SLiCAP time-domain analysis

The following time-domain analysis functions are implemented in SLiCAP:

• doImpulse() returns an expression for the unit-impulse response of a transfer. It is obtained as the Inverse Laplace Transform of the Laplace transfer function.

• doStep() returns an expression for the unit-step response of a transfer. It is obtained as the Inverse Laplace Transform of the product of \(\frac{1}{s}\) and Laplace transfer function.

• doTime() returns an expression for the time domain response, considering the Laplace values of all independent sources in the circuit.

Important

SliCAP calculates all time-domain responses using the Inverse Laplace Transform. If it fails to do so, it returns a non-evaluated expression.

SLiCAP output displayed on this manual page, is generated with the script: ttime.py, imported by Manual.py.

#!/usr/bin/env python3
# -*- coding: utf-8 -*-
"""
ttime.py: SLiCAP scripts for the HTML help file
"""
import SLiCAP as sl
import sympy as sp
# Circuit definition
ACcoupling = sl.makeCircuit("kicad/ACcoupling/ACcoupling.kicad_sch")

# Define parameter values
params = {"R_a": "1k",
          "R_b": "1k",
          "R_s": "50",
          "C_c": "1n",
          "f_s": "10M",
          "V_p": 2,
          "V_B": 5}

The circuit diagram of the circuit ACcoupling:

The source signal signal is a sinusoidal voltage with an amplitude of \(V_p\) V and a frequency of \(f_s\) Hz. It is specified with its Laplace Transform:

\[V_\mathrm{V2} = V_p\frac{2 \pi f_s}{s^2 + \left( 2 \pi f_s \right)^2}\]

The power supply voltage steps from zero to \(V_B\):

\[V_\mathrm{V1} = V_B\frac{1}{s}\]

Unit-impulse response

Symbolic unit-impulse response

# Obtain the symbolic unit-impulse response of the power supply transfer
power_impulse_sym = sl.doImpulse(ACcoupling, source="V1").impulse

This yields (rendered output from formatter):

\[h_{p}{\left(t \right)} = \frac{R_{b} \left(\frac{\left(R_{a} R_{s} + R_{b} R_{s}\right) \delta\left(t\right)}{R_{a} R_{b} + R_{a} R_{s} + R_{b} R_{s}} + \frac{\left(R_{a}^{2} R_{b} + R_{a} R_{b}^{2}\right) e^{- \frac{t \left(R_{a}^{2} R_{b} + R_{a}^{2} R_{s} + R_{a} R_{b}^{2} + 2 R_{a} R_{b} R_{s} + R_{b}^{2} R_{s}\right)}{C_{c} R_{a}^{2} R_{b}^{2} + 2 C_{c} R_{a}^{2} R_{b} R_{s} + C_{c} R_{a}^{2} R_{s}^{2} + 2 C_{c} R_{a} R_{b}^{2} R_{s} + 2 C_{c} R_{a} R_{b} R_{s}^{2} + C_{c} R_{b}^{2} R_{s}^{2}}}}{C_{c} R_{a}^{2} R_{b}^{2} + 2 C_{c} R_{a}^{2} R_{b} R_{s} + C_{c} R_{a}^{2} R_{s}^{2} + 2 C_{c} R_{a} R_{b}^{2} R_{s} + 2 C_{c} R_{a} R_{b} R_{s}^{2} + C_{c} R_{b}^{2} R_{s}^{2}}\right)}{R_{a} + R_{b}}\]

Numeric unit-impulse response

# Numeric unit-step response of the signal transfer
signal_impulse_num = sl.doImpulse(ACcoupling, pardefs=params, source="V2").impulse
print(signal_impulse_num)

This yields:

-1652892.56198347*exp(-1818181.81818182*t)

Unit-step response

Symbolic unit-step response

# Obtain the symbolic unit-step response of the signal transfer
signal_step_sym = sl.doStep(ACcoupling, source="V2").stepResp

This yields (rendered output from formatter):

\[a_{s}{\left(t \right)} = \frac{C_{c} R_{a} R_{b} e^{- \frac{t \left(R_{a} + R_{b}\right)}{C_{c} R_{a} R_{b} + C_{c} R_{a} R_{s} + C_{c} R_{b} R_{s}}}}{C_{c} R_{a} R_{b} + C_{c} R_{a} R_{s} + C_{c} R_{b} R_{s}}\]

Numeric unit-step response

power_step_num  = sl.doStep(ACcoupling, pardefs=params, source="V1").stepResp
print(power_step_num)

This yields:

0.5 - 0.454545454545455*exp(-1818181.81818182*t)

