RLC Chart

Author:	Ken Kundert
Version:	1.0.0
Released:	2022-01-25

What?

rlc_chart is library that renders impedance charts in SVG with the normal impedance versus frequency log-log grids, but they also include capacitance and inductance grids. They can be used to directly read component values from a plot of impedance. This is explained in Introduction to Phasors.

Consider the impedance of a leaky capacitor that has series resistance and inductance parasitics along with a shunt resistor as represented by the following circuit:

You can use the various grids on this graph to determine the values of the various components: C = 1 nF, L = 10 μH, Rs = 2 Ω, Rp = 500 kΩ, and f₀ = 1.6 MHz. You can do this in other ways, but they involve manual calculation. In addition, an RLC chart is a convenient way of sharing or publishing your findings.

Using an RLC chart is often enough to allow you to build a linear model for common two terminal components.

How?

Here is an example of how to use rlc_chart:

from rlc_chart import RLC_Chart
from math import log10 as log, pi as π

Rs = 2
Rp = 500_000
C = 1e-9
L = 10e-6
fmin = 1
fmax = 1e8
zmin = 1
zmax = 1e6
mult = 10**((log(fmax) - log(fmin))/400)
f = fmin
freq = []
impedance = []
j2π = 2j*π

# Compute impedance of component
# z1 = (Rs + 1/(jωC + jωL)     Rs=2Ω, C=1nF, L=10μH
# z2 = Rp                      Rp=500kΩ
# z = z1 ‖ z2
while(f <= 1.01*fmax):
    jω = j2π*f
    z1 = Rs + 1/(jω*C) + jω*L
    z2 = Rp
    z = z1 * z2 / (z1 + z2)
    freq += [f]
    impedance += [abs(z)]
    f *= mult

with RLC_Chart('lcr-chart.svg', fmin, fmax, zmin, zmax) as chart:
    chart.add_trace(freq, impedance)

Most of the code builds the two arrays that represent the trace. The impedance array is expected to contain positive real values. In this case it is the magnitude that is being plotted, though it is also common to call add_trace twice to show both the real and imaginary parts of the impedance.

RLC_Chart

The RLC_Chart class constructor takes the following required arguments:

filename:

Path to the output SVG file.

fmin:

The minimum frequency value (left-most value on the chart). This value is always rounded down the next lower multiple of 10. So for example, if you give 25 Hz as fmin, then 10 Hz is used.

fmax:

The maximum frequency value (right-most value on the chart). This value is always rounded up the next higher multiple of 10. So for example, if you give 75 MHz as fmax, then 100 MHz is used.

zmin:

The minimum impedance value (bottom-most value on the chart). This value is

always rounded down the next lower multiple of 10. So for example, if you give 150 mΩ zmin, then 100 mΩ is used.

zmax:

The maximum impedance value (top-most value on the chart). This value is always rounded up the next higher multiple of 10. So for example, if you give 800 kΩ as zmax, then 1 MΩ is used.

In addition, the following keyword arguments are optional.

axes:

Specifies which axes are desired, where the choices are f for frequency, z for impedance, c for capacitance, and l for inductance. axes is a string that contains any or all of the four characters, or none at all. If the characters are lower case, then only the major grid lines are drawn, and if given as upper case letters, both the major and minor grid lines are drawn. The default is "FZRC".

The visual clutter in the chart can be reduced by eliminating unneeded grid lines.

trace_width:

The width of a trace. The default is 0.025 inches.

trace_color:

The default color of the trace. You can use one of the named SVG colors, or you can use 'rgb(R,G,B)' where R, G, and B are integers between 0 and 255 that specify the intensity of red, blue, and green components of the color.

major_line_width:

The width of a major division line. The default is 0.01 inches.

minor_line_width:

The width of a minor division line. The default is 0.005 inches.

outline_line_width:

The width of grid outline. The default is 0.015 inches.

outline_line_color:

The color of the grid outline. The default is 'black'.

fz_grid_color:

The color of the frequency and impedance grid lines. The default is 'grey'.

cl_grid_color:

The color of the capacitance and inductance grid lines. The default is 'grey'.

background:

The background color of the grid. The default is 'white'.

minor_divs:

The minor divisions to include. The default is '123456789'. Other common values are '1', '13', '125', and '12468'.

decade:

The size of one decade square. The default is 1 inch. With this value, a grid that is 6 decades wide and 4 decades high is 6" by 4".

left_margin:

The size of the left margin. The default is 1 inch.

right_margin:

The size of the right margin. The default is 1 inch.

top_margin:

The size of the top margin. The default is 1 inch.

bottom_margin:

The size of the bottom margin. The default is 1 inch.

font_family:

The text font family. The default is "sans-serif".

font_size:

The text font size. The default is 12.

text_color:

The text color size. The default is "black".

text_offset:

The separation between the axis labels and the grid. The default is 0.15 inches.

pixels_per_unit:

This is a scaling factor that allows you specify your dimensions in what ever units you wish. A value of 96, the default, means that you are specifying your units in inches. A value of 37.8 allows you specify values in centimeters.

