# Overview¶

This document provides an overview of Lcapy’s capabilities.

## Introduction¶

Lcapy is a Python package for linear circuit analysis. It will only solve linear, time invariant networks. In other words, networks comprised of basic circuit components (R, L, C, etc.) that do not vary with time.

Networks and circuits can be described using netlists or combinations of network elements. These can be drawn semi-automatically.

As well as performing circuit analysis, Lcapy can output the systems of equations for modified nodal analysis and state-space analysis.

Lcapy cannot directly analyse non-linear devices such as diodes or transistors although it does support simple opamps without saturation. Nevertheless, it can draw them! Lcapy can generate text-book quality schematics using vector graphics (unlike the bit-mapped graphics used in this document).

Lcapy uses SymPy (symbolic Python) for its values and expressions and thus the circuit analysis can be performed symbolically. See http://docs.sympy.org/latest/tutorial/index.html for the SymPy tutorial.

## Preliminaries¶

• Before you can use Lcapy you need to install the Lcapy package (see Installation) or set PYTHONPATH to find the Lcapy source files.

• Then fire up your favourite python interpreter, for example, ipython:

>>> ipython --pylab

• Alternatively, you can use a Jupyter notebook.

## Conventions¶

Lcapy uses the passive sign convention. Thus for a passive device (R, L, C), current flows into the positive node, and for a source (V, I), current flows out of the positive node. ## Expressions¶

Lcapy defines a number of symbols corresponding to different domains (see Domain variables):

• t – time (real)
• f – frequency (real)
• s – complex (s-domain) frequency
• omega – angular frequency (real)

Expressions can be formed using these symbols, for example, a time-domain expression can be created using:

>>> from lcapy import t, delta, u
>>> v = 2 * t * u(t) + 3 + delta(t)
>>> i = 0 * t + 3


and a s-domain expression can be created using:

>>> from lcapy import s, j, omega
>>> H = (s + 3) / (s - 4)
>>> H
s + 3
─────
s - 4


For steady-state causal signals, the s-domain can be converted to the angular frequency domain by substituting $$\mathrm{j} \omega$$ for $$s$$

>>> from lcapy import s, j, omega
>>> H = (s + 3) / (s - 4)
>>> A = H(j * omega)
>>> A
j⋅ω + 3
───────
j⋅ω - 4


Also note, real numbers are approximated by rationals.

Lcapy expressions have a number of attributes (see Attributes) including:

• numerator, N – numerator of rational function
• denominator, D – denominator of rational function
• magnitude – magnitude
• angle – angle
• real – real part
• imag – imaginary part
• conjugate – complex conjugate
• expr – the underlying SymPy expression
• val – the expression as evaluated as a floating point value (if possible)

and a number of generic methods (see Methods) including:

• simplify() – attempt simple simplification of the expression
• rationalize_denominator() – multiply numerator and denominator by complex conjugate of denominator
• divide_top_and_bottom(expr) – divides numerator and denominator by expr.
• multiply_top_and_bottom(expr) – multiplies numerator and denominator by expr.
• evaluate() – evaluate at specified vector and return floating point vector

Here’s an example of using these attributes and methods:

>>> from lcapy import s, j, omega
>>> H = (s + 3) / (s - 4)
>>> A = H(j * omega)
>>> A
j⋅ω + 3
───────
j⋅ω - 4
>>> A.rationalize_denominator()
2
ω  - 7⋅j⋅ω - 12
───────────────
2
ω  + 16
>>> A.real
2
ω  - 12
───────
2
ω  + 16
>>> A.imag
-7⋅ω
───────
2
ω  + 16
>>> A.N
j⋅ω + 3
>>> A.D
j⋅ω - 4
>>> A.phase
⎛       2     ⎞
atan2⎝-7⋅ω, ω  - 12⎠
>>> A.magnitude
__________________
╱  4       2
╲╱  ω  + 25⋅ω  + 144
─────────────────────
2
ω  + 16


Each domain has specific methods, including:

• fourier – Convert to Fourier domain
• laplace – Convert to Laplace (s) domain
• time – Convert to time domain

Lcapy defines a number of functions (see Mathematical functions) that can be used in expressions, including:

• u – Heaviside’s unit step
• H – Heaviside’s unit step
• delta – Dirac delta
• cos – cosine
• sin – sine
• sqrt – square root
• exp – exponential
• log10 – logarithm base 10
• log – natural logarithm

## Simple circuit components¶

The basic circuit components are two-terminal (one-port) devices:

• I – current source
• V – voltage source
• R – resistance
• G – conductance
• C – capacitance
• L – inductance

These are augmented by generic s-domain components:

• Z – impedance

Here are some examples of their creation:

>>> from lcapy import *
>>> R1 = R(10)
>>> C1 = C(10e-6)
>>> L1 = L('L_1')


### Simple circuit element combinations¶

Here’s an example of resistors in series

>>> from lcapy import *
>>> R1 = R(10)
>>> R2 = R(5)
>>> Rtot = R1 + R2
>>> Rtot
R(10) + R(5)
>>> Rtot.simplify()
R(15)


Here R(10) creates a 10 ohm resistor and this is assigned to the variable R1. Similarly, R(5) creates a 5 ohm resistor and this is assigned to the variable R2. Rtot is the name of the network formed by connecting R1 and R2 in series. Calling the simplify method will simplify the network and combine the resistors into a single resistor equivalent.

Here’s an example of a parallel combination of resistors. Note that the parallel operator is | instead of the usual ||.

