Signal-flow graphs are another method for visually representing a system. Signal Flow Diagrams are especially useful, because they allow for particular methods of analysis, such as Mason's Gain Formula.
Signal flow diagrams typically use curved lines to represent wires and systems, instead of using lines at right-angles, and boxes, respectively. Every curved line is considered to have a multiplier value, which can be a constant gain value, or an entire transfer function. Signals travel from one end of a line to the other, and lines that are placed in series with one another have their total multiplier values multiplied together (just like in block diagrams).
Signal flow diagrams help us to identify structures called "loops" in a system, which can be analyzed individually to determine the complete response of the system.
If we have computed our delta values (above), we can then use Mason's Gain Rule to find the complete gain of the system:
Where M is the total gain of the system, represented as the ratio of the output gain (yout) to the input gain (yin) of the system. Mk is the gain of the kth forward path, and Δk is the loop gain of the kth loop.
Solving a signal-flow graph by systematic reduction : Two interlocking loopsEdit
This example shows how a system of five equations in five unknowns is solved using systematic reduction rules. The independent variable is . The dependent variables are , , , , . The coefficients are labeled .
Solving a signal-flow graph by systematic reduction: Three equations in three unknownsEdit
This example shows how a system of three equations in three unknowns is solved using systematic reduction rules.
The independent variables are , , . The dependent variables are , , . The coefficients are labeled . The steps for solving follow:
Electrical engineering: Construction of a flow graph for a RC circuitEdit
This illustration shows the physical connections of the circuit. Independent voltage source S is connected in series with a resistor R and capacitor C. The example is developed from the physical circuit equations and solved using signal-flow graph techniques. Polarity is important:
S is a source with the positive terminal at N1 and the negative terminal at N3
R is a resistor with the positive terminal at N1 and the negative terminal at N2
C is a capacitor with the positive terminal at N2 and the negative terminal at N3.
The unknown variable of interest is the voltage across capacitor C.
Approach to the solution:
Find the set of equations from the physical network. These equations are acausal in nature.
Branch equations for the capacitor and resistor. The equations will be developed as transfer functions using Laplace transforms.
We then look at the inventory of equations, and the signals that each equation relates:
The next step consists in assigning to each equation a signal that will be represented as a node. Each independent source signal is represented in the signal-flow graph as a source node, therefore no equation is assigned to the independent source . There are many possible valid signal flow graphs from this set of equations. An equation must only be used once, and the variables of interest must be represented.
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 = 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 variables in purple.