Verilog-A Nonlinear and Dynamic systems

Introduction¶

Verilog-A is a high-level analog hardware description language that enables designers to model and simulate complex electronic systems. Engineering systems such as amplifiers, oscillators, power converters, sensors, and communication blocks are governed by nonlinear and time-varying behavior, often requiring a modeling language with strong support for differential equations, nonlinear responses, and dynamic conditions. Verilog-A excels in these areas by providing dedicated analog operators, numerical constructs, and solver-friendly modeling styles.

Nonlinear Functions in Verilog-A¶

Most analog and mixed-signal devices inherently exhibit nonlinear behavior. For example:

• MOSFET saturation characteristics
• Diode exponential conduction
• BJT transconductance variations
• RF mixers and modulators
• Nonlinear filters
• Output saturation of operational amplifiers

To model these phenomena, Verilog-A supports a wide set of nonlinear built-in functions including:

Diode nonlinearity¶

Id = Is * (exp(V(d,a)/Vt) - 1);
I(d, a) <+ Id;

However, pure exponential functions may cause numerical convergence issues. Therefore, solvers recommend bounded functions like limexp() instead:

Id = Is * (limexp(V(d,a)/Vt) - 1);

Soft limiting to avoid solver failure¶

Hard clipping (e.g., using if conditions) introduces discontinuities.

if (Vin <= Vscale/2)
    Vo = 0;
else
    Vo = Vsat;

Instead, smooth nonlinear boundaries are better:

Vo = Vsat * tanh(Vin/Vscale);

Time-Dependent Functions¶

Dynamic systems evolve with time — requiring memory and derivative behavior. Verilog-A includes direct support for differential equations, delays, and frequency-dependent functions.

Key-Dynamic operators

Capacitor model¶

I(p, n) <+ C * ddt(V(p, n));

This directly reflects :

$$I=C\cfrac{dV}{dt}$$

Inductor model¶

V(p, n) <+ L * ddt(I(p, n));

Frequency-dependent behavior¶

Laplace operator representation of first-order RC filter:

V(out) <+ laplace_nd(V(in), {1}, {R*C,1});

State-dependent Nonlinear Dynamics¶

Dynamic nonlinearities involve both voltage dependence and derivatives:

I(p, n) <+ C(V(p,n)) * ddt(V(p,n));

Nonlinear capacitor modeling:

I(p, n) <+ C(V(p,n)) * ddt(V(p,n))*(1+vc2*V(p,n)+vc2*V(p,n)**2);

Time-dependent functions enable modeling of:

• Filters and Compensators
• VCOs and PLLs
• Switching converters
• RF front-ends
• Neural networks and nonlinear control systems

Handling Initial Conditions¶

Initial conditions heavily influence dynamic simulations such as transient startup. Verilog-A provides tools to manage these states explicitly.

Methods for initial conditions¶

Initialization Block

analog begin
    @(initial_step) begin
        state = Vinit;
    end
end

Integrator with initial voltage

V(out) <+ idt(input_signal, Vinitial);

This solves the common startup convergence failure seen in standalone integrators.

Conditions Where Initialization is Crucial

• Oscillator waveform startup
• Power-on reset modeling
• Charge-based transistor models
• Sequential analog systems
• Initial capacitor/inductor energy

Handling initial conditions properly ensures:

• Faster convergence
• Realistic transient response
• Accurate switching behavior

Preventing floating states¶

Floating nodes can produce numerical instability. Methods to avoid this include :

• Adding small parasitic R or C
• Providing DC paths in dynamic models
• Ensuring charge conservation using ddt()

Best practices summary¶