Capacitor

Linear or nonlinear capacitor with optional tolerance, operational limits and fault behavior

Library:
Simscape / Electrical / Passive / Basic

Description

The Capacitor block lets you model linear, nonlinear (table-based), and frequency-dependent capacitors, including polar capacitors. Optionally, you can also model the following effects:

You can turn these modeling options on and off independently of each other. When all the additional options are turned off, the component behavior is identical to the Simscape™ Foundation library Capacitor block.

In its simplest form, the Capacitor block models a linear capacitor, described with the following equation:

$I = C \frac{d V}{d t}$

where:

I is the current.
C is the capacitance.
V is the voltage.
t is the time.

To model a nonlinear or polar capacitor, set the Capacitance model parameter to Lookup table and provide a lookup table of capacitance-voltage values:

For polar capacitors, where this lookup table is asymmetric with respect to the applied terminal voltage, set the Symmetric C-V table parameter to No - use C-V data as-is.
For other types of nonlinear capacitor, ensure symmetry of the capacitance with regards to the applied terminal voltage by setting the Symmetric C-V table parameter to Yes - use voltage magnitude when computing C.

To model a frequency-dependent capacitor with resistive and dielectric losses, set the Capacitance model parameter to Dielectric relaxation (Debye). The Debye relaxation model considers a collection of noninteracting dipoles in the frequency domain. The result is in terms of a complex permittivity. The real (𝜖′) and imaginary (𝜖′′) parts of the complex permittivity are given by the equations:

$ε' = ε_{\infty} + \frac{ε_{s} - ε_{\infty}}{1 + ω^{2} τ^{2}}$

$ε'' = \frac{(ε_{s} - ε_{\infty}) ω τ}{1 + ω^{2} τ^{2}},$

where 𝜔 is the radial frequency, 𝜖_∞ is the real permittivity at very high frequency, 𝜖_𝑠 is the real permittivity at low frequency and 𝜏 is the relaxation time constant.

In the time-domain, the characteristic equation for a capacitor in the Debye model is

$τ i = - q + C_{s} (α τ \dot{ν} + ν),$

where 𝐶_𝑠 is the low-frequency capacitance, 𝛼 = 𝜖_∞/𝜖_𝑠, 𝑞 is the charge, 𝑖 is the current and 𝑣 is the voltage across the capacitor.

Most datasheets do not provide explicit values for the complex permittivity nor for the relaxation time; however, often the dissipation factor is provided at two frequencies. The parameters 𝛼 and 𝜏 may be derived from these values as described by the equations:

$τ = \frac{D F_{1} D F_{2} (ω_{2}^{2} - ω_{1}^{2}) + \sqrt{D F_{1}^{2} D F_{2}^{2} {(ω_{2}^{2} - ω_{1}^{2})}^{2} + 4 ω_{1} ω_{2} (D F_{2} ω_{2} - D F_{1} ω_{1}) (D F_{1} ω_{2} - D F_{2} ω_{1})}}{2 ω_{1} ω_{2} (D F_{2} ω_{2} - D F_{1} ω_{1})}$

$α = \frac{ω_{1} τ - D F_{1}}{ω_{1} τ (1 + D F_{1} ω_{1} τ)},$

where 𝜔₁ and 𝜔₂ are two different frequencies, and DF₁ and DF₂ are the dissipation factors calculated at said frequencies, respectively.

The argument of the square root must be positive for a valid parameterization in the Debye model.

Tolerances

You can apply tolerances to the nominal value you provide for the Capacitance parameter. Datasheets typically provide a tolerance percentage for a given capacitor type. The table shows how the block applies tolerances and calculates capacitance based on the selected Tolerance application option.

Option	Capacitance Value
`None — use nominal value`	C
`Random tolerance`	Uniform distribution: C · (1 – tol + 2· tol· `rand`) Gaussian distribution: C · (1 + tol · `randn` / nSigma)
`Apply maximum tolerance value`	C · (1 + tol )
`Apply minimum tolerance value`	C · (1 – tol )

In the table,

C is the Capacitance parameter value, nominal capacitance.
tol is the fractional tolerance, Capacitance tolerance (%) /100.
nSigma is the value you provide for the Number of standard deviations for quoted tolerance parameter.
rand and randn are standard MATLAB^® functions for generating uniform and normal distribution random numbers.

