS 212, UIUC