Mathematical Modeling

Modeling of Electrical Systems

Basic Components of electrical systems are as follows.

Modeling of electrical system is based on Kirchhoff’s Law, i.e either

using Nodal Analysis or Mesh Analysis depending on which one is

easier for each case.

LRC Circuit

Consider the following LRC circuit,

Using KVL (Kirchhoff’s Voltage Law) we get,

Assuming zero initial conditions and transform the eqns into Laplace,

Then the transfer function of the circuit is

Cascaded RLC Circuit

Using KVL (Kirchhoff’s Voltage Law) we get,

Assuming zero initial conditions and transform the eqns into Laplace,

Subtitude eqn(3-70) into eqn(3-69) to eliminate I1 from eqn(3-69),

then the transfer function is:

This approach becomes quite difficult as the circuit becomes more

complex. The simpler approach is by using SFG.

Modeling of Electrical Network Using SFG

Example 1

Consider the RC circuit,

Using KVL and KCL (Kirchhoff’s Current Law) we can write,

Draw SFG,

The transfer function,

Example 2

Consider the cascaded RC circuit,

Using KVL and KCL (Kirchhoff’s Current Law) we can write,

Draw SFG and apply Mason gain formula,

Example 3

Using KVL and KCL we can write,

Draw SFG and apply Mason gain formula,

Active Circuit

Active circuit consists of operational amplifier or op amps, normally

used to amplify signal in sensor circuit and also in filter or controller.

The equation that govern the op amps is,

Inverting Amplifier

Non-Inverting Amplifier

Example 1

Consider the following circuit that could be simplified as the next

circuit.

The second circuit is similar to the inverting amplifier, hence

Example 2

Consider the following lead or lag controller circuit,

From Figure (a) that is similar to inverting amplifier,

To get the non-inverting circuit, the sign inverter as in Figure (b) is

normally used, where

Modeling of Mechanical Systems

Basic elements and laws of mechanical systems are as follows:

1. Spring

where k is the spring constant .

2. Damper and Dashpot

where c = damping coefficient

3. Mass and Inertia

where m, J = mass/ inertia

4. Lever Mechanism

without the fixed point,

and if a = b,

5. Gear

The number of teeth on the gear surface is proportional to the

gears radii

The distance traveled along the surface of each gear is the same.

The work done by the gear is equal to that of the other,

The gear ratio,

Example 1 – Damper-Spring-Mass system

Force equation according to Newton’s Law

In Laplace domain,

Therefore, transfer function

Example 2: Dynamic Absorber (Two-mass system)

Mass m:

or

Mass m1

or

In Laplace Domain

or

Draw SFG

Using Mason’s Gain formula,

Example 3 – Rotational Drive system

Assume , then force eqns.,

and

In Laplace domain,

or

Draw SFG

Apply Mason’s gain formula

Example 4 – System with Gears

Force Eqns.,

In Laplace Domain,

Draw SFG

The Transfer Functions,

Modeling of Biomedical Instruments/ DevicesMost of the biomedical instruments / devices are made of a

combination of mechanical and electrical components. Some of those

simple instruments will be considered in this section.

Example 1 – Dialysate weight measuring circuit (for Peritoneal

Dialysis)

A simple schematic diagram for Peritoneal Dialysis system is shown

bellow. To provide a mobile system, the control hardware, dialysate

supply and spent dialysate are commonly mounted on a wheeled

stand.

It is required to measure the weight of the dialysate supply and spent

dialysate to determine the amount of fluid and waste removed from

the body. The measuring circuit consists of the following circuit

attached to a spring-loaded mechanism to change the position of a

potentiometer.

The transfer function of circuit is given by a standard non-inverting

amplifier,

Example 2 – An Integrator for EMG signals

It is frequently of interest to quantify the amount of EMG

(Electromyogram) activity. Such quantification often assumes the

form of taking the absolute value of EMG and integrating it, as shown

in the block diagram bellow,

The transfer function of the integrator is,

Example 3 – An ECG amplifier circuit

The block diagram of an electrocardiograph (ECG) is shown bellow.

The transfer function of the first op amp,

Since the second op-amp also has the same resisters so it has the same

transfer function. The third op-amp will have the transfer function of,

Therefore, the overall transfer function will be,

Example 4 – Signal conditioning, low pass filter for ECG signal

A low pass filter is part of the conditioning circuit for ECG and one of

the common filters is Sallen and Key quadratic low-pass filter as in

the following diagram.

Note that

KCF at V1

KCF at V2

Hence the transfer function is,

Example 5 – Strain gauge to study the strength of artificial bone

A fine wire is cemented on a carrier (e.g thin paper, bakelite or teflon). The carrier is then bonded to subject being measured

Stress tends to elongate the wire, hence increase its length and decrease cross-sectional area. Thus resistance is:

ρ = resistivity of the material in Ωm

A strain gauge is normally glued to an artificial bone to study the

effect of applying a load to the bone.

Initially the circuit is balance (as R1 = R2), thus the output voltage is

zero. However, if the active gauge is stretched or compressed the

gauge resistance will vary and produce some voltage. The transfer

function is,

Where r is the resistance due to the strain of the gauge,

Where k is the gauge factor and strain is the change in length divide

original length,