In addition, many SVG parameters can be passed into RLC_Chart, in which case they are simply passed on to svgwrite.

Generally, RLC_Chart is the argument of a with statement. If you choose not to do this, then you must explicitly call the close method yourself. Other than close, RLC_Chart provides several other methods, described next.

Methods

add_trace()

This method adds a trace to the graph. It may be called multiple times to add additional traces. There are two required arguments:

frequency:

An array of positive real values representing the frequency values of the points that when connected make up the trace.

impedance:

An array of positive real values representing the impedance values of the points that when connected make up the trace.

Each of these arrays can be in the form of a Python list or a numpy array, and they must be the same length.

It is also possible to specify additional keyword arguments, which are passed on to svgwrite and attached to the trace. This can be used to specify trace color and style. For example, specify stroke to specify the trace color.

to_x()

Given a frequency, to_x returns the corresponding canvas X coordinate. This can be used to add SVG features to your chart like labels.

to_y()

Given an impedance, to_y returns the corresponding canvas Y coordinate. This can be used to add SVG features to your chart like labels.

add_line()

Given a start and end value and a component value (r, l, c, or f), add_line draws a line on the chart. If you specify r, the start and end values are frequencies and the line is horizontal with the impedance being r. If you specify f, the start and end values are impedances and the line is vertical and the frequency is f. If you specify either c or l the start and end values are frequencies and the lines are diagonal and the impedance values are either 2π f l or 1/(2π f c).

It is also possible to specify additional keyword arguments, which are passed on to svgwrite and attached to the line. This can be used to specify line color and style. For example, specify stroke to specify the line color.

Attributes

HEIGHT

The height of the canvas, which includes the height of the grid plus the top and bottom margins. Realize that in SVG drawings, the 0 Y value is at the top of the drawing. Thus HEIGHT when used as a Y coordinate represents the bottom of the canvas.

WIDTH

The width of the canvas, which includes the width of the grid plus the left and right margins. The 0 X value is at the left of the drawing and WIDTH when used as an X coordinate represents the right of the canvas.

Labeling

The chart object returned by RLC_Chart is a svgwrite Drawing object, and so you can call its methods to add SVG features to your chart. This can be used to add labels to your charts. Here is an example that demonstrates how to add labels and lines. It also demonstrates how to read impedance data from a CSV file:

from rlc_chart import RLC_Chart
from inform import fatal, os_error
from pathlib import Path
from math import pi as π
import csv

fmin = 100
fmax = 10e9
zmin = 0.01
zmax = 1e6
cmod = 1e-9
lmod = 700e-12
rmod = 20e-3
j2π = 2j*π

def model(f):
    jω = j2π*f
    return 1/(jω*cmod) + rmod + jω*lmod

frequency = []
z_data = []
r_data = []
z_model = []
r_model = []
try:
    contents = Path('C0603C102K3GACTU_imp_esr.csv').read_text()
    data = csv.DictReader(contents.splitlines(), delimiter=',')
    for row in data:
        f = float(row['Frequency'])
        z = model(f)
        frequency.append(f)
        z_data.append(float(row['Impedance']))
        r_data.append(float(row['ESR']))
        z_model.append(abs(z))
        r_model.append(z.real)

    with RLC_Chart('C0603C102K3GACTU.svg', fmin, fmax, zmin, zmax) as chart:

        # add annotations
        svg_text_args = dict(font_size=22, fill='black')

        # capacitance annotations
        chart.add(chart.text(
            "C = 1 nF",
            insert = (chart.to_x(150e3), chart.to_y(1.5e3)),
            **svg_text_args
        ))
        chart.add_line(1e3, 190.23e6, c=1e-9)

        # inductance annotations
        chart.add(chart.text(
            "L = 700 pH",
            insert = (chart.to_x(6e9), chart.to_y(30)),
            text_anchor = 'end',
            **svg_text_args
        ))
        chart.add_line(190.232e6, 10e9, l=700e-12)

