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Algebraic Loop

Demonstration of PathSim's automatic handling of algebraic loops.

You can also find this example as a single file in the GitHub repository.

What is an Algebraic Loop?

An algebraic loop occurs when the output of a block depends on its own input in the same timestep. This creates an implicit equation that must be solved:

MATHDISPLAY0ENDMATH

PathSim automatically detects algebraic loops and resolves them at each timestep using accelerated fixed-point iterations through wrapping looping connections with the ConnectionBooster class.

System Description

In this example:

A source generates a sinusoidal signal: MATHINLINE2ENDMATH
An amplifier multiplies by gain MATHINLINE3ENDMATH
An adder sums the source and amplifier output
The amplifier input comes from the adder output, creating a loop

This creates the algebraic equation: MATHDISPLAY0ENDMATH

Which has the analytical solution: MATHDISPLAY1ENDMATH

PathSim automatically detects and solves algebraic loops using accelerated fixed-point iteration trough the ConnectionBooster class. Tolerances for the loop solver are configurable via the `tolerance_fpi` parameter.

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System Parameters

We define:

Timestep: 0.1 s
Algebraic feedback gain: MATHINLINE0ENDMATH

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Block Diagram

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Connections

Notice that the output of the Add block connects to the input of Amp, and the output of Amp connects back to the second input of Add. This creates the algebraic loop.

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Simulation

PathSim automatically detects the algebraic loop and solves it using fixed-point iteration at each timestep. No special configuration is needed!

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12:43:14 - INFO - LOGGING (log: True)
12:43:14 - INFO - BLOCKS (total: 4, dynamic: 0, static: 4, eventful: 0)
12:43:14 - INFO - GRAPH (nodes: 4, edges: 6, alg. depth: 1, loop depth: 2, runtime: 0.467ms)
12:43:14 - INFO - STARTING -> TRANSIENT (Duration: 5.00s)
12:43:14 - INFO - --------------------   2% | 0.0s<0.1s | 340.0 it/s
12:43:14 - INFO - ####----------------  21% | 0.0s<0.0s | 6406.3 it/s
12:43:14 - INFO - ########------------  40% | 0.0s<0.0s | 6770.9 it/s
12:43:14 - INFO - ############--------  60% | 0.0s<0.0s | 6789.0 it/s
12:43:14 - INFO - ################----  80% | 0.0s<0.0s | 7532.9 it/s
12:43:14 - INFO - #################### 100% | 0.0s<--:-- | 6844.3 it/s
12:43:14 - INFO - FINISHED -> TRANSIENT (total steps: 51, successful: 51, runtime: 28.28 ms)

Results

The plot shows:

src (blue): Input signal = MATHINLINE0ENDMATH
amp (orange): Amplifier output = MATHINLINE1ENDMATH
add (green): Adder output = MATHINLINE2ENDMATH

The algebraic loop is solved correctly at each timestep!

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Verification

Let's verify the solution is correct by checking that the adder output matches the analytical solution.

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Maximum error: 2.22e-16