Linear first order system

Principle

A time invariant (see Linear systems: general) linear first order system is formally described by a linear first order differential equation with the time as independent variable. The output of the system is given by its input and the system characteristics. The latter form the time constant τ (see Halftime and time constant). The solution of the equation comprises an exponential function with time constant τ: e^–t/τ.

The most used analogue of this mathematical system is the electric analogue, an electric closed circuit comprising a resistor R, a capacitor C (see Electrical resistance, capacitance and inductance) and a voltage V_s (or current) source (Fig. 1). They are called RC filters. There are two types: the low pass (or high frequency cut off) filter (Fig. 1a) and the high pass (or low frequency cut off) filter (Fig. 1b). They are used to diminish the high and the low frequencies in a signal, respectively.

Fig. 1 Electrical first order systems with left the low pass and right the high pass system (or filter).

There are 3 types of special input signals. Normalized they are:

· the unit step function (with amplitude 1);

· the unit impulse function, infinite short and amplitude infinite, such that the time integral (area) is one;

· the sinusoidal signal (amplitude 1).

Fig. 2 Step responses measured over the capacitor and over the resistor of Fig.1a and b respectively.

Step response

The step response to the step function with amplitude V_s of the low (a) and high (b) pass filters of Fig. 1 are given in Fig. 2. They are:

· low pass filter (Fig. 1a, 2a): V_{C, step} = V_s(1- e^–t/τ); (1a)

· high pass filter (Fig. 1b, 2b): V_{R, step} = V_se^–t/τ, (1b)

where τ = RC.

Impulse response

Since the impulse is the time derivative of the step, the impulse response is the time derivative of the step response:

V_{C, impulse} = V_sτ^-1e^–t/τ; (2a)

V_{R, impulse} = −V_sτ^-1e^–t/τ(t>0), with an impulse to +∞ at t = 0. (2b)

Sinus response

The response to a continuing sinusoidal input is always a continuing sinusoidal output of the same frequency, but generally with a different amplitude and phase. The sinus response is dependent on the frequency, which is often expressed as angular frequency ω. Notice that ω = 2πf with f the frequency in Hz. The input/output ratio of the sinus amplitude as a function of frequency, A(ω), is visualized in the amplitude characteristic. The phase shift of the output sinus related to the phase of the input as a function of frequency, φ(ω), is visualized in the phase characteristic. Fig. 3 and 4 present both for the low and high pass filter. Expressed in equations it holds that:

A_{low pass}(ω) = 1/(1+ ω/ω₀)^0.5 (3a)

A_{high pass}(ω) = 1/(1+ ω₀/ω)^0.5 (3b)

φ _{low pass}(ω) = -arctanω/ω₀ (4a)

φ_{high pass}(ω) = arctanω₀/ω, (4b)

where ω₀=1/τ. Derivations (there are several) can be found in any basic textbook on physics. Notice that A_{low pass} and A_{high pass} are mutually mirrored along the vertical axis at ω/ω₀ = 1, and that φ_{high pass} is shifted +90⁰ with respect to φ_{low pass}.

Fig. 3 Frequency characteristics of low pass first order system with a normalized frequency axis. The frequency characteristics presented as log-log and log-lin plots are called Bode plots.

Fig. 4 As Fig. 3 for a high pass first order system.

A first order filter can also be made with a resistor and self-inductance. When the latter takes the place of the capacitance in Fig. 1a, a high pass filter is obtained, since high frequencies hardly pass the inductance, whereas for low frequencies the inductance is practically a short-circuit. Similarly, a low pass filter is obtained when the place of the capacitance in Fig. 1b is taken by the inductance. In practice this approach is a little more complicated because a coil also has a resistance (see Electrical resistance, capacitance and inductance). Therefore LC filters are seldom used.

Application

General

RC filters are applied in any electronic device. Their analogues exist in fluid dynamics and acoutics and phase-less 1D or 2D analogues exist in optics (lenses).

In numerical form they are used to filter digital signals in the time and spatial domain (2D, 3D or 3D-t).

Modeling

They are widely used in computational modeling of systems, also in medical and biological application.

A classical example of a low pass filter in neurobiology is the decay of an action potential. Its characteristic frequency ω₀ = 1/τ = 1/r_mc_m, where r_m is the resistance across the membrane and c_m is the capacitance of the membrane. r_m is a function of the number of open ion channels and c_m is a function of the properties of the lipid bilayer.

More Info

In real notation, A and φ can be combined in a polar plot, in which A is the length of the vector and φ is the angle between the vector and the horizontal axis. For the low (high) pass filter, it is a semicircle in the right lower (upper) half-plane. ω follows this trajectory clockwise. Fig. 5 presents the polar plot for the low pass system.

Fig. 5 Polar plot of frequency response of low pass 1^st order system.

In jω notation, the transfer characteristic, comprising the real A(ω) and φ(ω), is denoted by H(jω). H(jω) can be found easily with the impedances of R and C (see Electrical resistance, capacitance and inductance). It always holds that H(jω) = output impedance/Σimpedances along the closed circuit, provided that the voltage source is ideal (impedance zero). Consequently, H(jω) of the low pass system is:

(5a)

It holds that:

A(ω) =│H(jω) │; (5b)

φ(ω),= argH(jω) = Im{(jω)}/Re{H(jω)}. (5c)

The Laplace back transform yields the impulse response h(t):

L^-1(H(jω) = h(t) (6)

The Laplace operator s is s = α + jω, but in calculations of the transfer characteristics α = 0. So, calculations based on (5) and (6) use the imaginary frequency jω. However the complex s-plane can be used, not only to calculate in an easy why A(ω) and φ(ω), but also to visualize the action of poles and zero’s (see Linear system analysis). This is explained with the low pass 1^st order system, which H(jω) is

H(jω) = ω₀/(jω + ω₀). ω₀ is a pole (root) of the denominator of H(jω).

Fig. 6 Pole-zero plot of low pass 1^st order system, in Laplace notation H(s) = ω₀/(s + ω₀).

With poles and zero’s (a zero is the root of the nominator) indicated in the complex s plane, this figure is called the pole-zero diagram. For the low pass 1^st order system, the diagram, comprising only a pole at the left real axis, is presented in Fig. 6. A(ω) is found by moving from the origin along the positive jω axis (vertical arrow) and calculating the reciprocal of the length of the vector. This yields (3a).

φ(ω), given by (4a), is found by turning the oblique side to the real axis (clockwise is a negative phase).

For the high pass system Fig. 7 holds. It has also a zero in the origin. In Laplace notation it is:

H(s) = s/(s + ω₀), (7)

with the nominator yielding the zero. The s in the nominator with its root in the origin has the action of pure differentiation. For sine waves this means adding +90^o to the phase plot of the low pass system of Fig. 3b and adding a straight line with a slope of +1 through the point (ω₀,1) to the amplitude plot of Fig. 3a. Notice that adding in this log/log plot means multiplication by ω. Now, the Bode plots of Fig. 4 are obtained. The inset of Fig. 7 gives the contribution of the pole and zero in A(ω) and φ(ω).

Fig. 7 Pole-zero plot of the high pass 1^st order system H(jω) = jω/(jω + ω₀).

In general, multiplying some H(s) by the operator s yields differentiation of the output signal (any signal) and dividing H(s) by s means integration. This is related to the notion that the step response is the integral of the impulse response.

See for a more complete description of the Laplace approach in system theory Linear system analysis.