>>> from lcapy import *
>>> Rtot = R(10) | R(5)
>>> Rtot
R(10) | R(5)
>>> Rtot.simplify()
R(10/3)


The result can be performed symbolically, for example,

>>> from lcapy import *
>>> Rtot = R('R_1') | R('R_2')
>>> Rtot
R(R_1) | R(R_2)
>>> Rtot.simplify()
R(R_1*R_2/(R_1 + R_2))
>>> Rtot.simplify()
R(R₁) | R(R₂)


Here’s another example using inductors in series

>>> from lcapy import *
>>> L1 = L(10)
>>> L2 = L(5)
>>> Ltot = L1 + L2
>>> Ltot
L(10) + L(5)
>>> Ltot.simplify()
L(15)


Finally, here’s an example of a parallel combination of capacitors

>>> from lcapy import *
>>> Ctot = C(10) | C(5)
>>> Ctot
C(10) | C(5)
>>> Ctot.simplify()
C(15)


### Impedances¶

Let’s consider a series R-L-C network

>>> from lcapy import *
>>> n = R(4) + L(10) + C(20)
>>> n
R(4) + L(10) + C(20)
>>> n.Z(omega)
ⅉ
10⋅ⅉ⋅ω + 4 - ────
20⋅ω
>>> n.Z(s)
2         1
10⋅s  + 4⋅s + ──
20
────────────────
s


Notice the result is a rational function of s. Remember impedance is a frequency domain concept. A rational function can be formatted in a number of different ways, for example,

>>> n.Z(s).ZPK()
⎛      ____    ⎞ ⎛      ____    ⎞
⎜    ╲╱ 14    1⎟ ⎜    ╲╱ 14    1⎟
10⋅⎜s - ────── + ─⎟⋅⎜s + ────── + ─⎟
⎝      20     5⎠ ⎝      20     5⎠
────────────────────────────────────
s

>>> n.Z(s).standard()
1
10⋅s + 4 + ────
20⋅s


Here ZPK() prints the impedance in ZPK (zero-pole-gain) form while standard() prints the rational function as the sum of a polynomial and a strictly proper rational function.

The corresponding parallel R-L-C network yields

>>> from lcapy import *
>>> n = R(5) | L(20) | C(10)
>>> n
R(5) | L(20) | C(10)
>>> n.Z(s)
s
──────────────────
⎛ 2   s     1 ⎞
10⋅⎜s  + ── + ───⎟
⎝     50   200⎠

>>> n.Z(s).ZPK()
s
──────────────────────────────────
⎛     1    7⋅j⎞ ⎛     1    7⋅j⎞
10⋅⎜s + ─── - ───⎟⋅⎜s + ─── + ───⎟
⎝    100   100⎠ ⎝    100   100⎠
>>> n.Z(s).canonical()
s
──────────────────
⎛ 2   s     1 ⎞
10⋅⎜s  + ── + ───⎟
⎝     50   200⎠
>>> n.Y(s)
2
200⋅s  + 4⋅s + 1
────────────────
20⋅s


Notice how n.Y(s) returns the s-domain admittance of the network, the reciprocal of the impedance n.Z(s).

The frequency response can be evaluated numerically by specifying a vector of frequency values. For example:

>>> from lcapy import *
>>> from numpy import linspace
>>> n = Vstep(20) + R(5) + C(10, 0)
>>> vf = linspace(0, 4, 400)
>>> Isc = n.Isc(f).evaluate(vf)


Note, in this example, the initial capacitor voltage is specified to be zero. If this initial condition is unspecified, the short circuit current cannot be determined.

Then the frequency response can be plotted. For example,

>>> from matplotlib.pyplot import figure, show
>>> fig = figure()
>>> ax.loglog(f, abs(Isc), linewidth=2)
>>> ax.set_xlabel('Frequency (Hz)')
>>> ax.set_ylabel('Current (A/Hz)')
>>> ax.grid(True)
>>> show()


A simpler approach is to use the plot method:

>>> from lcapy import *
>>> from numpy import linspace
>>> n = Vstep(20) + R(5) + C(10, 0)
>>> vf = linspace(0, 4, 400)
>>> n.Isc(f).plot(vf, log_scale=True) Here’s a complete example Python script to plot the impedance of a series R-L-C network:

from lcapy import *
from numpy import logspace
from matplotlib.pyplot import savefig

N = R(10) + C(1e-4) + L(1e-3)

vf = logspace(0, 5, 400)
N.Z(f).magnitude.plot(vf)

savefig('series-RLC3-Z.png') ## Simple transient analysis¶

Let’s consider a series R-C network in series with a DC voltage source

>>> from lcapy import *
>>> n = Vstep(20) + R(5) + C(10, 0)
>>> n
Vstep(20) + R(5) + C(10, 0)
>>> Voc = n.Voc(s)
>>> Voc
20
──
s
>>> n.Isc(s)
4
────────
s + 1/50
>>> isc = n.Isc.transient_response()
>>> isc
⎧   -t
⎪   ───
⎨    50
⎪4⋅e     for t ≥ 0
⎩


Here n is network formed by the components in series, and n.Voc(s) is the open-circuit s-domain voltage across the network. Note, this is the same as the s-domain value of the voltage source. n.Isc(s) is the short-circuit s-domain voltage through the network. The method transient_response converts this to the time-domain. Note, since the capacitor has the initial value specified, this network is analysed as an initial value problem and thus the result is not known for $$t<0$$. If the initial capacitor voltage is not specified, the network cannot be analysed.

Of course, the previous example can be performed symbolically,

>>> from lcapy import *
>>> n = Vstep('V_1') + R('R_1') + C('C_1', 0)
>>> n
Vstep(V₁) + R(R₁) + C(C₁, 0)
>>> Voc = n.Voc(s)
>>> Voc
V₁
──
s
>>> n.Isc(s)
V₁
──────────────
⎛      1  ⎞
R₁⋅⎜s + ─────⎟
⎝    C₁⋅R₁⎠
>>> isc = n.Isc.transient_response()
>>> isc
⎧     -t
⎪    ─────
⎪    C₁⋅R₁
⎨V₁⋅e
⎪─────────  for t ≥ 0
⎪    R₁
⎩


The transient response can be evaluated numerically by specifying a vector of time values.

