| Matrix or Vector | Shape Functions | Integration Points |
|---|---|---|
| Stiffness Matrix | None (lumped) | None |
| Damping Matrix | None (lumped, harmonic analysis only) | None |
| Load Vector | None (lumped) | None |
The piezoelectric circuit element, CIRCU94, simulates basic linear electric circuit components that can be directly connected to the piezoelectric FEA domain. For details about the underlying theory, see Wang and Ostergaard([324]). It is suitable for the simulation of circuit-fed piezoelectric transducers, piezoelectric dampers for vibration control, crystal filters and oscillators etc.
CIRCU94 contains 5 linear electric circuit element options:
| a. Resistor | (KEYOPT(1) = 0) |
| b. Inductor | (KEYOPT(1) = 1) |
| c. Capacitor | (KEYOPT(1) = 2) |
| d. Current Source | (KEYOPT(1) = 3) |
| e. Voltage Source | (KEYOPT(1) = 4) |
Options a, b, c, d are defined by two nodes I and J (see figure above), each node having a VOLT DOF. The voltage source is also characterized by a third node K with CURR DOF to represent an auxiliary charge variable.
The finite element equations for the resistor, inductor, capacitor and current source of CIRCU94 are derived using the nodal analysis method (McCalla([189])) that enforces Kirchhoff's Current Law (KCL) at each circuit node. To be compatible with the system of piezoelectric finite element equations (see Piezoelectrics), the nodal analysis method has been adapted to maintain the charge balance at each node:
 | (13–145) |
where:
| [K] = stiffness (capacitance) matrix |
| {V} = vector of nodal voltages (to be determined) |
| {Q} = load vector of nodal charges |
The voltage source is modeled using the modified nodal analysis method (McCalla([189])) in which the set of unknowns is extended to include electric charge at the auxiliary node K, while the corresponding entry of the load vector is augmented by the voltage source amplitude. In a transient analysis, different integration schemes are employed to determine the vector of nodal voltages.
For a resistor, the generalized trapezoidal rule is used to approximate the charge at time step n+1 thus yielding:
 | (13–146) |
 | (13–147) |
 | (13–148) |
where:
| θ = first order time integration parameter (input on TINTP command) |
| Δt = time increment (input on DELTIM command) |
| R = resistance |
|
|
|
The constitutive equation for an inductor is of second order with respect to the charge time-derivative, and therefore the Newmark integration scheme is used to derive its finite element equation:
 | (13–149) |
 | (13–150) |
where:
| L = inductance |
|
|
|
| α, δ = Newmark integration parameters (input on TINTP command |
A capacitor with nodes I and J is represented by
 | (13–151) |
 | (13–152) |
where:
| C = capacitance |
|
|
For a current source, the [K] matrix is a null matrix, while the charge vector is updated at each time step as
 | (13–153) |
where:
|
|
Note that for the first substep of the first load step in a transient analysis, as well as on the transient analysis restart, all the integration parameters (θ, α, δ) are set to 1. For every subsequent substep/load step, ANSYS uses either the default integration parameters or their values input using the TINTP command.
In a harmonic analysis, the time-derivative is replaced by jω, which produces
 | (13–154) |
for a resistor,
 | (13–155) |
for an inductor, and
 | (13–156) |
where:
| j = imaginary unit |
| ω = angular frequency (input on HARFRQ command) |
The element charge vector {Q} is a null vector for all of the above components.
For a current source, the [K] matrix is a null matrix and the charge vector is calculated as
 | (13–157) |
where:
|
| IS = source current amplitude |
| φ = source current phase angle (in radians) |
Note that all of the above matrices and load vectors are premultiplied by -1 before being assembled with the piezoelectric finite element equations that use negative electric charge as a through variable (reaction "force") for the VOLT degree of freedom.