EAS327: Response of a Linear System

The general result is given last (you may already have studied most of the material that follows)...

A.1 Introduction

The "Step Response" of many common sensors is characterised by a falling rate of approach to a new equilibrium. Mathematically, the response y(t) shown on the figure can be written as:

Δy/Δt ~ (Y₂ - y)

where Y₂ (= constant) is the final value of y that will ultimately be "achieved." We may introduce a constant of proportionality τ, to obtain an equation:

Δy/Δt =(Y₂ - y)/τ

where it is obvious that the proportionality constant τ must have the units of [time]. This constant τ is called the "time constant" of the system, and it is an immensely important attribute of the sensor/system. In Section A.3 we do the Calculus to show how to integrate this equation to obtain an explicit formula for the system response. If the step change in input takes place at time t=0, the output is given by

y(t) = Y₂ + (Y₁ - Y₂ ) exp [ - t /τ ]

We can now speak of the fraction of the total "adjustment" (Y₁ - Y₂ ) that will have taken place at various times (t).

t (time since step)	% adjustment
τ	63
2τ	86
3τ	95

But why do many sensors have this type of response? Lets take an example. Suppose that at t=0 we plunge a mercury-in-glass thermometer into a bucket of hot water. The output of the thermometer is a column-length (y), and prior to t=0, let us say y=Y₁ which corresponds to the (cool) air temperature, T_a. As the sketch shows, the final column length is longer, Y₂. And that final column length is only achieved when the entire volume of mercury has come to the temperature (T_H) of the hot water. The heat to warm the mercury must come from the water... by a process of heat transport. So

• a finite amount of heat must be added to the thermometer - it has "heat capacity."
• the process of heat addition proceeds at a rate determined by the temperature difference (T_H - T^s) between the hot water and the surface of the glass.
• as time (t) increases that temperature difference decreases - decreasing the rate of warming of the thermometer.

Consider the accompanying schematic of a signal y(t), seen over an interval of time (t₁ <= t <= t₂) during which its mean value (average) was Y. We can "decompose" y(t) as:

y(t) = Y + y'(t)

where y'=y'(t) is called the "fluctuation." The fluctuation is simply the instantaneous deviation from the mean (Y).

Now suppose the fastest fluctuations in the signal have period T_min and frequency f_max=1/T_min. Our sensor has time constant τ.

If we want our sensor to "show," ie. reveal, or measure, the fastest fluctuations, then we must ensure that

τ < = T_min /10

ie. make sure we use an instrument with a time-constant at least 10 times smaller than the fastest periods of interest.

But IF we only care to measure the average value of the signal, or the very slow changes, we can safely allow τ>> T_min, a choice which will result in our "seeing" all changes with periods T>>τ.

This implies that we can use a "slow" sensor to achieve averaging of a variable signal, as an alternativeto use of a faster sensor, and subsequent digital averaging after the signal has been aquired (digitised, logged).

A.3 DOING THE CALCULUS TO DETERMINE THE RESPONSE OF OUR LINEAR, FIRST-ORDER SYSTEM

Our system governing equation is

Δy/Δt =(Y₂ - y)/τ

and if we let the time interval Δt tend towards zero, we obtain a differential equation:

dy/dt =(Y₂ - y)/τ

This is a quite generic equation, which characterises many many physical systems... and there are a number of interesting points to be made about it:

• it is an "ordinary differential equation" (o.d.e.). That is to say, it is a differential equation, rather than an algebraic equation, because it contains a derivative; and it is "ordinary" because the "dependent variable" y (the variable whose behaviour we want to know - whose behaviour is described by the equation) depends on only a single independent variable, the time (t). If y depended on other variables, say y=y(t,z), then it would be governed by a partial (rather than ordinary) differential equation.
• It is a "first order" o.d.e., because there are no derivatives of second- or higher order... the "highest" derivative appearing is dy/dt
• It is a "linear" o.d.e. because the dependent variable (y) appears only to the first power, ie. y¹ but not y² etc.
• We can identify a "driving variable" which is "forcing" our response-variable y... this is the variable Y₂
• as it is "first-order", the behaviour of "y" can only be known completely if we specify a single additional condition (equation)

• We re-write the equation as

  dy/dt + y/τ = Y₂/τ

  The equation is said to be "inhomogeneous" due to the presence of the forcing term (the "inhomogeneity") on the right hand side (rhs).

• We write down the corresponding "auxiliary equation," by dropping the inhomogeneity,to obtain:

  dy/dt + y/τ = 0

  We term the solution to this equation the "complimentary function" (CF). There are standard methods for finding y^CF, and for this type of o.d.e., we take a trial solution of form

  y^CF = c₁ e^{k t}

  where k is an arbitrary constant. We substitute into the auxiliary equation, noting that

  d/dt ( e^kt)=k e^kt.

  For the equation to be true, we must have that k = - 1/τ.

  So, our complimentary function (the first part of our solution) is:

  y^CF = c₁ e^{- t / τ}

• Now we have to find the "particular integral," y^PI, which has to be a solution (ie. any solution) of the complete equation. It is evident by inspection that

  y^PI = Y₂/τ

  is a solution.

• Thus our general solution is y(t)=y^PI + y^CF = Y₂/τ + c₁ e^{- t / τ}

• The unknown c1 is determined by imposing our initial condition, namely, that y=Y₁ at t=0. Substituting, we evaluate c₁ and arrive at the result that:

  y(t) = Y₂ + (Y₁ - Y₂ ) exp [ - t /τ ]

Referring back to the "system diagram", we had an input x(t), and an output y(t). There exists a General (formal) result for the relationship between input and output.

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Last modified 13 March 2016 (last prior modification 27 Mar. 2003).