>>> from lcapy import *
>>> from numpy import linspace
>>> n = Vstep(20) + R(5) + C(10, 0)
>>> t = linspace(0, 100, 400)
>>> isc = n.Isc.transient_response(t)


Then the transient response can be plotted. Alternatively, the plot method can be used.

from lcapy import *
from numpy import linspace
from matplotlib.pyplot import savefig

N = Vstep(20) + R(10) + C(1e-4, 0)

vt = linspace(0, 0.01, 1000)
N.Isc(t).plot(vt)

savefig('series-VRC1-isc.png')


This produces: Here’s a complete example Python script of the short-circuit current through an underdamped series RLC network:

from lcapy import Vstep, R, L, C, t
from matplotlib.pyplot import savefig
from numpy import linspace

a = Vstep(10) + R(0.1) + C(0.4) + L(0.2, 0)

vt = linspace(0, 10, 1000)
a.Isc(t).plot(vt)

savefig('series-VRLC1-isc.png') ## Transformations¶

A one-port network can be represented as a Thevenin network (a series combination of a voltage source and an impedance) or as a Norton network (a parallel combination of a current source and an admittance).

Here’s an example of a Thevenin to Norton transformation:

>>> from lcapy import *
>>> T = Vdc(10) + R(5)
>>> n = T.norton()
>>> n
G(1/5) | Idc(2)


Similarly, here’s an example of a Norton to Thevenin transformation:

>>> from lcapy import *
>>> n = Idc(10) | R(5)
>>> T = n.thevenin()
>>> T
R(5) + Vdc(50)


## Two-port networks¶

The basic circuit components are one-port networks. They can be combined to create a two-port network. The simplest two-port is a shunt:

-----+----
|
+-+-+
|   |
|OP |
|   |
+-+-+
|
-----+----


A more interesting two-port network is an L section (voltage divider):

  +---------+
--+   OP1   +---+----
+---------+   |
+-+-+
|   |
|OP2|
|   |
+-+-+
|
----------------+----

This is comprised from any two one-port networks. For example,
>>> from lcapy import *
>>> R1 = R('R_1')
>>> R2 = R('R_2')
>>> n = LSection(R1, R2)
>>> n.Vtransfer
R_2/(R_1 + R_2)


Here n.Vtransfer determines the forward voltage transfer function V_2(s) / V_1(s).

The open-circuit input impedance can be found using:
>>> n.Z1oc
R₁ + R₂

The open-circuit output impedance can be found using:
>>> n.Z2oc
R₂

The short-circuit input admittance can be found using:
>>> n.Y1sc
1
──
R₁

The short-circuit output admittance can be found using:
>>> n.Y2sc
R₁ + R₂
───────
R₁⋅R₂


### Two-port combinations¶

Two-port networks can be combined in series, parallel, series at the input with parallel at the output (hybrid), parallel at the input with series at the output (inverse hybrid), but the most common is the chain or cascade. This connects the output of the first two-port to the input of the second two-port.

For example, an L section can be created by chaining a shunt to a series one-port.

>>> from lcapy import *
>>> n = Series(R('R_1')).chain(Shunt(R('R_2')))
>>> n.Vtransfer
R_2/(R_1 + R_2)


### Two-port matrices¶

Two-port networks can be represented by six two by two matrices, A, B, G, H, Y, Z. Each has their own merits (see http://en.wikipedia.org/wiki/Two-port_network).

Consider an L section comprised of two resistors:
>>> from lcapy import *
>>> n = LSection(R('R_1'), R('R_2')))

The different matrix representations can be shown using:
>>> n.A
⎡R₁ + R₂    ⎤
⎢───────  R₁⎥
⎢   R₂      ⎥
⎢           ⎥
⎢  1        ⎥
⎢  ──     1 ⎥
⎣  R₂       ⎦
>>> n.B
⎡ 1    -R₁  ⎤
⎢           ⎥
⎢-1   R₁    ⎥
⎢───  ── + 1⎥
⎣ R₂  R₂    ⎦
>>> n.G
⎡   1       -R₂  ⎤
⎢───────  ───────⎥
⎢R₁ + R₂  R₁ + R₂⎥
⎢                ⎥
⎢   R₂     R₁⋅R₂ ⎥
⎢───────  ───────⎥
⎣R₁ + R₂  R₁ + R₂⎦
>>> n.H
⎡R₁  1 ⎤
⎢      ⎥
⎢    1 ⎥
⎢-1  ──⎥
⎣    R₂⎦
>>> n.Y
⎡1      -1   ⎤
⎢──     ───  ⎥
⎢R₁      R₁  ⎥
⎢            ⎥
⎢-1   R₁ + R₂⎥
⎢───  ───────⎥
⎣ R₁   R₁⋅R₂ ⎦
>>> n.Z
⎡R₁ + R₂  R₂⎤
⎢           ⎥
⎣  R₂     R₂⎦


Note, some of the two-port matrices cannot represent a network. For example, a series impedance has a non specified Z matrix and a shunt impedance has a non specified Y matrix.

## Transfer functions¶

Transfer functions can be created for netlists using the transfer() method. Here’s an example for an RC low-pass filter

>>> cct = Circuit("""
... R 1 2
... C 2 0""")
>>> H = cct.transfer(1, 0, 2, 0)
>>> H(s)
1
─────────────
⎛     1 ⎞
C⋅R⋅⎜s + ───⎟
⎝    C⋅R⎠


Transfer functions can also be created in a similar manner to Matlab, either using lists of numerator and denominator coefficients:

>>> from lcapy import *
>>> H1 = tf(0.001, [1, 0.05, 0])
>>> H1
0.001
───────────────
2
1.0⋅s  + 0.05⋅s


from lists of poles and zeros (and optional gain):

>>> from lcapy import *
>>> H2 = zp2tf([], [0, -0.05])
>>> H2
0.001
───────────────
2
1.0⋅s  + 0.05⋅s


or symbolically:

>>> from lcapy import *
>>> H3 = 0.001 / (s**2 + 0.05 * s)
>>> H3
0.001
───────────────
2
1.0⋅s  + 0.05⋅s


In each case, parameters can be expressed numerically or symbolically, for example,

>>> from lcapy import *
>>> H4 = zp2tf(['z_1'], ['p_1', 'p_2'])
>>> H4
s - z₁
───────────────────
(-p₁ + s)⋅(-p₂ + s)


## Parameterization¶

Transfer functions (or any d-domain expression) can be parameterized with the parameterize() method (see Parameterization). This returns a tuple. The first element is the parameterized expression and the second element is a dictionary of substitutions.