        # resistance annotations
        chart.add(chart.text(
            "ESR = 20 mΩ",
            insert = (chart.to_x(100e3), chart.to_y(25e-3)),
            text_anchor = 'start',
            **svg_text_args
        ))
        chart.add_line(100e3, 1e9, r=20e-3)

        # resonant frequency annotations
        chart.add(chart.text(
            "f₀ = 190 MHz",
            insert = (chart.to_x(190.23e6), chart.to_y(400)),
            text_anchor = 'middle',
            **svg_text_args
        ))
        chart.add_line(1e-2, 300, f=190.23e6)

        # Q annotations
        chart.add(chart.text(
            "Q = 42",
            insert = (chart.to_x(10e6), chart.to_y(100e-3)),
            text_anchor = 'start',
            **svg_text_args
        ))
        chart.add_line(10e6, 190.23e6, r=836.66e-3)

        # title
        chart.add(chart.text(
            "C0603C102K3GACTU 1nF Ceramic Capacitor",
            insert = (chart.WIDTH/2, 36),
            font_size = 24,
            fill = 'black',
            text_anchor = 'middle',
        ))

        # add traces last, so they are on top
        chart.add_trace(frequency, z_data, stroke='red')
        chart.add_trace(frequency, r_data, stroke='blue')
        chart.add_trace(frequency, z_model, stroke='red', stroke_dasharray=(10,5))
        chart.add_trace(frequency, r_model, stroke='blue', stroke_dasharray=(10,5))

except OSError as e:
    fatal(os_error(e))

This example demonstrates two different ways to specify the location of the label. The chart object provides the to_x and to_y methods that convert data values into coordinates within the grid. This is used to add labels on the traces. The chart object also provides the HEIGHT and WIDTH attributes. These can be used to compute coordinates within the entire canvas. This is used to add a title that is near the top.

The example also illustrates the use of add_line to add dimension lines to the chart.

In this figure the solid traces are the data and the dashed traces are the model. The red traces are the magnitude of the impedance, and the blue traces are the real part of the impedance, or the ESR.

Notice that in this chart the resistance at low frequencies drops with 1/f, just like the reactance. In this regard the data differs significantly from the model. This effect is referred to as dielectric absorption and it is both common and remarkable. You can read more about it, and how to model it, in Modeling Dielectric Absorption in Capacitors.

Examples

NumPy Arrays

The first example, given above in how, demonstrates how to generate an RLC chart by evaluating formulas in Python. Here the example is repeated reformulated to use NumPy arrays:

from rlc_chart import RLC_Chart
from inform import fatal, os_error
from numpy import logspace, log10 as log, pi as π

Rs = 2
Rp = 500e3
C = 1e-9
L = 10e-6
fmin = 1
fmax = 100e6
zmin = 1
zmax = 1e6
filename = "leaky-cap-chart.svg"
j2π = 2j*π

f = logspace(log(fmin), log(fmax), 2000, endpoint=True)
jω = j2π*f
z1 = Rs + 1/(jω*C) + jω*L
z2 = Rp
z = z1 * z2 / (z1 + z2)

try:
    with RLC_Chart(filename, fmin, fmax, zmin, zmax) as chart:
        chart.add_trace(f, abs(z.real), stroke='blue')
        chart.add_trace(f, abs(z.imag), stroke='red')
        chart.add_trace(f, abs(z))
except OSError as e:
    fatal(os_error(e))

CSV Data

The example given in labeling demonstrates how to read impedance data from a CSV (comma separated values) file and use it to create an RLC chart. It is rather long, and so is not repeated here.

Plotting Spectre Data

If you use the Spectre circuit simulator, you can use psf_utils with rlc_chart to extract models from simulation results. For example, here is the model of an inductor given by its manufacturer:

subckt MCFE1412TR47_JB (1 2)
    R1 (1 7) resistor  r=0.036
    L5 (2 8) inductor  l=20u
    C2 (7 8) capacitor c=10.6p
    R2 (8 2) resistor  r=528
    C1 (7 9) capacitor c=28.5p
    R5 (9 2) resistor  r=3.7
    L0 (7 3) inductor  l=0.27u
    L1 (3 4) inductor  l=0.07u
    L2 (4 2) inductor  l=0.11u
    L3 (3 5) inductor  l=0.39u
    L4 (4 6) inductor  l=0.35u
    R3 (5 4) resistor  r=3.02158381422266
    R4 (6 2) resistor  r=43.4532529473926
ends MCFE1412TR47_JB