Note

If you choose the Random tolerance option and you are in "Fast Restart" mode, the random tolerance value is updated on every simulation if at least one between the fractional tolerance, tol, or the Number of standard deviations for quoted tolerance, nSigma, is set to Run-time and is defined with a variable (even if you do not modify that variable).

Operating Limits

You can specify operating limits in terms of maximum working voltage and the maximum (instantaneous) power dissipation in the series resistance and in the parallel conductance of the capacitor.

For polar capacitors, you can define the working voltage range in such a way that the block provides a warning, or an error, if the polarity of the applied voltage becomes incorrect.

When an operating limit is exceeded, the block can either generate a warning or stop the simulation with an error. For more information, see Operating Limits.

Faults

Instantaneous changes in capacitor parameters are unphysical. Therefore, when the Capacitor block enters the faulted state, the capacitance, resistance, and conductance transition to their faulted values over a period of time, according to the following formula:

CurrentValue = FaultedValue – ( FaultedValue – UnfaultedValue ) · sech (∆t / τ) (1)

where:

∆t is the time since the onset of the fault condition.
τ is the user-defined time constant associated with the fault transition.

The block can trigger the start of fault transition:

At a specific time
When terminal voltage is outside the permissible voltage range for longer than a specific time interval

You can enable or disable these trigger mechanisms separately, or use them together if more than one trigger mechanism is required in a simulation. When more than one mechanism is enabled, the first mechanism to trigger the fault transition takes precedence. In other words, component fails no more than once per simulation.

You can also choose whether to issue an assertion when a fault occurs, by using the Reporting when a fault occurs parameter. The assertion can take the form of a warning or an error. By default, the block does not issue an assertion.

Variables

Use the Variables section of the block interface to set the priority and initial target values for the block variables prior to simulation. For more information, see Set Priority and Initial Target for Block Variables.

The Capacitor voltage variable lets you specify a high-priority target for the initial capacitor voltage at the start of simulation.

Ports

Conserving

expand all

`+` — Positive terminal
electrical

Electrical conserving port associated with the capacitor positive voltage.

`-` — Negative terminal
electrical

Electrical conserving port associated with the capacitor negative voltage.

Parameters

expand all

Main

`Capacitance model` — Capacitor type
`Constant` (default) | `Lookup table` | `Dielectric relaxation (Debye)`

Select the type of capacitor:

Constant — Model a linear capacitor, with nominal capacitance defined by the Capacitance parameter value. This is the default.
Lookup table — Model a nonlinear capacitor, where the nominal capacitance value changes based on the value of applied terminal voltage.
Dielectric relaxation (Debye) —Model a frequency-dependent capacitor with resistive and dielectric losses.

`Capacitance` — Nominal capacitance
`1e-6` `F` (default)

The nominal capacitance value for linear capacitor, or the low-frequency capacitance in the Debye parameterization.

Dependencies

This parameter is visible only when you select Constant or Dielectric relaxation (Debye) for the Capacitance model parameter.

`Capacitance values` — Vector of capacitance values
`[1e-05, 1e-06]` `F` (default) | positive vector

The vector of capacitance values, for table lookup based on the corresponding voltage value. Capacitance values must be greater than 0. The vector length must be the same as the voltage vector length.

Dependencies

This parameter is visible only when you select Lookup table for the Capacitance model parameter.

`Corresponding voltage values` — Input vector of voltage values
`[0, 10]` `V` (default)

The input vector of voltage values for table-based capacitance calculation. The vector length must be greater than or equal to 2, and the values must be strictly monotonic, either increasing or decreasing.

Dependencies

This parameter is visible only when you select Lookup table for the Capacitance model parameter.

`Symmetric C-V table` — Table data
`Yes - use voltage magnitude when computing C` (default) | `No - use C-V data as-is`

Specify how to use the table data:

Yes - use voltage magnitude when computing C — Use this option to ensure symmetry of the capacitance with regards to the applied terminal voltage. This is the default.
No - use C-V data as-is — Use this option to model polar capacitors. For example, with default parameter values for table-based capacitance, applied voltage of –10 V would produce nominal capacitance of 1e-6 F. However, if you select No - use C-V data as-is for the Symmetric C-V table parameter, the resulting capacitance value is 1e-5 F, because the block uses the nearest input value for extrapolation.

Dependencies

This parameter is visible only when you select Lookup table for the Capacitance model parameter.