Here’s a second order example:

>>> H2 = 3 / (s**2 + 2*s + 4)
>>> H2
3
────────────
2
s  + 2⋅s + 4
>>> H2p, defs = H2.parameterize()
>>> H2p
K
───────────────────
2               2
ω₀  + 2⋅ω₀⋅s⋅ζ + s

>>> defs
{K: 3, omega_0: 2, zeta: 1/2}


## Partial fraction analysis¶

Lcapy can be used for converting rational functions into partial fraction form. Here’s an example:

>>> from lcapy import *
>>> G = 1 / (s**2 + 5 * s + 6)
>>> G.partfrac()
1       1
- ───── + ─────
s + 3   s + 2


Here’s an example of a not strictly proper rational function,

>>> from lcapy import *
>>> H = 5 * (s + 5) * (s - 4) / (s**2 + 5 * s + 6)
>>> H.partfrac()
70      90
5 + ───── - ─────
s + 3   s + 2


The rational function can also be printed in ZPK form:

>>> H.ZPK()
5⋅(s - 4)⋅(s + 5)
─────────────────
(s + 2)⋅(s + 3)


Here it is obvious that the poles are -2 and -3. These can also be found using the poles function:

>>> H.poles()
{-3: 1, -2: 1}


Here the number after the colon indicates how many times the pole is repeated.

Similarly, the zeros can be found using the zeros function:

>>> H.zeros()
{-5: 1, 4: 1}


Lcapy can also handle rational functions with a delay.

## Inverse Laplace transforms¶

Lcapy can perform inverse Laplace transforms. Here’s an example for a strictly proper rational function:

>>> from lcapy import s
>>> H = 5 * (s - 4) / (s**2 + 5 * s + 6)
>>> H.partfrac()
35      30
───── - ─────
s + 3   s + 2

>>> H.inverse_laplace()
⎧      -2⋅t       -3⋅t
⎨- 30⋅e     + 35⋅e      for t ≥ 0
⎩


or alternatively

>>> H(t)
⎧      -2⋅t       -3⋅t
⎨- 30⋅e     + 35⋅e      for t ≥ 0
⎩


Note that the unilateral inverse Laplace transform can only determine the result for $$t \ge 0$$. If you know that the system is causal, then use:

>>> H(t, causal=True)
⎛      -2⋅t       -3⋅t⎞
⎝- 30⋅e     + 35⋅e    ⎠⋅Heaviside(t)


The Heaviside function is also known as the unit step. Alternatively, you can force the result to be causal

>>> H(t).force_causal()
⎛      -2⋅t       -3⋅t⎞
⎝- 30⋅e     + 35⋅e    ⎠⋅Heaviside(t)


or remove the condition that $$t \ge 0$$,

>>> H(t).remove_condition()
-2⋅t       -3⋅t
- 30⋅e     + 35⋅e


When the rational function is not strictly proper, the inverse Laplace transform has Dirac deltas (and derivatives of Dirac deltas):

>>> from lcapy import s
>>> H = 5 * (s - 4) / (s**2 + 5 * s + 6)
>>> H.partfrac()
70      90
5 + ───── - ─────
s + 3   s + 2
>>> H.inverse_laplace(causal=True)
⎛      -2⋅t       -3⋅t⎞
⎝- 90⋅e     + 70⋅e    ⎠⋅Heaviside(t) + 5⋅DiracDelta(t)


Here’s another example of a strictly proper rational function with a repeated pole:

>>> from lcapy import s
>>> H = 5 * (s + 5) / ((s + 3) * (s + 3))
>>> H.ZPK()
5⋅(s + 5)
─────────
2
(s + 3)
>>> H.partfrac()
5        10
───── + ────────
s + 3          2
(s + 3)
>>> H.inverse_laplace(causal=True)
⎛      -3⋅t      -3⋅t⎞
⎝10⋅t⋅e     + 5⋅e    ⎠⋅Heaviside(t)


Rational functions with delays can also be handled:

>>> from lcapy import s
>>> import sympy as sym
>>> T = sym.symbols('T')
>>> H = 5 * (s + 5) * (s - 4) / (s**2 + 5 * s + 6) * sym.exp(-s * T)
>>> H.partfrac()
⎛      70      90 ⎞  -T⋅s
⎜5 + ───── - ─────⎟⋅e
⎝    s + 3   s + 2⎠
>>> H.inverse_laplace(causal=True)
⎛      2⋅T - 2⋅t       3⋅T - 3⋅t⎞
⎝- 90⋅e          + 70⋅e         ⎠⋅Heaviside(-T + t) + 5⋅DiracDelta(-T + t)


Lcapy can convert s-domain products to time domain convolutions, for example,

>>> from lcapy import expr
>>> expr('V(s) * Y(s)')(t, causal=True)
t
⌠
⎮ v(t - τ)⋅y(τ) dτ
⌡
0


Here the function expr converts a sring argument to an Lcapy expression.

