Lumped elements and time invariance
Every circuit is, at bottom, an electromagnetic wave problem. The lumped-element model is the approximation that lets you ignore that, and knowing when it stops holding is as important as knowing how to use it.
01Every circuit is, at bottom, a wave problem
Maxwell’s equations make no distinction between a wire and an antenna: any conductor carrying a varying current radiates and propagates fields at the speed of light. Solving a circuit that way —field by field, point by point in space— is exact but, for the vast majority of the circuits that get designed, a waste of effort. The complete circuit theory that begins in this course is a deliberate approximation: one of the great shortcuts of electrical engineering, and it pays to know exactly what is being approximated.
The approximation is called the lumped-element model (lumped elements): instead of fields that vary with position and time, the current is assumed to be the same throughout a conductor at every instant, and an entire resistor, capacitor or inductor can be treated as a single point with one relationship between voltage and current. That assumption is false, strictly speaking, for any real component. It is an excellent approximation when the circuit is much smaller than the wavelength of the signals it handles.
02The lumped-element condition: the λ/10 rule
The practical criterion used throughout engineering: a circuit can be treated as lumped elements if its largest physical dimension is less than one tenth of the wavelength of the signal passing through it. Beyond that, the propagation delay inside the circuit itself is no longer negligible and you need transmission-line theory —part of the courses that follow this one.
| Signal | Frequency | \( \lambda \) | \( \lambda/10 \) | Is a desktop circuit lumped? |
|---|---|---|---|---|
| Household mains | 50 Hz | 6,000 km | 600 km | Without any doubt |
| Audio | 20 kHz | 15 km | 1.5 km | Without any doubt |
| AM radio | 1 MHz | 300 m | 30 m | Yes, with no special care |
| FM / VHF | 100 MHz | 3 m | 30 cm | Yes, but it is already worth taking care with the trace layout |
| Wi-Fi 2.4 GHz | 2.4 GHz | 12.5 cm | 1.25 cm | Not always: a trace several centimeters long is already a transmission line |
| 60 GHz radar (automotive, 5G mmWave) | 60 GHz | 5 mm | 0.5 mm | No: any trace longer than half a millimeter needs transmission-line design |
The same physical board, with the same layout, can be “a circuit” at 1 MHz and “an accidental antenna” at 2.4 GHz. The model does not change because the designer decides so: it changes because the ratio between the physical size and the wavelength has changed.
03What it means for a circuit to be time-invariant
A system is time-invariant if its behavior does not depend on when it is excited: applying the same input today or tomorrow gives exactly the same output, only shifted in time. Formally, if an input \( x(t) \) produces an output \( y(t) \), a time-invariant system responds to \( x(t-t_0) \) with \( y(t-t_0) \), for any shift \( t_0 \).
| Circuit | Time-invariant? | Why |
|---|---|---|
| An RC filter with fixed components | Yes | The values of \( R \) and \( C \) do not change with time |
| A potentiometer someone is turning | No | The value of the resistance depends on the instant at which you look at it |
| A mixer with a sampling clock | No | It multiplies the signal by something that varies periodically in time, on purpose |
| An amplifier that warms up and drifts | Approximately yes, in the short term | Thermal drift is a change of parameters that is slow compared with the signal being amplified |
Almost all of this course —and Circuit Theory II, which follows— assumes linear time-invariant (LTI) circuits, because that is the class of systems where the frequency response, the Laplace transform and systematic solution methods work without caveats. A circuit that changes its parameters on purpose —like the mixer— is a tremendously valuable tool, but it belongs to a different kind of analysis.
04When the ideal model breaks down: self-resonance
An ideal capacitor is a lumped element by definition: its impedance is \( 1/j\omega C \), which falls with frequency without limit. A real capacitor does not. Its leads and its internal construction add a parasitic series inductance, \( L_{ESL} \), and a parasitic series resistance, \( R_{ESR} \). The result is a series RLC circuit, not a pure capacitor, with a self-resonant frequency above which the capacitor starts to behave, electrically, like an inductor.
