# 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 reduces 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 to what every 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 measure as a two port 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 = []
Zind1 = []
Zind2 = []
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)
Zind1.append(abs(1/y12))
with RLC_Chart('tfm201610alm.svg', 100e3, 1e9, 0.1, 1000) as chart:
chart.add_trace(freq, Zind1, stroke='red')
chart.add_trace(freq, Zind2, stroke='blue')
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.