`Frequencies for specifying dissipation factors [f1 f2]` — Frequencies for dissipation factors calculation
`[1, 10]` `kHz` (default)

Frequencies at which the dissipation factors [DF1 DF2] are calculated, in kHz.

Dependencies

This parameter is visible only when you select Dielectric relaxation (Debye) for the Capacitance model parameter.

`Dissipation factors (%) at f1 and f2 [DF1 DF2]` — Dissipation factors at f1 and f2
`[.8, 1.2]` (default)

Ratio between the equivalent series resistance and the capacitive reactance, or the tangent of the loss angle. Dissipation factors are a common metric for capacitors.

Dependencies

This parameter is visible only when you select Dielectric relaxation (Debye) for the Capacitance model parameter.

`Capacitance tolerance (%)` — Capacitor tolerance
`5` (default)

The capacitor tolerance as defined on the manufacturer datasheet. For table-based capacitors, this tolerance is applied to the entire table at once.

`Tolerance application` — Tolerance application
`None — use nominal value` (default) | `Random tolerance` | `Apply maximum tolerance value` | `Apply minimum tolerance value`

Select how to apply tolerance during simulation:

None — use nominal value — The block does not apply tolerance, uses the nominal capacitance value. This is the default.
Random tolerance — The block applies random offset to the capacitance value, within the tolerance value limit. You can choose Uniform or Gaussian distribution for calculating the random number by using the Tolerance distribution parameter.
Apply maximum tolerance value — The capacitance is increased by the specified tolerance percent value.
Apply minimum tolerance value — The capacitance is decreased by the specified tolerance percent value.

`Tolerance distribution` — Tolerance distribution type
`Uniform` (default) | `Gaussian`

Select the distribution type:

Uniform — Uniform distribution. This is the default.
Gaussian — Gaussian distribution.

Dependencies

This parameter is visible only when you select Random tolerance for the Tolerance application parameter.

`Number of standard deviations for quoted tolerance` — Number of standard deviations for quoted tolerance
`4` (default)

Number of standard deviations for calculating the Gaussian random number.

Dependencies

This parameter is visible only when you select Gaussian for the Tolerance distribution parameter.

`Series resistance` — Series resistance
`1e-6` `Ohm` (default)

Simulation of some circuits may require the presence of the small series resistance. Equivalent series resistance (ESR) is sometimes specified on manufacturer datasheets. If not, you can define this resistance for a linear capacitor via the dissipation factor (DF), which is also shown on many datasheets. The relationship is DF = 2π· f· C· ESR, where f is signal frequency. For a Debye capacitor, the Dissipation factors (%) at f1 and f2 [DF1 DF2] are corrected for this additional series resistance prior to computing 𝛼 and 𝜏.

`Parallel conductance` — Parallel leakage path
`0` `1/Ohm` (default)

Parallel leakage path associated with the capacitor. For capacitors connected in series, the presence of a small parallel conductance can help with convergence.

`Averaging period for power logging` — Averaging period for power logging
`0` `s` (default)

Averaging period for power logging, in s.

Note

If this parameter is set to 0, the output is the instantaneous power.

Dependencies

This parameter is visible only when you select Dielectric relaxation (Debye) for the Capacitance model parameter.

Operating Limits

`Enable operating limits` — Enable operating limits
`No` (default) | `Yes`

Select Yes to enable reporting when the operational limits are exceeded. The associated parameters become visible on the Operating Limits tab to let you select the reporting method and specify the operating limits in terms of power and working voltage.

`Reporting if operating limits exceeded` — Report behavior
`Warn` (default) | `Error`

Select what happens when an operating limit is exceeded:

Warn — The block issues a warning. This is the default.
Error — Simulation stops with an error.

`Working voltage range` — Working voltage range
`[-25, 25]` `V` (default)

Range of voltage values allowed for normal block operation, specified as a vector of size 2.

`Power rating` — Maximum instantaneous power dissipation
`1` `W` (default)

Maximum instantaneous power dissipation in the resistance and conductance elements associated with the capacitor.

Faults

`Enable faults` — Enable faults
`No` (default) | `Yes`

Select Yes to enable faults modeling. The associated Faults parameters become visible to let you select the reporting method and specify the trigger mechanism (temporal or behavioral). You can enable these trigger mechanisms separately or use them together. The default value is No.

