page_title: Signal-flow graph
section_title: Mechatronics :  Position servo with multi-loop feedback
hierarchical_section_title: Signal-flow graph / Linear signal-flow graph examples / Mechatronics :  Position servo with multi-loop feedback
caption_attribution_description: English: Angular position servo and signal flow graph. θC = desired angle command, θL = actual load angle, KP = position loop gain, VωC = velocity command, VωM = motor velocity sense voltage, KV = velocity loop gain, VIC = current command, VIM = current sense voltage, KC = current loop gain, VA = power amplifier, VM = voltage applied to motor inductance, L = motor inductance, IM = motor current, RM = motor resistance, RS = current sense resistance, KM = motor torque constant (Nm/amp) , T = torque, M = momment of inertia of all rotating components α = angular acceleration, ω = angular velocity, β = mechanical damping, GM = motor back EMF constant, GT = tachometer conversion gain constant,. There is one forward path (shown in a different color) and six feedback loops. The drive shaft assumed to be stiff enough to not treat as a spring. Constants are shown in black and variable in purple.
context_page_description: A signal-flow graph or signal-flowgraph, invented by Claude Shannon, but often called a Mason graph after Samuel Jefferson Mason who coined the term, is a specialized flow graph, a directed graph in which nodes represent system variables, and branches represent functional connections between pairs of nodes. Thus, signal-flow graph theory builds on that of directed graphs, which includes as well that of oriented graphs. This mathematical theory of digraphs exists, of course, quite apart from its applications.
  SFGs are most commonly used to represent signal flow in a physical system and its controller, forming a cyber-physical system. Among their other uses are the representation of signal flow in various electronic networks and amplifiers, digital filters, state-variable filters and some other types of analog filters. In nearly all literature, a signal-flow graph is associated with a set of linear equations.
context_section_description: This example is representative of a SFG (signal-flow graph) used to represent a servo control system and illustrates several features of SFGs.  Some of the loops (loop 3, loop 4 and loop 5) are extrinsic intentionally designed feedback loops.  These are shown with dotted lines.  There are also intrinsic loops (loop 0, loop1, loop2) that are not intentional feedback loops, although they can be analyzed as though they were.  These loops are shown with solid lines.  Loop 3 and loop 4 are also known as minor loops because they are inside a larger loop.
  The forward path begins with θC, the desired position command.  This is multiplied by KP which could be a constant or a function of frequency.  KP incorporates the conversion gain of the DAC and any filtering on the DAC output.  The output of KP is the velocity command VωC which is multiplied by KV which can be a constant or a function of frequency.  The output of KV is the current command, VIC which is multiplied by KC which can be a constant or a function of frequency.  The output of KC is the amplifier output voltage, VA. The current, IM, though the motor winding is the integral of the voltage applied to the inductance.  The motor produces a torque, T, proportional to IM.  Permanent magnet motors tend to have a linear current to torque function.  The conversion constant of current to torque is KM.  The torque, T, divided by the load moment of inertia, M, is the acceleration, α, which is integrated to give the load velocity ω which is integrated to produce the load position, θLC.
  The forward path of loop 0 asserts that acceleration is proportional to torque and the velocity is the time integral of acceleration.  The backward path says that as the speed increases there is a friction or drag that counteracts the torque.  Torque on the load decreases proportionately to the load velocity until the point is reached that all the torque is used to overcome friction and the acceleration drops to zero. Loop 0 is intrinsic.
  Loop1 represents the interaction of an inductor's current with its internal and external series resistance.  The current through an inductance is the time integral of the voltage across the inductance.  When a voltage is first applied, all of it appears across the inductor.  This is shown by the forward path through .  As the current increases, voltage is dropped across the inductor internal resistance RM and the external resistance RS.  This reduces the voltage across the inductor and is represented by the feedback path -(RM + RS).  The current continues to increase but at a steadily decreasing rate until the current reaches the point at which all the voltage is dropped across (RM + RS). Loop 1 is intrinsic.
  Loop2 expresses the effect of the motor back EMF.  Whenever a permanent magnet motor rotates, it acts like a generator and produces a voltage in its windings.  It does not matter whether the rotation is caused by a torque applied to the drive shaft or by current applied to the windings.  This voltage is referred to as back EMF.  The conversion gain of rotational velocity to back EMF is GM.  The polarity of the back EMF is such that it diminishes the voltage across the winding inductance. Loop 2 is intrinsic.
  Loop 3 is extrinsic.  The current in the motor winding passes through a sense resister.  The voltage, VIM, developed across the sense resister is fed back to the negative terminal of the power amplifier KC.  This feedback causes the voltage amplifier to act like a voltage controlled current source.  Since the motor torque is proportional to motor current, the sub-system VIC to the output torque acts like a voltage controlled torque source.  This sub-system may be referred to as the "current loop" or "torque loop".  Loop 3 effectively diminishes the effects of loop 1 and loop 2.
  Loop 4 is extrinsic.  A tachometer (actually a low power dc generator) produces an output voltage VωM that is proportional to is angular velocity.  This voltage is fed to the negative input of KV.  This feedback causes the sub-system from VωC to the load angular velocity to act like
caption: Angular position servo and signal flow graph.  θC = desired angle command, θL = actual load angle, KP = position loop gain, VωC = velocity command, VωM = motor velocity sense voltage, KV = velocity loop gain, VIC = current command, VIM = current sense voltage, KC = current loop gain, VA = power amplifier output voltage, LM = motor inductance, VM = voltage across motor inductance, IM = motor current, RM = motor resistance, RS = current sense resistance, KM = motor torque constant (Nm/amp), T = torque, M = moment of inertia of all rotating components α = angular acceleration, ω = angular velocity, β = mechanical damping, GM = motor back EMF constant, GT = tachometer conversion gain constant,.  There is one forward path (shown in a different color) and six feedback loops.  The drive shaft assumed to be stiff enough to not treat as a spring.  Constants are shown in black and variables in purple.
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