Circuit Analysis: Why the Lights Dim When the Kettle Starts

You switch the kettle on and, for an instant, the kitchen lights dip. Nothing is wrong. The kettle draws about 10 amps, those amps flow through the same wires that feed the lights, and every wire has a little resistance, so a few volts quietly go missing on the way. The lights see the leftover.

Feel your phone charger after an hour: warm. Some of the energy it handles never reaches the battery; it turns into heat in the charger's own resistances. If a 20 W charger is 90% efficient, 2 W of heat is the price. Both mysteries have the same answer, and it fits on a postcard. That postcard is circuit analysis.

Ohm and the heat in a wire

Picture water in a pipe. Voltage is the pressure pushing, current is how much flows past a point each second, and resistance is how narrow and rough the pipe is. A narrow pipe needs more pressure for the same flow. Georg Ohm published in 1827 that for metals the relation is proportional.

The second line is the one a graduate carries: heat in a resistance grows with the square of the current. Double the current and you get four times the heat. A 60 W load at 5 V needs 12 A. Push that through a cable of 0.2 Ω and the cable burns 12² × 0.2 = 28.8 W, nearly half the power you meant to deliver. The same 60 W at 20 V needs 3 A and wastes 3² × 0.2 = 1.8 W, 3%. This is why fast chargers negotiate their voltage upward: a fixed power, a lower current, less heat in the cable.

Kirchhoff's laws, series and parallel, and the divider

Gustav Kirchhoff published two rules in 1845, and every circuit calculation is built from them. Both are bookkeeping. Current law: charge does not pile up at a junction, so the current in equals the current out. Voltage law: go once around any loop and the voltage rises and falls add up to zero, because a point can only have one voltage.

Two resistors in series share one current, so their resistances add and the voltage splits in proportion. Two in parallel share one voltage, so the currents add, and the combined resistance is always smaller than either. Series is a queue; parallel is extra checkout lanes.

With 5 V across two equal 10 kΩ resistors, the middle is 2.5 V. That is how a sensor, a volume knob or a battery-level gauge turns a resistance into a voltage. Now the catch every beginner meets once: connect something to the middle and it joins in parallel with R₂. A 10 kΩ load turns the lower half into 5 kΩ, and the output falls to 1.67 V, a third lower. Measuring or loading a node changes it.

Node analysis: one equation per junction

Real circuits are not one divider. For the rest, the method is mechanical and it is the one every program uses. Pick a ground, call it 0 V. Give every other junction a voltage, an unknown. Then write Kirchhoff's current law at each: the currents leaving through every resistor add to zero (or to whatever a current source pushes in). Each current is a voltage difference divided by a resistance, so every equation is linear.

That gives one equation per unknown, which is a system you solved in Linear Algebra (34/04). Fifty nodes means fifty equations in a matrix, and the circuit simulator that designs a chip solves exactly that, millions of times. The matrix is symmetric, with each node's total conductance on the diagonal and the minus-conductances between neighbours off it: you can write it by looking at the picture.

In the circuit below, with Vs = 12 V and a 2 mA source, the two equations (in mS and mA) are 2A − 0.5B = 12 and −0.5A + 1.5B = 2. They give A = 6.91 V and B = 3.64 V. The readout checks that the currents leaving each node cancel.

Thévenin, Norton and maximum power

Here is the idea that makes big circuits manageable. Take any pile of sources and resistors with two terminals sticking out. From outside, the whole pile behaves like one ideal voltage source V_th in series with one resistance R_th. Léon Thévenin published this in 1883 (Hermann von Helmholtz had it earlier, in 1853), and the Norton version, a current source in parallel, is the same thing turned over. You find V_th by asking what the terminals read with nothing attached, and R_th by asking how much the voltage falls when you draw current.

The wall socket is the clearest example. Its Thévenin voltage is 230 V (or 120 V) and its Thévenin resistance is the wiring back to the transformer, plausibly a few tenths of an ohm. Flicker standards test appliances against a reference loop impedance of about 0.4 Ω plus a small reactive part (IEC 61000-3-3, as listed by test-equipment makers); here we use 0.5 Ω. A 2.4 kW kettle is 22 Ω, so it draws 230 ÷ 22.5 = 10.2 A. The share of the 230 V that reaches the kettle is 22 ÷ 22.5, so the kettle sees about 225 V and everything else on that wire loses the missing 5 V while it runs. That is the dimming lights.

There is a second lesson hiding here. How much power can a source with internal resistance give to a load? Too small a load and the current is large but the load's voltage is tiny; too large and the voltage is high but the current vanishes. The best is the middle:

At that match half the power is wasted inside the source. A radio receiver or a speaker amplifier chasing every last bit of a weak signal accepts this trade. A power station never does: it wants a load far larger than its own resistance so that almost nothing is lost, which is also why the mains is built with the lowest resistance the money allows.