`Reporting when a fault occurs` — Report behavior
`None` (default) | `Warn` | `Error`

Choose whether to issue an assertion when a fault occurs:

None — The block does not issue an assertion. This is the default.
Warn — The block issues a warning.
Error — Simulation stops with an error.

`Faulted capacitance as % of unfaulted` — Faulted capacitance as % of unfaulted
`100` (default)

Relative change in the capacitance when the block is in the faulted state, as compared to the unfaulted state. For table-based capacitances, the relative change is applied to all elements of the vector. The default value is 100, which means that the faulted capacitance is equal to the unfaulted capacitance.

`Faulted series resistance` — Faulted series resistance
`1e-3` `Ohm` (default)

Equivalent series resistance of the capacitor when the block is in the faulted state.

`Faulted parallel conductance` — Faulted parallel conductance
`0` (default) | `1/Ohm`

Parallel leakage conductance of the capacitor when the block is in the faulted state.

`Fault transition time constant` — Fault transition time constant
`1e-3` `s` (default)

Time constant associated with the transition to the faulted state, as described in Faults.

`Enable temporal fault trigger` — Time-based fault triggering
`No` (default) | `Yes`

Select Yes to enable time-based fault triggering.

`Simulation time for fault event` — Simulation time for fault event
`1` `s` (default)

Set the simulation time at which you want the block to start entering the fault state.

Dependencies

This parameter is visible only when you select Yes for the Enable temporal fault trigger parameter.

`Enable behavioral fault trigger` — Behavioral fault triggering
`No` (default) | `Yes`

Select Yes to enable behavioral fault triggering.

`Permissible voltage range` — Minimum and maximum permissible voltage
`[-100, 100]` `V` (default)

Specify the minimum and maximum permissible voltage. If the voltage value is outside this range for longer than the Time to fail when exceeding voltage range parameter value, then the block starts entering the fault state.

Dependencies

This parameter is visible only when you select Yes for the Enable behavioral fault trigger parameter.

`Time to fail when exceeding voltage range` — Time to fail when exceeding voltage range
`1` `s` (default)

Set the maximum length of time that the voltage can be outside the permissible voltage range without triggering the fault.

Dependencies

This parameter is visible only when you select Yes for the Enable behavioral fault trigger parameter.

Model Examples

AM Radio Receiver

A simplified AM radio receiver. A single tone signal at 2kHz is transmitted with a carrier frequency of 600kHz. The variable capacitor, Cres, in the resonant circuit is used in order to sweep through a certain frequency span. When the resonance passes through 600kHz, the signal is picked up and amplified by a two-stage Class A RF power amplifier. The signal is finally extracted by a diode detector, where it would normally be passed on to an audio amplifier (not included here). The Scope displays the final output, the value of the resonant capacitance, and the received and amplified signals.

Documentation

Capacitor

Description

Tolerances

Operating Limits

Faults

Variables

Ports

Conserving

+ — Positive terminal electrical

- — Negative terminal electrical

Parameters

Main

Capacitance model — Capacitor type Constant (default) | Lookup table | Dielectric relaxation (Debye)

Capacitance — Nominal capacitance 1e-6 F (default)

Dependencies

Capacitance values — Vector of capacitance values [1e-05, 1e-06] F (default) | positive vector

Dependencies

Corresponding voltage values — Input vector of voltage values [0, 10] V (default)

Dependencies

Symmetric C-V table — Table data Yes - use voltage magnitude when computing C (default) | No - use C-V data as-is

Dependencies

Frequencies for specifying dissipation factors [f1 f2] — Frequencies for dissipation factors calculation [1, 10] kHz (default)

Dependencies

Dissipation factors (%) at f1 and f2 [DF1 DF2] — Dissipation factors at f1 and f2 [.8, 1.2] (default)

Dependencies

Capacitance tolerance (%) — Capacitor tolerance 5 (default)

Tolerance application — Tolerance application None — use nominal value (default) | Random tolerance | Apply maximum tolerance value | Apply minimum tolerance value

Tolerance distribution — Tolerance distribution type Uniform (default) | Gaussian

Dependencies

Number of standard deviations for quoted tolerance — Number of standard deviations for quoted tolerance 4 (default)

Dependencies

Series resistance — Series resistance 1e-6 Ohm (default)

Parallel conductance — Parallel leakage path 0 1/Ohm (default)

