Fractional Impedance/Admittance Pole (fracpole)

Fractional Impedance Pole takes as input a start frequency, a stop frequency, a negative slope, the unity intercept point, and the number of lumps, and synthesizes a RC circuit that models a fractional impedance pole over the given frequency range. The circuit is a one-port that exhibits poles and zeros that are real and that are spaced evenly in a logarithmic sense over the frequency range. The impedance exhibited by the one port approximates a fractional pole slope between -1 and 0 in the frequency range. In other words, if the impedance is plotted on a log-log scale, it will have a negative slope equal to the fraction specified. The user requested half a pole, the slope will be -1/2, etc. Of course it is a lumped approximation, so the slope will not be exact, but it will slowly oscillate about the desired value.

o----+------+------+--...--+------+

| | | | |

/ / / ... / |

R0 R1 R2 Rn |

| | | | |

| --- --- ... --- ---

| --- --- --- ---

| | C1 | C2 | Cn | Cinf

o----+------+------+--...--+------+

This model is converted to model a Fractional Admittance Pole by combining it with a gyrator.

o---+ +----+------+------+--...--+------+

| | | | | | |

|+---------+| |

|| || / / / ... / |

|| Gyrator || R0 R1 R2 Rn |

|| || | | | | |

|| || | --- --- ... --- ---

|+---------+| | --- --- --- ---

| | | | C1 | C2 | Cn | Cinf

o---+ +--+-+------+------+--...--+------+

GND

It can be used to model skin-effect loss in an inductor and dielectric absorption in a capacitor, and can be used to shape white noise into flicker noise.

Profile

In both fractional impedance/admittance pole cases, what happens outside the range of the approximation is specifies the parameter profiles of fracpole. It is a code that consists of a pair of letters. The first letter represents the low frequency behavior and the second represents the high frequency behavior. The letters are either f or d, f represents flat or a zero-pole slope, and d represents down or a one-pole slope. Depending on the profile chosen, either R0 or Cinf may or may not be present in the synthesized RC circuit.

Internal Nodes

Internal nodes are created for each RC pair in the synthesized RC circuit, and they are labeled as InstanceName:int_1, InstanceName:int_2, ... In the case of fractional admittance pole, one more additional internal node, InstanceName:int_gyr, is created on the opposite side of gyrator from the terminals of the device.

Current

The current reported by fracpole device is the static terminal current. In the case of fractional impedance pole, this current does not include the dynamic part. To find the total current of fractional impedance pole, either use current probe in series with it or use the option useprobes=yes.

This device is supported within altergroups.

Fractional Impedance Pole approximates

Z(s) = Coef*s^Slope

for s=2*pi*freq*j with f0 < freq < f1, and -1 < Slope < 0.

Fractional Admittance Pole approximates

Z(s) = Coef*s^Slope

(or Y(s) = s^(-Slope)/Coef)

for s=2*pi*freq*j with f0 < freq < f1, and 0 < Slope < 1.

Sample Instance Statement

without model:

fp (1 2) fracpole f0=1 f1=1M coef=1k dec=1.0 slope=-0.5 profile=ff

with model:

fp (1 2) fpModel f0=1 f1=1M coef=1.5k lumps=12 slope=-0.5 profile=dd

Sample Model Statement

model fpModel fracpole f0=1 f1=1M coef=1k dec=1.0 slope=-0.5 profile=ff

Instance Syntax

Name  1  2 ModelName parameter=value ...

Name  1  2 fracpole parameter=value ...

Instance Parameters

`f0=1.0 Hz`	Low frequency limit for the approximation.
`f1=1.0e6 Hz`	High frequency limit for the approximation.
`coef=1.0` Ω	Unity intercept point for ideal impedance at 2pifreq=1.
`slope=-0.5`	Slope of the impedance when plotted on a log-log scale.
`lumps`	Number of lumps used in the approximation.
`dec=1.0`	Number of lumps per decade used in the approximation.
`profile=dd if abs(slope)`
`m=1.0`	Multiplicity factor.
`ic (V)`	Initial condition for fractional impedance pole devices with `df` or `dd` profile.
`rforce=1.0` Ω	Resistance used when forcing initial conditions.

Model Syntax

model modelName fracpole parameter=value ...

Model Parameters

`f0=1.0 Hz`	Low frequency limit for the approximation.
`f1=1.0e6 Hz`	High frequency limit for the approximation.
`coef=1.0` Ω	Unity intercept point for ideal impedance at 2pifreq=1.
`slope=-0.5`	Slope of the impedance when plotted on a log-log scale.
`lumps`	Number of lumps used in the approximation.
`dec=1.0`	Number of lumps per decade used in the approximation.
`profile=dd if abs(slope)`

Operating-Point Parameters

`v (V)`	Voltage at operating point.
`i (A)`	Current. See comment on current above.
`pwr (W)`	Power dissipation.

Spectre Circuit Simulator Components and Device Models Reference
Product Version 23.1, June 2023