This model is overly complicated and so expensive to simulate. It requires 13 extra unknowns that the simulator must compute (7 internal nodes and 6 inductor currents). The impedance of this subcircuit is extracted by grounding one end and driving the other with a 1 A magnitude AC source. Spectre is then run on the circuit to generate a ASCII PSF file. Then, the RLC chart for this subcircuit can be generated with:

from psf_utils import PSF
from inform import Error, os_error, fatal
from rlc_chart import RLC_Chart

try:
    psf = PSF('MCFE1412TR47_JB.ac')
    sweep = psf.get_sweep()
    z_ckt = psf.get_signal('1')
    z_mod = psf.get_signal('2')

    with RLC_Chart('MCFE1412TR47_JB.svg', 100, 1e9, 0.01, 1000) as chart:
        chart.add_trace(sweep.abscissa, abs(z_ckt.ordinate), stroke='red')
        chart.add_trace(sweep.abscissa, abs(z_mod.ordinate), stroke='blue')

    with RLC_Chart('MCFE1412TR47_JB.rxz.svg', 100, 1e9, 0.01, 1000) as chart:
        chart.add_trace(sweep.abscissa, abs(z_ckt.ordinate.real), stroke='green')
        chart.add_trace(sweep.abscissa, abs(z_ckt.ordinate.imag), stroke='orange')
        chart.add_trace(sweep.abscissa, abs(z_mod.ordinate.real), stroke='blue')
        chart.add_trace(sweep.abscissa, abs(z_mod.ordinate.imag), stroke='red')

except Error as e:
    e.terminate()
except OSError as e:
    fatal(os_error(e))

The RLC chart shows that the above subcircuit can be replaced with:

subckt MCFE1412TR47_JB (1 2)
    L   (1 2) inductor l=442.24nH r=36mOhm
    C   (1 2) capacitor c=27.522pF
    R   (1 2) resistor r=537.46_Ohm
ends MCFE1412TR47_JB

This version only requires one additional unknown, the inductor current, and so is considerably more efficient.

Here is the RLC chart of both showing the difference, which are inconsequential.

The differences are a bit more apparent if the real and imaginary components of the impedance are plotted separately.

The differences are significant only in the loss exhibited above resonance, which is usually not of concern.

Plotting S-Parameter Data

You may find that the data on a two-terminal component is given as a two-port S-parameter data file. The following example shows how to read a TouchStone two-port S-parameter data file, convert the S-parameters into Z-parameters, and then plot Z12 on an RLC chart:

#!/usr/bin/env python3
# Convert S-Parameters of Inductor measured as a two port to Impedance

from inform import fatal, os_error
from rlc_chart import RLC_Chart
from cmath import rect
from pathlib import Path

y11 = []
y12 = []
y21 = []
y22 = []
Zind = []
freq = []
z0 = 50

try:
    data = Path('tfm201610alm_r47mtaa.s2p').read_text()
    lines = data.splitlines()
    for line in lines:
        line = line.strip()
        if line[0] in '!#':
            continue
        f, s11m, s11p, s12m, s12p, s21m, s21p, s22m, s22p = line.split()
        s11 = rect(float(s11m), float(s11p)/180)
        s12 = rect(float(s12m), float(s12p)/180)
        s21 = rect(float(s21m), float(s21p)/180)
        s22 = rect(float(s22m), float(s22p)/180)
        Δ = (1 + s11)*(1 + s22) - s12*s21
        y11 = ((1 - s11)*(1 + s22) + s12*s21) / Δ / z0
        y12 = -2*s12 / Δ / z0
        y21 = -2*s21 / Δ / z0
        y22 = ((1 + s11)*(1 - s22) + s12*s21) / Δ / z0
        f = float(f)
        if f:
            freq.append(f)
            Zind.append(abs(1/y12))

    with RLC_Chart('tfm201610alm.svg', 100e3, 1e9, 0.1, 1000) as chart:
        chart.add_trace(freq, Zind, stroke='red')

except OSError as e:
    fatal(os_error(e))

Here is the resulting RLC chart for a 470 nH inductor where the S-parameters were downloaded from the TDK website.

Supposedly, this data is for a 470 nH inductor, but the actual value appears to be 257 nH, which is well outside the expected 20% tolerance. Perhaps there is some mix-up in the data files on the website.

Installing

To install use:

pip install --user rlc_chart

Releases

Latest development release

Version: 1.0.0

Released: 2022-01-25

1.0 (2022-01-25)

• Promote to full release.
• Cosmetic changes to README.
• Added test script (figures/run_tests).

0.1 (2021-03-25)

• Initial release.