Averaging period for power logging — Averaging period for power logging 0 s (default)

Dependencies

Operating Limits

Enable operating limits — Enable operating limits No (default) | Yes

Reporting if operating limits exceeded — Report behavior Warn (default) | Error

Working voltage range — Working voltage range [-25, 25] V (default)

Power rating — Maximum instantaneous power dissipation 1 W (default)

Faults

Enable faults — Enable faults No (default) | Yes

Reporting when a fault occurs — Report behavior None (default) | Warn | Error

Faulted capacitance as % of unfaulted — Faulted capacitance as % of unfaulted 100 (default)

Faulted series resistance — Faulted series resistance 1e-3 Ohm (default)

Faulted parallel conductance — Faulted parallel conductance 0 (default) | 1/Ohm

Fault transition time constant — Fault transition time constant 1e-3 s (default)

Enable temporal fault trigger — Time-based fault triggering No (default) | Yes

Simulation time for fault event — Simulation time for fault event 1 s (default)

Dependencies

Enable behavioral fault trigger — Behavioral fault triggering No (default) | Yes

Permissible voltage range — Minimum and maximum permissible voltage [-100, 100] V (default)

Dependencies

Time to fail when exceeding voltage range — Time to fail when exceeding voltage range 1 s (default)

Dependencies

Model Examples

AM Radio Receiver

Extended Capabilities

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See Also

Simscape Electrical Documentation

Support

`+` — Positive terminal
electrical

`-` — Negative terminal
electrical

`Capacitance model` — Capacitor type
`Constant` (default) | `Lookup table` | `Dielectric relaxation (Debye)`

`Capacitance` — Nominal capacitance
`1e-6` `F` (default)

`Capacitance values` — Vector of capacitance values
`[1e-05, 1e-06]` `F` (default) | positive vector

`Corresponding voltage values` — Input vector of voltage values
`[0, 10]` `V` (default)

`Symmetric C-V table` — Table data
`Yes - use voltage magnitude when computing C` (default) | `No - use C-V data as-is`

`Frequencies for specifying dissipation factors [f1 f2]` — Frequencies for dissipation factors calculation
`[1, 10]` `kHz` (default)

`Dissipation factors (%) at f1 and f2 [DF1 DF2]` — Dissipation factors at f1 and f2
`[.8, 1.2]` (default)

`Capacitance tolerance (%)` — Capacitor tolerance
`5` (default)

`Tolerance application` — Tolerance application
`None — use nominal value` (default) | `Random tolerance` | `Apply maximum tolerance value` | `Apply minimum tolerance value`

`Tolerance distribution` — Tolerance distribution type
`Uniform` (default) | `Gaussian`

`Number of standard deviations for quoted tolerance` — Number of standard deviations for quoted tolerance
`4` (default)

`Series resistance` — Series resistance
`1e-6` `Ohm` (default)

`Parallel conductance` — Parallel leakage path
`0` `1/Ohm` (default)

`Averaging period for power logging` — Averaging period for power logging
`0` `s` (default)

`Enable operating limits` — Enable operating limits
`No` (default) | `Yes`

`Reporting if operating limits exceeded` — Report behavior
`Warn` (default) | `Error`

`Working voltage range` — Working voltage range
`[-25, 25]` `V` (default)

`Power rating` — Maximum instantaneous power dissipation
`1` `W` (default)

`Enable faults` — Enable faults
`No` (default) | `Yes`

`Reporting when a fault occurs` — Report behavior
`None` (default) | `Warn` | `Error`

`Faulted capacitance as % of unfaulted` — Faulted capacitance as % of unfaulted
`100` (default)

`Faulted series resistance` — Faulted series resistance
`1e-3` `Ohm` (default)

`Faulted parallel conductance` — Faulted parallel conductance
`0` (default) | `1/Ohm`

`Fault transition time constant` — Fault transition time constant
`1e-3` `s` (default)

`Enable temporal fault trigger` — Time-based fault triggering
`No` (default) | `Yes`

`Simulation time for fault event` — Simulation time for fault event
`1` `s` (default)

`Enable behavioral fault trigger` — Behavioral fault triggering
`No` (default) | `Yes`

`Permissible voltage range` — Minimum and maximum permissible voltage
`[-100, 100]` `V` (default)

`Time to fail when exceeding voltage range` — Time to fail when exceeding voltage range
`1` `s` (default)

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.