Control Systems: Mason’s Gain Formula, Signal Flow Graphs and Servomotors

Posted on Oct 31, 2026 in Electrical Engineering

Stability Margins in Control Systems

  • Gain Margin: The factor by which the system gain can be increased before the system becomes marginally stable.

  • Phase Margin: The amount of additional phase lag required at the gain crossover frequency to bring the system to the verge of instability.

  • Gain Crossover Frequency: The frequency at which the magnitude of the open-loop transfer function is exactly 1 (or 0 dB).

  • Phase Crossover Frequency: The frequency at which the phase angle of the open-loop transfer function is exactly 180 degrees.


Mason’s Gain Formula

Mason’s gain formula is:


Where:

  • Pk = gain of the k-th forward path.

  • Δ = graph determinant.

  • Δk = determinant after removing loops touching the k-th forward path.

The determinant is calculated using:

Δ = 1 − ∑Li + ∑LiLj − ∑LiLjLm + ⋯

Here, Li represents an individual loop gain. Only products of non-touching loops are included in the positive terms.

Steps to Solve the Given Graph

  1. Identify all forward paths from R(s) to C(s).

  2. Multiply the branch gains along each path to obtain Pk.

  3. Identify every individual closed loop.

  4. Find the gain of each loop, including its sign.

  5. Identify all pairs and higher combinations of non-touching loops.

  6. Calculate Δ.

  7. Calculate Δk for each forward path.

  8. Substitute into Mason’s formula.

    Stability Comment

    • Positive gain margin and positive phase margin generally indicate adequate relative stability for a conventional single-loop system.

    • Zero margin indicates the verge of instability.

    • A negative margin indicates instability or insufficient stability margin under the usual frequency-response interpretation.


Signal Flow Graphs

A Signal Flow Graph is a graphical representation of the relationships between variables in a system.

Steps to Draw an SFG

  1. Identify all system variables.

  2. Represent each variable by a node.

  3. Draw directed branches to show the direction of signal flow.

  4. Write the transfer gain on each branch.

  5. Add feedback branches wherever feedback exists.

  6. Identify forward paths and closed loops.

  7. Use Mason’s gain formula to calculate the overall transfer function.

    A position control system using an AC servomotor rotates a load to a specific target angle.

    • Block Diagram Flow: It consists of a reference potentiometer (input), an error detector, a servo amplifier, a 2-phase AC servomotor, a gear train, and the load, which is mechanically coupled to a feedback potentiometer.

    • Operation: The input angle is converted into a reference voltage. The output angle is converted into a feedback voltage. The error detector compares these two voltages. If there is a difference (an error), the resulting error voltage is amplified to drive the control phase of the AC servomotor. The motor spins the load via the gear train until the output position matches the input position, reducing the error to zero and stopping the motor.


Servomotors, Synchros and Stepper Motors

  • DC Servomotor Control Modes: Armature control provides fast dynamic response with constant field current, whereas field control provides high output power at lower speeds.

    DC Servomotor Torque Dynamics: Features a high torque-to-inertia ratio, allowing the motor to accelerate, decelerate, and reverse direction almost instantaneously.

  • AC Servomotor Phase Shift: Requires a 90-degree phase difference between its fixed reference phase and variable control phase to generate a revolving magnetic field.

    AC Servomotor Reliability: Lacks mechanical brushes and commutators, eliminating electrical sparking, brush wear, and high maintenance overhead.

  • Synchro Transmitter (Tx) Function: Converts the physical angular position of its input shaft into proportional electrical voltage signals across its three stator windings.

    Synchro Receiver (Rx) Function: Matches the stator electrical signals from the transmitter to create an identical magnetic field, forcing its rotor to align synchronously.

    Stepper Motor Open-Loop Advantage: Operates accurately without feedback sensors by moving in precise, discrete angular increments per input electrical pulse.

  • Variable Reluctance (VR) Stepper and Permanent Magnet (PM) Stepper.

    Hybrid Stepper Motor: Combines VR and PM designs for high step resolution, torque, and accuracy.