Flow-Pressure Characteristics of Hydraulic Half-Bridges
A comprehensive analysis of hydraulic resistance networks and their applications in hydraulic systems, including hydraulic motors and pumps.
A half-bridge hydraulic resistance network features a single output control port. Let the inlet pressure be Po, the flow rate through the output control port to the actuator be qv, and the outlet pressure be p. The flow rate qv is a function of the variable hydraulic resistance opening y and the outlet pressure p. Plotting the functional relationship qv = f(p, y) results in what is known as the characteristic curve of the half-bridge hydraulic resistance network. For convenient comparison of different types of hydraulic resistance network characteristics, curves plotted using dimensionless parameters are referred to as dimensionless characteristic curves, which are essential for designing efficient hydraulic motors and pumps.
In analyzing the characteristics of the half-bridge hydraulic resistance network in this section, it is assumed that the variable hydraulic resistance is formed by a spool valve, and that the spool valve has a full-circumference orifice whose flow area has a linear relationship with the spool displacement y. This relationship is critical for understanding the performance of various hydraulic components, including hydraulic motors and pumps.
The characteristic curve of a half-bridge hydraulic resistance network describes the functional relationship between the control signal y and the hydraulic parameters p and q at the output control port. Any one of these three can serve as a parameter, with the other two acting as independent and dependent variables. For the convenience of derivation, the flow rate qv flowing out from the output control port is set as positive. When the spool displacement y = 0, it is specified that the two hydraulic resistance opening lengths of the Type A half-bridge are equal, both being yo. When the spool has a displacement y, the axial opening length of hydraulic resistance R₁ is yo + y, and the axial opening length of hydraulic resistance R₂ is yo - y. The output port flow of the Type A half-bridge is denoted by qv, so for the Type A half-bridge:
The flow equation for Type A half-bridge:
Where:
- qv₁ – flow through the first hydraulic resistance R₁
- qv₂ – flow through the second hydraulic resistance R₂
- b(yo + y), b(yo - y) – hydraulic conductances of R₁ and R₂
- b = Cₐπd√(2/ρ), where Cₐ is the flow coefficient, d is the spool orifice diameter
Figure 1: Schematic representation of Type A half-bridge hydraulic network
Types of Half-Bridge Networks
Type B Half-Bridge
Similarly, for the Type B half-bridge, the flow equation is expressed differently. This configuration is commonly analyzed in the context of various hydraulic systems, including those utilizing hydraulic motors and pumps. The specific flow equation for Type B half-bridge is:
In this equation, c represents the hydraulic conductance of a fixed hydraulic resistance, where c = byo. The direction of spool displacement is shown in Figures 2-3 and 2-4, with special attention to the fact that the direction of y in Type C half-bridges differs from that in Type A and B half-bridges. Specifically, the resistance value of hydraulic resistance R₁ in Type C half-bridges increases with increasing y, a crucial distinction for applications involving hydraulic motors and pumps.
Type C Half-Bridge
For the Type C half-bridge, the flow equation takes the following form, which is important for understanding its behavior in systems with hydraulic motors and pumps:
Each half-bridge type exhibits unique characteristics that make them suitable for different applications in hydraulic systems. Understanding these differences is essential when designing systems that incorporate hydraulic motors and pumps, as the performance characteristics directly affect overall system efficiency and response.
Dimensionless Parameters
To describe equations (2-1) to (2-3) using dimensionless parameters, the valve pre-opening amount yo is specified as the reference for valve opening, the constant inlet pressure Po as the reference for control pressure, and the reference for control flow is calculated according to the maximum flow. This normalization allows for easier comparison between different system configurations, including those with various sizes of hydraulic motors and pumps.
Type A Half-Bridge
Maximum flow occurs when R₂ valve port is fully closed, y = yo, and control valve port pressure is 0:
Type B Half-Bridge
Maximum flow occurs when R₂ valve port is fully closed and control valve port pressure is 0:
Type C Half-Bridge
Reference flow is the same as for Type B half-bridge:
Dividing both sides of equation (2-1) by qvAmax and 2byo√Po respectively, we get the dimensionless form that facilitates comparison across different scales of hydraulic systems, including those with varying capacities of hydraulic motors and pumps:
Using y/yo as the parameter, y/yo as the coordinate, and p/Po as the ordinate, the dimensionless characteristic curve of the Type A half-bridge is obtained. This normalization technique is particularly valuable when scaling designs for different applications involving hydraulic motors and pumps.
Characteristic Curves
Figure 2: Dimensionless characteristic curve for Type A half-bridge
Figure 3: Dimensionless characteristic curve for Type B half-bridge
Figure 4: Dimensionless characteristic curve for Type C half-bridge
In Figure 2, the values of qv/qvAmax are 0.50, 0.25, 0, -0.25, and -0.50 respectively, with y/yo ranging from -1 to 1 and p/Po ranging from 0 to 1. Similar dimensionless forms can be derived for Type B and Type C half-bridges by dividing equations (2-2) and (2-3) by their respective maximum flows. These curves are invaluable for designing and analyzing hydraulic systems, including those incorporating hydraulic motors and pumps.
Using the same method as for the Type A half-bridge hydraulic resistance network, the dimensionless characteristic curves for Type B and Type C half-bridges can be obtained, as shown in Figures 3 and 4. It can be seen from these figures that the dimensionless characteristic curves of Type B and Type C half-bridges are mirror images of each other, a relationship that is important to consider when selecting the appropriate half-bridge type for specific applications involving hydraulic motors and pumps.