It can also recognise integrations and differentiations of arbitrary functions, for example,

>>> from lcapy import s, t
>>> (s * 'V(s)')(t, causal=True)
d
──(v(t))
dt

>>> ('V(s)' / s)(t, causal=True)
t
⌠
⎮ v(τ) dτ
⌡
0


These expressions also be written as:

>>> from lcapy import expr, t
>>> expr('s * V(s)')(t, causal=True)
>>> expr('V(s) / s')(t, causal=True)


or more explicitly:

>>> from lcapy import expr
>>> expr('s * V(s)').inverse_laplace(causal=True)
>>> expr('V(s) / s').inverse_laplace(causal=True)


## Laplace transforms¶

Lcapy can also perform Laplace transforms. Here’s an example:

>>> from lcapy import s, t
>>> v = 10 * t ** 2 + 3 * t
>>> v.laplace()
3⋅s + 20
────────
3
s


There is a short-hand notation for the Laplace transform:

>>> v(s)
3⋅s + 20
────────
3
s


## Circuit analysis¶

The nodal voltages for a linear circuit can be found using Modified Nodal Analysis (MNA). This requires the circuit topology be entered as a netlist (see Netlists). This describes each component, its name, value, and the nodes it is connected to. This netlist can be read from a file or created dynamically, for example,

>>> from lcapy import Circuit
>>> cct = Circuit()
>>> cct.add('V1 1 0 step 10')


This creates a circuit comprised of a 10 V step voltage source connected to two resistors in series. The node named 0 denotes the ground which the other voltages are referenced to. Here’s a more compact way to specify the netlist:

>>> from lcapy import Circuit
>>> cct = Circuit("""
... V1 1 0 step 10
... Ra 1 2 3e
... Rb 2 0 1e3""")


The circuit has an attribute for each circuit element (and for each node starting with an alphabetical character). These can be interrogated to find the voltage drop across an element or the current through an element, for example,

>>> cct.V1.V
⎧   10⎫
⎨s: ──⎬
⎩   s ⎭
>>> cct.Rb.V
⎧    5 ⎫
⎨s: ───⎬
⎩   2⋅s⎭


Notice, how the displayed voltages are a dictionary. This represents the result as a superposition of a number of transform domains (DC, AC, Laplace, etc.). In this case s denotes the Laplace domain result of the transient component. The full Laplace response is returned using

>>> cct.V1.V(s)
10
──
s


The time domain response is found using:

>>> cct.V1.V(t)
10


Alternatively, this can be achieved using the lowercase v attribute:

>>> cct.V1.v
10


The current through a component is obtained with the I attribute. For a source the current is assumed to flow out of the positive node, however, for a passive device (R, L, C) it is assumed to flow into the positive node.

The voltage between a node and ground can be determined with the node name as an index, for example,

>>> cct.V(t)
10
>>> cct.V(t)
5
─
2


Since Lcapy uses SymPy, circuit analysis can be performed symbolically. This can be achieved by using symbolic arguments or by not specifying a component value. In the latter case, Lcapy will use the component name for its value. For example,

>>> cct = Circuit("""
... V1 1 0 step Vs
... R1 1 2
... C1 2 0""")
>>> cct.V(s)
V_s
──────────────────
⎛ 2     s  ⎞
C₁⋅R₁⋅⎜s  + ─────⎟
⎝     C₁⋅R₁⎠

>>> : cct.V(t)
⎛            -t  ⎞
⎜           ─────⎟
⎜           C₁⋅R₁⎟
⎝V_s - V_s⋅e     ⎠⋅Heaviside(t)


### Transform domains¶

Lcapy analyses a linear circuit using a number of transform domains and the principle of superposition. Voltage and current signals are decomposed into a DC component, one or more AC components (one for each angular frequency), a transient component, and noise components (one for each noise source).

For example, consider:

>>> Voc = (Vdc(10) + Vac(20) + Vstep(30) + Vnoise(40)).Voc
>>> Voc
⎧                   30        ⎫
⎨dc: 10, n1: 40, s: ──, ω₀: 20⎬
⎩                   s         ⎭


Here the open-circuit voltage is decomposed into four parts (stored in a dictionary). The DC component is keyed by ‘dc’, the transient component is keyed by ‘s’ (since this is analysed in the Laplace or s-domain), the noise components are keyed by noise identifiers of the form ‘nx’ (where x is an integer), and the ac components are keyed by the angular frequency. The different parts of a decomposition can also be accessed using attributes, for example,

>>> Voc.s
30
──
s


Note, this only returns the Laplace transform of the transient component of the decomposition. The full Laplace transform of the open-circuit voltage (ignoring the noise component) can be obtained using:

>>> from lcapy import s
>>> Voc(s)
⎛    2      2⎞
20⋅⎝2⋅ω₀  + 3⋅s ⎠
─────────────────
⎛  2    2⎞
s⋅⎝ω₀  + s ⎠


Similarly, the time-domain representation (ignoring the noise component) can be determined using:

>>> from lcapy import t
>>> Voc(t)
20⋅cos(ω₀⋅t) + 30⋅u(t) + 10


### Initial value problems¶

The initial voltage difference across a capacitor or the initial current through an inductor can be specified as an additional argument. For example,

>>> cct = Circuit("""
... V1 1 0 step Vs
... C1 2 1 C1 v0
... L1 2 0 L1 i0""")
>>> cct.V(s)
⎛        i₀          ⎞
(V_s + v₀)⋅⎜- ────────────── + s⎟
⎝  C₁⋅V_s + C₁⋅v₀    ⎠
─────────────────────────────────
2     1
s  + ─────
C₁⋅L₁


Note, the component values need to be specified as well as the initial value; thus C1 2 1 C1 v0 and not C1 2 1 v0 since the latter specifies the capacitance to be v0.

When an initial condition is detected, the circuit is analysed in the s-domain as an initial value problem. The values of sources are ignored for $$t<0$$ and the result is only defined for $$t\ge 0$$.