You choose the capacitance and the package type —each with its typical parasitic inductance— and the plot shows the real impedance against the ideal one, on a logarithmic scale.
05Where it shows up in electronics
This is the real reason, not an anecdote, that computer and microcontroller boards carry several decoupling capacitors in parallel with different values —typically 10 µF, 100 nF and sometimes 1 nF right next to the chip itself—: each value has its own \( f_0 \), and together the three cover a much wider frequency range than any one alone. The one with the largest capacitance filters the low-frequency noise; the smallest, with a lower relative \( L_{ESL} \), is still capacitive at the highest frequencies, where the other one has already stopped being useful.
The λ/10 rule of section 2 and the self-resonance of this section are the same idea applied at two different scales. One says when the whole circuit stops being “a point” compared with the signal; the other, when a single component stops behaving like the ideal element that its symbol says it is. Both are the same question: is the lumped-element approximation still valid here?
06In the lab
Calculate the wavelength of a 433 MHz signal (the remote-control band) and determine whether a board 8 cm on a side can be treated as lumped elements at that frequency.
Using this page’s lab, for a microcontroller that switches at 48 MHz, determine which combination of capacitance and package keeps the capacitor in its capacitive region at that frequency and at its third harmonic.
Explain why a circuit with a relay that opens and closes during the analysis cannot be treated as time-invariant, and which tools of this course stop applying while the relay is switching.
07Common mistakes
- Thinking a circuit “is” or “is not” lumped based on its physical size alone. It depends on the ratio between the size and the wavelength of the signal, not on the size in isolation.
- Assuming a capacitor conducts better and better at higher frequency, without limit. Only up to its self-resonance; above it, it behaves like an inductor.
- Choosing the decoupling capacitor by its capacitance alone, ignoring the package. The package determines the parasitic inductance, which determines up to what frequency it remains useful.
- Confusing time invariance with the circuit “never changing.” It means that its parameters do not change during the interval of interest, nor as a function of the signal being analyzed.
- Applying LTI circuit tools —phasors, Laplace— to a circuit with an element that varies in time on purpose. A mixer or a circuit with a sampling clock needs a different treatment.
08Self-assessment
What does the lumped-element model approximate?
It ignores the propagation of electromagnetic fields inside the circuit itself, treating the current as uniform throughout a conductor at every instant.
What is the practical rule for knowing whether a circuit is lumped?
That its largest physical dimension be much smaller than one tenth of the wavelength of the signal passing through it.
What does it mean for a system to be time-invariant?
That its response to an input does not depend on when that input is applied: the same input, shifted in time, gives the same output shifted by the same amount.
Why does a real capacitor stop behaving like a capacitor at high frequency?
Because its parasitic series inductance dominates above its self-resonant frequency, and from there the impedance starts to grow with frequency instead of falling.
What determines the self-resonant frequency of a real capacitor?
Its capacitance \( C \) and its parasitic series inductance \( L_{ESL} \), according to \( f_0 = 1/(2\pi\sqrt{L_{ESL}C}) \).
Why does a board usually carry several decoupling capacitors of different values?
Because each value has its own self-resonant frequency, and combining them covers a wider frequency range than any one alone.
09Further reading
- Charles K. Alexander and Matthew N. O. Sadiku. Fundamentals of Electric Circuits. 7th ed., McGraw-Hill, 2021. The reference book for this whole course: the introductory chapter develops the lumped-element model with the same approach as this page.
- David M. Pozar. Microwave Engineering. 4th ed., Wiley, 2011. The formal treatment of when circuit theory stops being enough and transmission-line theory is needed, on the other side of the λ/10 rule.
- Henry W. Ott. Electromagnetic Compatibility Engineering. Wiley, 2009. The chapter on power-supply decoupling explains in detail, with manufacturer data, why capacitors of different value and package are combined.