Capacitors, inductors and the RC transient

Resistors have no memory. A capacitor stores charge, and its current is proportional to how fast its voltage changes (I = C dV/dt). An inductor stores energy in a magnetic field, and its voltage is proportional to how fast its current changes (V = L dI/dt). Put one of them with a resistor and Kirchhoff's law becomes a differential equation, the first one you solved in Differential Equations (34/03). The answer is the exponential everyone learns once:

The capacitor reaches 63% of the way in one τ, 95% in three, 99% in five. A 10 kΩ resistor with 100 µF has τ = 1 second. A capacitor charged to 5 V stores ½CV² = 1.25 mJ, and the resistor turns exactly as much into heat while charging it, whatever the resistance. That half-and-half loss is also why switching chips cost power (see Electronics, 34/35).

Feed the same RC a square wave and it becomes a filter. When τ is much shorter than a half-period the output follows the wave. When τ is much longer the capacitor never catches up and you get a nearly flat level: slow changes pass, fast ones are averaged away.

AC, phasors and resonance

The mains and every radio signal are sine waves, and a sine wave through an RLC circuit comes out a sine wave of the same frequency, just scaled and shifted in time. That changes the problem: you no longer need a differential equation, only the size and the shift. Engineers write that pair as a single arrow, a phasor, and the arrow's behaviour has a name, impedance.

Every resistor, coil and capacitor now obeys Ohm's law with a complex number. A coil's reactance ωL grows with frequency; a capacitor's 1 ÷ ωC shrinks. In a series RLC they point in opposite directions, so at one frequency they cancel completely. There the impedance is only R, the current peaks, and the circuit rings at its natural frequency:

With L = 10 mH and C = 1 µF, f₀ is 1,592 Hz. Make R only 10 Ω and Q is 10: the circuit answers loudly within about 160 Hz of that note and ignores everything else. This is a radio tuner, and it is also a filter: take the output across the resistor and you have a band-pass (the tuner); across the capacitor, a low-pass; across the coil, a high-pass. It is the mechanical mass-and-spring of Vibrations (34/17) in another medium, with Q playing the part of the damping ratio.

What a graduate just sees

• Heat is I² × R. Doubling the current quadruples the loss, so power travels at high voltage and low current.
• A source is never ideal: every battery, socket and sensor has an internal resistance, so loading it lowers its voltage.
• Series shares current, parallel shares voltage; parallel resistance is always smaller than the smallest branch.
• A circuit with n unknown voltages is n linear equations: write conductances down the diagonal.
• Any two-terminal mess is a Thévenin source. Matching the load to it gives maximum power and 50% efficiency.
• RC means τ = RC: 63% in one, 95% in three, 99% in five time constants.
• A coil and a capacitor trade energy back and forth at f₀; Q says how long it rings and how narrow the filter is.
• Impedance is resistance that depends on frequency, which is why the same circuit can pass bass and block treble.

Where it shows up next

• Electronics (34/35): diodes and transistors are added to the toolbox; the analysis is the same, with an extra source whose value depends on a voltage somewhere else.
• Signals & Systems (34/37): the phasor idea, one number for size and shift, grows into the Fourier and Laplace transforms; an RC filter is the first example of a system.
• Power Systems & Electric Machines (34/39): three-phase power and power factor are phasors with a bigger vocabulary; the kettle's voltage dip is how a grid engineer thinks about weak versus strong supplies.
• Vibrations (34/17) and Control Systems (34/20): the RLC circuit is the same differential equation as the mass-spring-damper, so what you learned there transfers in both directions.
• The real world: phone chargers, speaker crossovers, guitar tone knobs, radio tuners, the pull-up resistor next to every microcontroller pin, and the fuse in your plug.

If you want to go deeper

• 📘 Textbook: James Nilsson and Susan Riedel, Electric Circuits (Pearson; 12th edition): a long-standing classroom text, thorough on node and mesh methods.
• 📘 Textbook: Charles Alexander and Matthew Sadiku, Fundamentals of Electric Circuits (McGraw-Hill): many worked examples, widely used internationally.
• 📘 Gentler: Paul Horowitz and Winfield Hill, The Art of Electronics (Cambridge University Press; 3rd edition, 2015), whose first chapters are the clearest practical treatment of these ideas. Also Ben Eater's free video series on building circuits from scratch.
• ▶️ Free course: MIT OpenCourseWare 6.002 Circuits and Electronics (Spring 2007): the course the electrical engineering major usually opens with.
• 🌐 Easier first read in this library: Physics II: Electricity & Magnetism (34/07) for what charge and voltage are.