How to derive the transfer function of the regulating valve

As an actuator in an automatic control system, the dynamic characteristics of the regulating valve have an important impact on the performance of the entire control system. To accurately model and analyze the regulating valve, it is usually necessary to establish its transfer function. This article will introduce the basic working principle of the regulating valve and derive its linearized transfer function.

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One: Basic Structure and Working Principle of the Regulating Valve

The regulating valve consists of two parts: the actuator and the valve body. The actuator is usually a pneumatic or electric device that receives signals from the controller (such as 4~20mA current signal or 3~15psi pressure signal), converts the control signal into displacement or angular motion, thereby driving the valve core to move, changing the flow area, and regulating the fluid flow.

In control systems, we usually concern about the relationship between the output of the regulating valve (such as flow rate) and the input control signal. To simplify the analysis, the characteristics of the regulating valve are often linearized to establish its transfer function model.

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Two: Dynamic Model of the Regulating Valve

Assuming the regulating valve is a linear regulating valve driven by a pneumatic actuator. The actuator is typically composed of a diaphragm and a spring, forming a typical second-order system:

- Input signal: control signal $ u(t) $, usually voltage or pressure;
- Output signal: valve core displacement $ x(t) $;
- System characteristics: include mechanical components such as springs, dampers, and mass.

According to Newton's second law, the motion equation of the regulating valve can be expressed as:
$$
m \ddot{x}(t) + c \dot{x}(t) + k x(t) = K_u u(t)
$$
Among which:
- $ m $: Equivalent mass of the valve core,
- $ c $: Damping coefficient,
- $ k $: Spring stiffness,
- $ K_u $: Actuator gain.

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III. Derivation of the transfer function

Taking the Laplace transform of the differential equation (assuming the initial conditions to be zero):
$$
m s^2 X(s) + c s X(s) + k X(s) = K_u U(s)
$$
Simplified to:
$$
X(s)(m s^2 + c s + k) = K_u U(s)
$$
Therefore, the open-loop transfer function of the valve core displacement to the control signal is:
$$
G(s) = \frac{X(s)}{U(s)} = \frac{K_u}{m s^2 + c s + k}
$$

This transfer function describes the relationship between the input signal and the output displacement of the regulating valve in the frequency domain and is a typical second-order inertial system.

In practical applications, if the regulating valve responds quickly, the effect of mass can be ignored, and it can be simplified to a first-order system, then the transfer function is approximately:
$$
G(s) \approx \frac{K}{\tau s + 1}
$$
Among which:
- $ K $ is the static gain of the system,
- $ \tau $ is the time constant of the system.

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IV. Relationship between flow rate and displacement

In practical control, it is often more concerned about the fluid flow rate $ q(t) $ output by the regulating valve rather than the valve core displacement. Assuming that the flow rate is linearly related to the valve core position:
$$
q(t) = K_q x(t)
$$
Then, the transfer function of the flow rate to the control signal is:
$$
\frac{Q(s)}{U(s)} = K_q \cdot G(s) = \frac{K_q K_u}{m s^2 + c s + k}
$$

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V. Conclusion

From the above derivation, it can be seen that the dynamic characteristics of the regulating valve can be represented by a second-order or simplified first-order transfer function. This mathematical model provides a theoretical basis for the simulation, analysis, and design of the regulating valve in the control system. At the same time, in practical applications, it is necessary to combine specific regulating valve models and working conditions for parameter identification and correction to improve the accuracy and practicality of the model.

Understanding the derivation method of the transfer function of the regulating valve is helpful for comprehending its dynamic response characteristics and provides strong support for the optimization design of the control system.