Zero Flow Analysis
Let us first examine the curve where qv/qvmax = 0. A zero outlet flow from the half-bridge hydraulic resistance network indicates that the half-bridge hydraulic resistance network has only pressure output without flow output. Some actuators controlled by hydraulic resistance networks have very small displacements, such as the pilot hydraulic resistance network of pressure control valves, which is used to control the main valve spool movement. Because the main valve spool displacement is very small, the required flow is very small, and its output flow is basically zero during normal operation, mainly controlling the main valve through the outlet pressure. Therefore, studying the pressure-spool displacement characteristics when the output flow of the half-bridge hydraulic resistance network is zero is of great practical significance for various applications, including those with hydraulic motors and pumps.
Type A Half-Bridge at Zero Flow
For the Type A half-bridge, as can be seen from Figure 2, when the value of y/yo increases from -yo to yo, the outlet pressure increases, but not linearly. In a region near y/yo = 0, the outlet dimensionless pressure p/Po increases approximately linearly with the increase of y/yo, which can be derived from equation (2-4). When qv/qvAmax = 0, equation (2-4) becomes:
Squaring both sides and rearranging:
When the value of y is small, the y² term can be neglected, giving:
It is evident that the outlet dimensionless pressure p has an approximately linear relationship with the dimensionless spool displacement y, a characteristic that is useful in control system design for equipment using hydraulic motors and pumps.
Type B Half-Bridge at Zero Flow
For the Type B half-bridge, when the dimensionless outlet flow is zero, the outlet dimensionless pressure p also increases with the increase of y, but it is a non-linear monotonic increase. Near y = 0, the relationship between the outlet dimensionless pressure p and the dimensionless spool displacement y is also non-linear. This can be derived from equation (2-5). When qvB/qvBmax = 0, equation (2-5) becomes:
Squaring both sides and rearranging:
When the value of y is small, neglecting the y² term, we get:
From equation (2-8), it can be seen that the outlet dimensionless pressure p of the Type B half-bridge has a non-linear relationship with the dimensionless spool displacement y, a factor that must be considered when integrating these components with hydraulic motors and pumps.
Type C Half-Bridge at Zero Flow
Similarly, through equation (2-6), the functional relationship between the output control port pressure p and y when the output port flow of the Type C half-bridge is zero can be obtained as:
When the value of y is small, neglecting the y² term, we get:
It can be seen from equation (2-9) that the dimensionless outlet pressure p of the Type C half-bridge has a non-linear relationship with the dimensionless spool displacement y, and is a mirror image of the Type B half-bridge graph. This mirror relationship is particularly useful in symmetric hydraulic systems that employ both hydraulic motors and pumps.
Non-Zero Flow Characteristics
For the relationship between dimensionless outlet pressure and dimensionless spool displacement when the output control port flow of the half-bridge hydraulic resistance network is not zero, it can be seen from Figure 2-5 that near y = 0, the curve families with different output flows have similar outlet pressure-displacement characteristics. This similarity simplifies the initial design phase for systems incorporating hydraulic motors and pumps.
When the value of y approaches -1 or 1, the y² term can no longer be neglected, so the outlet dimensionless pressure p has a non-linear relationship with the dimensionless spool displacement y. For the Type A half-bridge, when the value of y approaches -1, the hydraulic resistance value of R₁ approaches infinity. If a positive outlet flow q is required at this time, oil must be sucked from the oil tank through R₂, making the dimensionless outlet pressure p < 0. In this case, equation (2-1) can no longer correctly describe the flow balance equation. When the value of y approaches 1, the hydraulic resistance value of R₂ approaches infinity. If a negative dimensionless outlet flow q is required at this time, there must be a flow from the outlet to the inlet through R₁, making the dimensionless outlet pressure p > 1. These two cases are not controlled by the inlet pressure po, so they are not shown in Figure 2-4. For example, the line qv/qvAmax = 0.5 only shows the segment y = 0~1, and the curve qv/qvAmax = -0.5 only shows the segment y = -1~0. The curves for dimensionless outlet pressure p < 0 or p > 1 are not shown, which is important knowledge for anyone working with hydraulic motors and pumps.
For the pressure-spool displacement characteristic curves of Type B and Type C half-bridge hydraulic resistance networks, when the value of y is closer to 1, the dimensionless outlet pressure p will also have p > 1 or p < 0, which are also not shown in the曲线图. Understanding these limitations is crucial for avoiding operating conditions that could damage system components, including hydraulic motors and pumps.
Practical Applications
In practical applications, the half-bridge hydraulic resistance network is mainly used to control the pressure at the outlet, and the outlet flow qv is generally small. Therefore, studying the pressure-spool displacement characteristics with the parameter qv/qvmax = 0 is of great significance. This is particularly true in systems where precise pressure control is required, such as in certain types of hydraulic motors and pumps where pressure regulation directly affects performance and efficiency.
It should be noted that the above discussion is based on constant pressure source supply, no leakage, no friction, and without considering the compressibility of the liquid, so it represents an ideal half-bridge hydraulic resistance network. Real-world applications must account for these factors, which can introduce additional complexities in system behavior. Engineers designing systems with hydraulic motors and pumps must therefore consider both the idealized characteristics and the real-world deviations.
Despite these idealizations, the analysis provides a valuable foundation for understanding and designing hydraulic control systems. The dimensionless approach allows engineers to scale designs appropriately for different applications, from small-scale precision controls to large industrial systems utilizing powerful hydraulic motors and pumps. By understanding the fundamental relationships between pressure, flow, and valve displacement, engineers can optimize system performance, improve efficiency, and reduce maintenance requirements in a wide range of hydraulic applications.
The characteristics of hydraulic half-bridge networks play a fundamental role in the design and operation of hydraulic control systems. By understanding the relationship between pressure, flow, and valve displacement, engineers can develop more efficient and reliable hydraulic systems, including those incorporating various types of hydraulic motors and pumps. The dimensionless analysis presented provides a powerful tool for comparing different configurations and scaling designs for specific applications.
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