### Transfer functions¶

Transfer functions can be found from the ratio of two s-domain quantities such as voltage or current with zero initial conditions. Here’s an example using an arbitrary input voltage V(s)

>>> from lcapy import Circuit
>>> cct = Circuit("""
... V1 1 0 {V(s)}
... R1 1 2
... C1 2 0 C1 0""")
>>> cct.V(s)
V(s)
───────────
C₁⋅R₁⋅s + 1

>>> H = cct.V(s) / cct.V(s)
>>> H
1
───────────
C₁⋅R₁⋅s + 1


The corresponding impulse response can found from an inverse Laplace transform:

>>> H.inverse_laplace(causal=True)
-t
─────
C₁⋅R₁
e     ⋅Heaviside(t)
───────────────────
C₁⋅R₁


or more simply using:

>>> H(t, causal=True)
-t
─────
C₁⋅R₁
e     ⋅Heaviside(t)
───────────────────
C₁⋅R₁


Transfer functions can also be created using the transfer method of a circuit. For example,

>>> from lcapy import Circuit
>>> cct = Circuit("""
... R1 1 2
... C1 2 0""")
>>> H = cct.transfer(1, 0, 2, 0)
>>> H
1
───────────
C₁⋅R₁⋅s + 1


In this example, the transfer method computes (V - V) / (V - V). In general, all independent sources are killed and so the response is causal.

>>> H(t)
-t
─────
C₁⋅R₁
e     ⋅Heaviside(t)
───────────────────
C₁⋅R₁


### State-space analysis¶

Lcapy can identify state variables and generate the state and output equations for state-space analysis. The state-space analysis is performed using the ss method of a circuit, e.g.,

>>> from lcapy import Circuit
>>> a = Circuit("""
... V 1 0 {v(t)}; down
... R1 1 2; right
... L 2 3; right=1.5, i={i_L}
... R2 3 0_3; down=1.5, i={i_{R2}}, v={v_{R2}}
... W 0 0_3; right
... W 3 3_a; right
... C 3_a 0_4; down, i={i_C}, v={v_C}
... W 0_3 0_4; right""")
>>> ss = a.ss This circuit has two reactive components and thus there are two state variables; the current through L and the voltage across C.

The state variable vector is shown using the x attribute:

>>> ss.x
⎡i_L(t)⎤
⎢      ⎥
⎣v_C(t)⎦


The initial values of the state variable vector are shown using the x0 attribute:

>>> ss.x0
⎡0⎤
⎢ ⎥
⎣0⎦


The independent source vector is shown using the u attribute. In this example, there is a single independent source:

>>> ss.u
[v(t)]


The output vector can either be the nodal voltages, the branch currents, or both. By default the nodal voltages are chosen. This vector is shown using the y attribute:

>>> ss.y
⎡v₁(t)⎤
⎢     ⎥
⎢v₂(t)⎥
⎢     ⎥
⎣v₃(t)⎦


The state equations are shown using the state_equations method:

>>> ss.state_equations()
⎡d         ⎤   ⎡-R₁  -1  ⎤
⎢──(i_L(t))⎥   ⎢───  ─── ⎥            ⎡1⎤
⎢dt        ⎥   ⎢ L    L  ⎥ ⎡i_L(t)⎤   ⎢─⎥
⎢          ⎥ = ⎢         ⎥⋅⎢      ⎥ + ⎢L⎥⋅[v(t)]
⎢d         ⎥   ⎢-1   -1  ⎥ ⎣v_C(t)⎦   ⎢ ⎥
⎢──(v_C(t))⎥   ⎢───  ────⎥            ⎣0⎦
⎣dt        ⎦   ⎣ C   C⋅R₂⎦


The output equations are shown using the output_equations method:

>>> ss.output_equations()
⎡v₁(t)⎤   ⎡0    0⎤            ⎡1⎤
⎢     ⎥   ⎢      ⎥ ⎡i_L(t)⎤   ⎢ ⎥
⎢v₂(t)⎥ = ⎢-R₁  0⎥⋅⎢      ⎥ + ⎢1⎥⋅[v(t)]
⎢     ⎥   ⎢      ⎥ ⎣v_C(t)⎦   ⎢ ⎥
⎣v₃(t)⎦   ⎣0    1⎦            ⎣0⎦


The A, B, C, and D matrices are obtained using the attributes of the same name. For example,

>>> ss.A
⎡-R₁   -1  ⎤
⎢───   ─── ⎥
⎢ L     L  ⎥
⎢          ⎥
⎢ 1    -1  ⎥
⎢───   ────⎥
⎣ C    C⋅R₂⎦

>>> ss.B
⎡1⎤
⎢─⎥
⎢L⎥
⎢ ⎥
⎣0⎦

>>> ss.C
⎡0    0⎤
⎢      ⎥
⎢-R₁  0⎥
⎢      ⎥
⎣0    1⎦

>>> ss.D
⎡1⎤
⎢ ⎥
⎢1⎥
⎢ ⎥
⎣0⎦


The Laplace transforms of the state variable vector, the independent source vector, and the output vector are accessed using the X, U, and Y attributes: For example,

>>> ss.X
⎡I_L(s)⎤
⎢      ⎥
⎣V_C(s)⎦


The s-domain state-transition matrix is given by the Phi attribute and the time-domain state-transition matrix is given by the phi attribute. For example,

>>> ss.Phi
⎡                  1                                                   ⎤
⎢             s + ────                                                 ⎥
⎢                 C⋅R₂                              -1                 ⎥
⎢  ──────────────────────────────    ──────────────────────────────────⎥
⎢   2   s⋅(C⋅R₁⋅R₂ + L)   R₁ + R₂      ⎛ 2   s⋅(C⋅R₁⋅R₂ + L)   R₁ + R₂⎞⎥
⎢  s  + ─────────────── + ───────    L⋅⎜s  + ─────────────── + ───────⎟⎥
⎢            C⋅L⋅R₂        C⋅L⋅R₂      ⎝          C⋅L⋅R₂        C⋅L⋅R₂⎠⎥
⎢                                                                      ⎥
⎢                                                      R₁              ⎥
⎢                                                  s + ──              ⎥
⎢                1                                     L               ⎥
⎢──────────────────────────────────    ──────────────────────────────  ⎥
⎢  ⎛ 2   s⋅(C⋅R₁⋅R₂ + L)   R₁ + R₂⎞     2   s⋅(C⋅R₁⋅R₂ + L)   R₁ + R₂  ⎥
⎢C⋅⎜s  + ─────────────── + ───────⎟    s  + ─────────────── + ───────  ⎥
⎣  ⎝          C⋅L⋅R₂        C⋅L⋅R₂⎠              C⋅L⋅R₂        C⋅L⋅R₂  ⎦


