Circuit models

W. Robert J. Funnell
Dept. BioMedical Engineering, McGill University

1. Lumped models

Type of differential equations Example
Lumped systems Ordinary R-L-C circuits
Distributed systems Partial Electromagnetic fields

In a ‘lumped’ model, the system characteristics are lumped into idealized discrete components with no (or negligible) spatial extent.

The only differentiation is with respect to time. Generic circuit

There is a well-developed theory for lumped-circuit analysis, originally developed for electrical circuits.

The foundations are

R-L-C components

The basic components are linear and time-invariant:

There are also voltage sources, current sources and transformers.

The components can also be nonlinear and/or time-varying.

2. Analogies

Analogies among electrical, mechanical & acoustical circuits:
Electrical Mechanical Acoustical
v voltage f force p pressure
i current u velocity U volume velocity
R resistance R resistance R resistance
L inductance m mass M mass
C capacitance 1/k compliance (spring) C compliance (volume)
v = iR f = Ru p = RU
v = L di/dt f = m du/dt p = M dU/dt
v =  1


i dt
f = k
u dt
p =  1


U dt

Transformers are required to convert between different domains in a circuit model.
Electrical Mechanical Acoustical
voltage force pressure
current velocity vol. velocity
resistor dashpot mesh
inductor mass tube
capacitor spring volume

Why does one circuit seem to be in parallel while the other two are in series?

In an electrical circuit, which is easier to measure: voltage or current?

In a mechanical circuit, which is easier to measure: force or velocity?

The electrical/mechanical analogy is sometimes made the other way around, by associating voltage with velocity rather than with force, and current with force rather than with velocity. The electrical/acoustical analogy may also be inverted.

There are advantages and disadvantages to both methods, and in fact the whole issue is more complicated than it first appears.


100-W audio L-pad with wire-wound resistor

In real life, circuit components are not ideal, e.g.,

3. Middle-ear models

Human ear

The middle ear lies between the external ear canal and the cochlea.

The middle ear includes

Human ear from inside

The middle ear also contains

Block diagram of middle-ear model Block diagram of middle ear Human ear

This block diagram, and the circuit model that we shall develop from it, apply equally well to human, cat and guinea-pig middle ears. Block diagram of middle ear

How to model the air cavities? Air cavities
Partial circuit model of middle ear

Represent air cavities by C's.
Represent passage between cavities by R & L. Air cavities
Partial circuit model of middle ear

How to model ear canal? Air cavities
Partial circuit model of middle ear

Represent volume by capacitor. Air cavities

This assumes that the input pressure and volume velocity are measured close to the eardrum. Partial circuit model of middle ear

How to model the malleus and incus?
Air cavities

Assume that they're fixed together. Partial circuit model of middle ear

Represent malleus/incus complex by R-L-C. Air cavities
Partial circuit model of middle ear

How to model the eardrum?
Bekesy low frequency

Conceptually divide it into 2 regions, à la Békésy.
Partial circuit model of middle ear Bekesy low frequency

Use one R-L-C branch for part of eardrum tightly coupled to malleus, and a second branch for the part which shunts energy directly to the cavities.
Partial circuit model of middle ear

How to model the stapes and cochlea? Air cavities
Complete circuit model of middle ear Air cavities

Represent stapes and cochlea each by R-L-C. Ignore incudostapedial joint and many other things.
Full circuit model

If experimental data consist of input impedance measurements, which components can be distinguished?

Combine components which cannot be distinguished.

Circuit model with variable elements Variable elements.

There are now 11 model parameters.

Casting of cavities Independently determine as many model parameters as possible.

This is a silicone-rubber casting of the air cavities of a guinea-pig middle ear. Measuring the cavity volumes gives us Ce, Cb1 and Cb2.

Input impedance, TM removed Estimate some model parameters by comparison with impedance measured with eardrum removed.

Input impedance, intact ear Estimate remaining model parameters by comparison with impedance of intact ear.

4. Problems with lumped models

Lack of direct connection between parameter values and anatomical or physiological properties. For example:

R. Funnell
Last modified: Sat, 2005 Feb 5 18:03:57

Equations converted from TEX to HTML by TTH. I have seen the equations show up properly under Windows using Mozilla 1.7, Firefox 1.0, Netscape 4 & 7, and IE 5 & 6. They do not show up properly for me under Windows using Netscape 6.2, nor under GNU/Linux using Mozilla 1.0.