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

The foundations are

• the current through a branch is well defined (current in = current out)
• the voltage across a branch is well defined (can be measured unambiguously)
• Kirchhoff's laws: The basic components are linear and time-invariant:

• resistor (Ohm's law)
• capacitor
• inductor

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 C �� i dt
 f = k �� u dt
 p = 1 C �� 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.

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

• resistors have non-zero inductance,
• inductors have non-zero resistance,
• springs have mass,
• cavities have nonuniform pressures, ...

## 3. Middle-ear models The middle ear lies between the external ear canal and the cochlea.

The middle ear includes

• the eardrum (tympanic membrane)
• 3 small bones (ossicles)
• air cavities The middle ear also contains

• two muscles
• various ligaments
• fibrocartilaginous ring
• mucosal folds (variable)
• joints

Block diagram of middle-ear model  This block diagram, and the circuit model that we shall develop from it, apply equally well to human, cat and guinea-pig middle ears. How to model the air cavities?  Represent air cavities by C's.
Represent passage between cavities by R & L.  How to model ear canal?  Represent volume by capacitor. This assumes that the input pressure and volume velocity are measured close to the eardrum. How to model the malleus and incus? Assume that they're fixed together. Represent malleus/incus complex by R-L-C.  How to model the eardrum? Conceptually divide it into 2 regions, à la Békésy.  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. How to model the stapes and cochlea?   Represent stapes and cochlea each by R-L-C. Ignore incudostapedial joint and many other things. If experimental data consist of input impedance measurements, which components can be distinguished?

Combine components which cannot be distinguished. Variable elements.

There are now 11 model parameters. 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. Estimate some model parameters by comparison with impedance measured with eardrum removed. 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:

• moment of inertia changes if axis of rotation changes
• mass parameter for stapes changes if its mode of vibration changes
• eardrum parameters change if its vibration pattern changes

R. Funnell