Design of C-element

C-element is a latch that synchronises the phases of its inputs. A symbol for a 2-input C-element and its timing diagram are shown in the figures below. Initially all the signals are in the low state. When both inputs in1 and in2 go high, the output out also switches to logical 1. It stays in this state until both inputs go low, at which stage the output switches to logical 0.

Note that the output of C-element is not fully determined by its inputs: When the inputs have different values the output keeps its previous value, i.e. it can be either 0 or 1. That is, C-element is not a combinational gate, but a latch with internal memory.

Let us model the C-element behaviour using the Signal Transition Graph (STG) formalism. Create a new STG work and translate the sequence of events in the timing diagram into a sequence of STG transitions. Basically you need to create a signal transition for each event of the timing diagram and capture the causality between these events by means of directed arcs between signal transitions. Recreate the following STG model in Workcraft.

Detailed instructions

The resulting STG should look as follows; save it as celement.stg.work – this file will be used later on to verify C-element circuit implementations.

Activate the simulation tool and exercise the obtained STG model. Click one of the enabled signal transitions (they are highlighted in orange) to fire (or execute) it, which changes the state (or marking) of the STG. Note that the sequence of fired transitions is recorded in the simulation trace that is somewhat similar to the original timing diagram. Check that the simulation traces correspond to the intended behaviour of C-element.

For example, simulate the following trace in2+, in1+, out+, in1-, in2-, out-, in1+, in2+ – you can copy this trace to clipboard and paste it in simulation tool by pressing . Then use the generator of trace diagram to convert the STG simulation trace to Digital Timing Diagram model. The produced timing diagram should resemble the initial informal specification and looks as follows (0 or 1 in brackets next to the signal name denotes its initial state).

Before proceeding to the synthesis of the C-element it is a good idea to verify that its specification meets essential requirements, e.g. that it is consistent, free from deadlocks, input-proper, and output-persistent. These properties can be checked via the Verification menu (see standard verification properties for details). Note that it is possible to check these properties either separately or jointly with a single click.

If the specification violates any of these properties then a trace leading to the problematic state will be reported. This trace can be simulated for better understanding the reported issues and for correcting them in the specification.

The STG specification can now be synthesised into an asynchronous circuit implementation with either MPSat or Petrify back-end via Synthesis menu. Workcraft automatically displays a circuit model derived from the synthesis output. For example, a complex-gate implementation obtained with MPSat (Synthesis→Complex gate [MPSat] menu item) would look as follows: you may get a slightly different synthesis results

The circuit can be subsequently saved in Verilog format via File→Export→Verilog netlist (*.v) menu. Note that complex-gate synthesis does not map gates into a cell library and all internal/output signals are represented by assign statements, as follows:

// Verilog netlist generated by Workcraft 3
module Untitled (out, in1, in2);
    input in1, in2;
    output out;
 
    assign out = (out | in2) & in1 | in2 & out;
 
    // signal values at the initial state:
    // !out !in1 !in2
endmodule

For the purposes of this exercise let us capture (a slightly modified) C-element schematics manually. By expanding the parentheses one can re-write the equation for signal out as follows:

  assign out = (in1 & in2) | (in1 & out) | (in2 & out);

This equation is directly implementable by an AND-OR complex gate computing the function Z = A*B + C*D + E*F, called AO222 in the default gate library shipped with Workcraft. Create a new Digital Circuit work and capture this C-element implementation manually.

• Activate functional generator and click in the desired position of the AND-OR complex gate.
• Activate selection tool .
• Select the only pin of the newly created function component.
• In the Property editor change the Name of the pin to Z and modify its Set function to A*B+C*D+E*F.

• Activate port generator . Pay attention to the hint at the bottom of the screen: Click to create an output port (hold Shift for input port).
• Click in intended location of the output port. Note that by default the port will be named out0 - you can change this name to out later.
• Hold Shift and click in the desired locations of the input ports – they will be automatically assigned the names in0 and in1.
• Switch to the selection tool, choose the output port, and change its Name to out in the Property editor..
• We also need to change the name of an input port. Let us use another Workcraft editor feature for this – double-click on the port in0 and rename it to in2 in place.

• Activate connection tool .
• Connect input ports in1 to the 1^st and 5^th pins of the complex gate (A and E respectively).
• Connect input ports in2 to the 2^nd and 3^rd pins of the complex gate (B and C respectively).
• Connect the output pin of the gate to the output port out and to the 4^th and 6^th inputs of the gate (D and F respectively).

Activate the simulation tool and simulate the captured complex gate implementation of the C-element. Ports, pins and wires are colour-coded: blue means low level and red means high level of the signal. Excited ports and gates are highlighted in orange.

Click one of the excited ports or gates to toggle its logical value. Similar to the STG simulation, the sequence of signal events is recorded in the simulation trace and can be subsequently replayed for analysing the circuit's behaviour or converted to a timing diagram for visualisation purpose.

To conduct formal verification the circuit has to be converted into an STG. Normally this is done silently by the tool, but it is possible to view the intermediate circuit-STG via Conversion→Signal Transition Graph. The resulting STG for this circuit is shown below. Note that your layout may be slightly different and that this STG will not be required in the rest of the tutorial.

Note that if the C-element's inputs are not restricted in any way then they can change in an unexpected manner and cause malfunction of the circuit. This can be confirmed by checking the circuit for output persistency Verification→Output persistency [MPSat] – the tool will warn you that the circuit is being verified without no environment STG and produce the following violation traces:

This indicates that output out exhibits non-persistent behaviour. Indeed, after the trace in2+, in1+ the output out is excited and ready to switch to logical 1, however either in1- or in2- would disable it. Play one of the reported traces to simulate this unexpected behaviour.

The circuit behaviour can be restricted by an environment STG as follows:

• In the circuit editor make sure that no components are selected (click on the editor canvas).
• In the property editor choose the Environment property and select the work file with the original STG specification – the celement.stg.work file.

Repeat the verification procedure to check that assuming a well behaved environment the circuit is output persistent, deadlock-free and conforms to the environment specification. All these verification steps can be performed simultaneously via Verification→Conformation, deadlock freeness, and output persistency (reuse unfolding) [MPSat].

Large gates like AO222 may not be available in the technology library and thus other implementations of C-element using smaller gates are of interest. However, it is very easy to make a mistake when designing asynchronous circuits, and so any such implementation has to be formally verified against the original STG specification.

Download the following C-element implementation due to Mayevsky [1] celement-decomposed.circuit.work and verify that it conforms to the STG specification, is free from deadlocks, input-proper, and output persistent.

Note that the correctness of this implementation depends on the isochronic forks assumption: the difference in arrival times of a signal to the ends of a wire fork is negligible compared to any gate delay. If this assumption is violated, the above implementation can exhibit non-persistent behaviour. Download the following C-element implementation celement-decomposed-hazard.circuit.work where a delay in a wire fork is made explicit (modelled by a buffer). Formally verify this model, inspect the violation trace, and explain what is going wrong in the circuit.

Download all the Workcraft models discussed in this tutorial here:

C-element models