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Proportional regulator(a.k.a static regulator)

Statical and dynamic characteristics

Proportional(also P or static) regulators control the output signal based on the change of the input signal from the given value of U prime. The law for regulation is given below:

Y = Kp*X

Kp = coeficient of proportion, can be changed with the setting of the regulator.

This law is realized through the amplification of the difference between the required and measured signal as shown on fig.2.1

Real regulators have a significant own inertia, Y is delayed from the theoretical expectations and changes based on the graphic from fig.2.1. The inertia is marked with the time constant T. The real law which is fulfilled by the P regulator is:

The reciprocal value of the coefficient of proportion Kp, in percent, is called the law of proportion D.

Quality of the systems with a P regulator

The amplitude-phase characteristic of an open system, consisting of an object and a regulator, is equal to the multiplication of the amplitude-phase characteristic of the object and coefficient of proportion of the regulator.

From the transmission function we see that the P regulators do not create dephasing of the amplitude-phase characteristic. Therefore the APC(amplitude-phase characteristic) of an open system is the same as of the object and is created from it by multiplying with Kp. According to Nyquist’s criteria, for one system to be stable, its necessary for the APC of an open system to not contain the point with coordinates -1, J0. If the object is from first order or second order, the APC can not cross the negative real axis at no value of the frequency omega(w). Since the P regulator does not bring de-phasing, the APC of the last systems will not have the mentioned point, as shown on part 1 of fig. 2.2.

The increase of the proportion coefiecient Kp, only makes the curve(APC) get closer to this point, which means that the increase of Kp leads to reaching the border of stability of the system. If the sistem for regulation is consisting of a P regulator and an object with a clean delay(part 2 of fig.2.2), the APC of the open system will cross the real axis and with the increase of Kp, the critical value will be reached, after this value of Kp, the system will be unstable. For better stability, we should aim for a lower Kp and a bigger zone of proportion.

Dynamic characteristics of a system with a P regulator and an object from 1st order.

If the system has an object with self-leveling from 1st order, the transient processes of the system after introducing a jump interference ‘z’, will look like:

z = interference

Kp = coefficient of proportion of the P regulator

Kob = object transfer coefficient

T = object time constant

If we accept that t = infinity:

this means that the deviation of the regulated signal after a long time from adding the interference, independent from the work of the regulator, is not 0. On fig 2.3 is shown the structural diagram of the system with automatic regulation(SAR). It consists of an object for regulation and a P regulator. At the input of the object, a jump interference ‘z’ is introduced with 0 starting conditions. This interference deviates the regulated signal ‘x’. The change of ‘x’ leads to the change of the regulating value ‘y’. But if in established mode we have a regulating value different than 0(y is different than 0), this means that the difference between the measured and required value of ‘x’ will also not be zero(it will take some time for the regulated signal ‘x’ to reach the required value), because the regulator is an amplifier of the signal E = x(required) – x(measured).

The value x(from infinity) is called the static error of the regulation. From this the name of the regulators static(P regulator). This means they have a static error. In practice its better to use the term ‘relative static error’ or ‘static event of the system’. The formula for the static error is given below:

When we replace with the given value of x(from infinity) we will receive:

From 2.8 we see that the bigger Kp is, the smaller the relative static error is. The selection of Kp is a result of a compromised decision, between the stability of the system and the quality of the regulation. Kp must be big enough to provide a small static error, improving the maximum dynamic deviation and the time for regulation.

The transfer function is:

Kp is the coefficient of amplification,

fig 3.3 fig.3.4
Practically Kp can be less or more than 1, since there is no delay between the input and output signal, Kp shows the ration between the input and output voltage. Fig 3.4 shows the transfer functions of the P regulator. The fast reaction time of this regulator is seen by the jump change of the output signal when we apply an input signal.

If on the input we have:

where Uout max is the maximum output signal of the regulator, then the regulator will be over regulated or saturated. This means that the regulator has passed its border of linear regulation and there is no longer a proportion between the input and output signal.

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