The system transfer functions are given by the G attribute and the impulse responses are given by the g attributes, for example:

>>> ss.G
⎡                 1                  ⎤
⎢                                    ⎥
⎢           2    s      1            ⎥
⎢          s  + ──── + ───           ⎥
⎢               C⋅R₂   C⋅L           ⎥
⎢   ──────────────────────────────   ⎥
⎢    2   s⋅(C⋅R₁⋅R₂ + L)   R₁ + R₂   ⎥
⎢   s  + ─────────────── + ───────   ⎥
⎢             C⋅L⋅R₂        C⋅L⋅R₂   ⎥
⎢                                    ⎥
⎢                 1                  ⎥
⎢────────────────────────────────────⎥
⎢    ⎛ 2   s⋅(C⋅R₁⋅R₂ + L)   R₁ + R₂⎞⎥
⎢C⋅L⋅⎜s  + ─────────────── + ───────⎟⎥
⎣    ⎝          C⋅L⋅R₂        C⋅L⋅R₂⎠⎦


The characteristic polynomial (system polynomial) is given by the P attribute, for example,

>>> ss.P
2   s⋅(C⋅R₁⋅R₂ + L)   R₁ + R₂
s  + ─────────────── + ────────
C⋅L⋅R₂        C⋅L⋅R₂


The roots of this polynomial are the eigenvalues of the system. These are given by the eigenvalues attribute. The corresponding eigenvectors are the columns of the modal matrix given by the M attribute. A diagonal matrix of the eigenvalues is returned by the Lambda attribute.

### Modified nodal analysis¶

Lcapy uses modified nodal analysis for its calculations. For reactive circuits it does this independently for the DC, AC, and transient components and uses superposition to combine the results. For resistive circuits, it can perform this in the time-domain.

Here’s an example with an independent source (V1) that as a DC component and an unknown component that is considered as a transient component:

>>> from lcapy import Circuit
>>> a = Circuit("""
... V1 1 0 {10 + v(t)}; down
... R1 1 2; right
... L1 2 3; right=1.5, i={i_L}
... R2 3 0_3; down=1.5, i={i_{R2}}, v={v_{R2}}
... W 0 0_3; right
... W 3 3_a; right
... C1 3_a 0_4; down, i={i_C}, v={v_C}
... W 0_3 0_4; right""")


The corresponding circuit for DC analysis can be found using the dc method:

>>> a.dc()
V1 1 0 dc {10}; down
R1 1 2; right
L1 2 3 L1; right=1.5, i={i_L}
R2 3 0_3; i={i_{R2}}, down=1.5, v={v_{R2}}
W 0 0_3; right
W 3 3_a; right
C1 3_a 0_4 C1; i={i_C}, down, v={v_C}
W 0_3 0_4; right


The equations used to solve this can be found with the equations method:

>>> ac.dc().equations()
⎛⎡1    -1            ⎤⎞
⎜⎢──   ───  0   1  0 ⎥⎟
⎜⎢R₁    R₁           ⎥⎟
⎡V₁  ⎤   ⎜⎢                   ⎥⎟   ⎡0 ⎤
⎢    ⎥   ⎜⎢-1   1             ⎥⎟   ⎢  ⎥
⎢V₂  ⎥   ⎜⎢───  ──   0   0  1 ⎥⎟   ⎢0 ⎥
⎢    ⎥   ⎜⎢ R₁  R₁            ⎥⎟   ⎢  ⎥
⎢V₃  ⎥ = ⎜⎢                   ⎥⎟  ⋅⎢0 ⎥
⎢    ⎥   ⎜⎢          1        ⎥⎟   ⎢  ⎥
⎢I_V1⎥   ⎜⎢ 0    0   ──  0  -1⎥⎟   ⎢10⎥
⎢    ⎥   ⎜⎢          R₂       ⎥⎟   ⎢  ⎥
⎣I_L1⎦   ⎜⎢                   ⎥⎟   ⎣0 ⎦
⎜⎢ 1    0   0   0  0 ⎥⎟
⎜⎢                   ⎥⎟
⎝⎣ 0    1   -1  0  0 ⎦⎠


Here V1, V2, and V3 are the unknown node voltages for nodes 1, 2, and 3. I_V1 is the current through V1 and I_L1 is the current through L1.

The equations are similar for the transient response:

>>> a.transient().equations()
-1
⎛⎡1    -1                      ⎤⎞
⎜⎢──   ───      0      1    0  ⎥⎟
⎜⎢R₁    R₁                     ⎥⎟
⎡V₁(s)  ⎤   ⎜⎢                             ⎥⎟   ⎡ 0  ⎤
⎢       ⎥   ⎜⎢-1   1                       ⎥⎟   ⎢    ⎥
⎢V₂(s)  ⎥   ⎜⎢───  ──       0      0    1  ⎥⎟   ⎢ 0  ⎥
⎢       ⎥   ⎜⎢ R₁  R₁                      ⎥⎟   ⎢    ⎥
⎢V₃(s)  ⎥ = ⎜⎢                             ⎥⎟  ⋅⎢ 0  ⎥
⎢       ⎥   ⎜⎢                 1           ⎥⎟   ⎢    ⎥
⎢I_V1(s)⎥   ⎜⎢ 0    0   C₁⋅s + ──  0   -1  ⎥⎟   ⎢V(s)⎥
⎢       ⎥   ⎜⎢                 R₂          ⎥⎟   ⎢    ⎥
⎣I_L1(s)⎦   ⎜⎢                             ⎥⎟   ⎣ 0  ⎦
⎜⎢ 1    0       0      0    0  ⎥⎟
⎜⎢                             ⎥⎟
⎝⎣ 0    1      -1      0  -L₁⋅s⎦⎠


### Other circuit methods¶

cct.Vdict Dictionary of node voltages

cct.Idict Dictionary of branch currents

cct.Isc(Np, Nm) Short-circuit current between nodes Np and Nm.

cct.Voc(Np, Nm) Open-circuit voltage between nodes Np and Nm.

cct.isc(Np, Nm) Short-circuit t-domain current between nodes Np and Nm.

cct.voc(Np, Nm) Open-circuit t-domain voltage between nodes Np and Nm.

cct.impedance(Np, Nm) s-domain impedance between nodes Np and Nm.

cct.kill() Remove independent sources.

cct.kill_except(sources) Remove independent sources except ones specified.

cct.transfer(N1p, N1m, N2p, N2m) Voltage transfer function V2/V1, where V1 = V[N1p] - V[N1m], V2 = V[N2p] - V[N2m].

cct.thevenin(Np, Nm) Thevenin model between nodes Np and Nm.

cct.norton(Np, Nm) Norton model between nodes Np and Nm.

cct.twoport(self, N1p, N1m, N2p, N2m) Create two-port component where I1 is the current flowing into N1p and out of N1m, I2 is the current flowing into N2p and out of N2m, V1 = V[N1p] - V[N1m], V2 = V[N2p] - V[N2m].

cct.remove(component) Remove component from net list.

cct.s_model() Convert circuit to s-domain model.

cct.pre_initial_model() Convert circuit to pre-initial model.

cct.ac() Create subnetlist for AC components of independent sources.

cct.dc() Create subnetlist for DC components of independent sources.

cct.transient() Create subnetlist for transient components of independent sources.

cct.laplace() Create subnetlist with Laplace representations of independent source values.

## Plotting¶

Lcapy expressions have a plot method; this differs depending on the domain. For example, the plot method for s-domain expressions produces a pole-zero plot. Here’s an example:

from lcapy import s, j, Hs
from matplotlib.pyplot import savefig

H = Hs((s - 2) * (s + 3) / (s * (s - 2 * j) * (s + 2 * j)))
H.plot()

savefig('tf1-pole-zero-plot.png') The plot method for f-domain and $$\omega$$ -domain expressions produce spectral plots, for example,

from lcapy import s, j, pi, f, Hs
from matplotlib.pyplot import savefig
from numpy import logspace

H = Hs((s - 2) * (s + 3) / (s * (s - 2 * j) * (s + 2 * j)))

A = H(j * 2 * pi * f)

fv = logspace(-1, 4, 400)
A.plot(fv, log_scale=True)

savefig('tf1-bode-plot.png') ## Schematics¶

Schematics can be generated from a netlist and from one-port networks. In both cases the drawing is performed using the LaTeX Circuitikz package. The schematic can be displayed interactively or saved to a pdf, png, or pgf file.

### Netlist schematics¶

Hints are required to designate component orientation and explicit wires are required to link nodes of the same potential but with different coordinates. For more details see Schematics.

Here’s an example:
>>> from lcapy import Circuit
>>> cct = Circuit("""
... V1 1 0 {V(s)}; down
... R1 1 2; right
... C1 2 0_2; down
... W1 0 0_2; right""")
>>> cct.draw('schematic.pdf')


Note, the orientation hints are appended to the netlist strings with a semicolon delimiter. The drawing direction is with respect to the first node. The component W1 is a wire. Nodes with an underscore in their name are not drawn with a closed blob.

>>> from lcapy import Circuit
>>> cct = Circuit('voltage-divider.sch')
>>> cct.draw('voltage-divider.pdf')


Here are the contents of the file ‘voltage-divider.sch’:

V1 1 0_1 dc V; down
R1 1 2 R1; right
R2 2 0 R2; down
P1 2_2 0_2; down
W1 2 2_2; right
W2 0_1 0; right
W3 0 0_2; right


Here, P1 defines a port. This is shown as a pair of open blobs.

Here’s the resulting schematic: Many other components can be drawn than can be simulated. This includes non-linear devices such as transistors and diodes and time varying components such as switches. For example, here’s a common base amplifier, This is described by the netlist:

Q1 3 0 2 pnp; up
R1 1 2;right
R2 4 0_4;down
P1 1 0_1;down
W 0_1 0;right
W 0 0_4;right
W 3 4;right


### Network schematics¶

One-port networks can be drawn with a horizontal layout. Here’s an example:

>>> from lcapy import R, C, L
>>> ((R(1) + L(2)) | C(3)).draw()


Here’s the result: The s-domain model can be drawn using:

>>> from lcapy import R, C, L
>>> ((R(1) + L(2)) | C(3)).s_model().draw()


This produces: Internally, Lcapy converts the network to a netlist and then draws the netlist. The netlist can be found using the netlist method, for example,

>>> from lcapy import R, C, L
>>> print(((R(1) + L(2)) | C(3)).netlist())


yields:

W 1 3; right, size=0.5
W 3 4; up, size=0.4
W 3 5; down, size=0.4
W 6 2; right, size=0.5
W 6 7; up, size=0.4
W 6 8; down, size=0.4
R 4 9 1; right
W 9 10; right, size=0.5
L 10 7 2 0; right
C 5 8 3 0; right


Note, the components have anonymous identifiers.

## Jupyter (IPython) notebooks¶

Jupyter notebooks allow interactive markup of python code and text. A number of examples are provided in the lcapy/doc/examples/notebooks directory. Before these notebooks can be viewed in a browser you need to start a Jupyter notebook server.

$cd lcapy/doc/examples/notebooks$ jupyter notebook


Alternatively, they can be viewed online at https://github.com/mph-/lcapy/tree/master/doc/examples/notebooks.