Time-domain response

Symbolic time-domain response

# Obtain the symbolic time-domain response
timeResult_sym = timeResult_num = sl.doTime(ACcoupling).time

This yields (rendered output from formatter):

\[V_{out} = \frac{R_{b} V_{B} \left(\frac{\left(2 \pi C_{c} R_{a}^{3} V_{p} f_{s} + 4 \pi C_{c} R_{a}^{2} R_{b} V_{p} f_{s} + 2 \pi C_{c} R_{a} R_{b}^{2} V_{p} f_{s}\right) \cos{\left(2 \pi f_{s} t \right)}}{4 \pi^{2} C_{c}^{2} R_{a}^{2} R_{b}^{2} V_{B} f_{s}^{2} + 8 \pi^{2} C_{c}^{2} R_{a}^{2} R_{b} R_{s} V_{B} f_{s}^{2} + 4 \pi^{2} C_{c}^{2} R_{a}^{2} R_{s}^{2} V_{B} f_{s}^{2} + 8 \pi^{2} C_{c}^{2} R_{a} R_{b}^{2} R_{s} V_{B} f_{s}^{2} + 8 \pi^{2} C_{c}^{2} R_{a} R_{b} R_{s}^{2} V_{B} f_{s}^{2} + 4 \pi^{2} C_{c}^{2} R_{b}^{2} R_{s}^{2} V_{B} f_{s}^{2} + R_{a}^{2} V_{B} + 2 R_{a} R_{b} V_{B} + R_{b}^{2} V_{B}} + \frac{0.5 \left(8 \pi^{3} C_{c}^{2} R_{a}^{3} R_{b} V_{p} f_{s}^{3} + 8 \pi^{3} C_{c}^{2} R_{a}^{3} R_{s} V_{p} f_{s}^{3} + 8 \pi^{3} C_{c}^{2} R_{a}^{2} R_{b}^{2} V_{p} f_{s}^{3} + 16 \pi^{3} C_{c}^{2} R_{a}^{2} R_{b} R_{s} V_{p} f_{s}^{3} + 8 \pi^{3} C_{c}^{2} R_{a} R_{b}^{2} R_{s} V_{p} f_{s}^{3}\right) \sin{\left(2 \pi t \left(f_{s}^{2}\right)^{0.5} \right)}}{\pi \left(4 \pi^{2} C_{c}^{2} R_{a}^{2} R_{b}^{2} V_{B} f_{s}^{2} + 8 \pi^{2} C_{c}^{2} R_{a}^{2} R_{b} R_{s} V_{B} f_{s}^{2} + 4 \pi^{2} C_{c}^{2} R_{a}^{2} R_{s}^{2} V_{B} f_{s}^{2} + 8 \pi^{2} C_{c}^{2} R_{a} R_{b}^{2} R_{s} V_{B} f_{s}^{2} + 8 \pi^{2} C_{c}^{2} R_{a} R_{b} R_{s}^{2} V_{B} f_{s}^{2} + 4 \pi^{2} C_{c}^{2} R_{b}^{2} R_{s}^{2} V_{B} f_{s}^{2} + R_{a}^{2} V_{B} + 2 R_{a} R_{b} V_{B} + R_{b}^{2} V_{B}\right) \left(f_{s}^{2}\right)^{0.5}} + \frac{\left(- 4 \pi^{2} C_{c}^{3} R_{a}^{3} R_{b}^{3} V_{B} f_{s}^{2} - 8 \pi^{2} C_{c}^{3} R_{a}^{3} R_{b}^{2} R_{s} V_{B} f_{s}^{2} - 4 \pi^{2} C_{c}^{3} R_{a}^{3} R_{b} R_{s}^{2} V_{B} f_{s}^{2} - 8 \pi^{2} C_{c}^{3} R_{a}^{2} R_{b}^{3} R_{s} V_{B} f_{s}^{2} - 8 \pi^{2} C_{c}^{3} R_{a}^{2} R_{b}^{2} R_{s}^{2} V_{B} f_{s}^{2} - 4 \pi^{2} C_{c}^{3} R_{a} R_{b}^{3} R_{s}^{2} V_{B} f_{s}^{2} - 2 \pi C_{c}^{2} R_{a}^{4} R_{b} V_{p} f_{s} - 2 \pi C_{c}^{2} R_{a}^{4} R_{s} V_{p} f_{s} - 4 \pi C_{c}^{2} R_{a}^{3} R_{b}^{2} V_{p} f_{s} - 6 \pi C_{c}^{2} R_{a}^{3} R_{b} R_{s} V_{p} f_{s} - 2 \pi C_{c}^{2} R_{a}^{2} R_{b}^{3} V_{p} f_{s} - 6 \pi C_{c}^{2} R_{a}^{2} R_{b}^{2} R_{s} V_{p} f_{s} - 2 \pi C_{c}^{2} R_{a} R_{b}^{3} R_{s} V_{p} f_{s} - C_{c} R_{a}^{3} R_{b} V_{B} - 2 C_{c} R_{a}^{2} R_{b}^{2} V_{B} - C_{c} R_{a} R_{b}^{3} V_{B}\right) e^{- \frac{t \left(4 \pi^{2} C_{c}^{2} R_{a}^{3} R_{b}^{2} V_{B} f_{s}^{2} + 8 \pi^{2} C_{c}^{2} R_{a}^{3} R_{b} R_{s} V_{B} f_{s}^{2} + 4 \pi^{2} C_{c}^{2} R_{a}^{3} R_{s}^{2} V_{B} f_{s}^{2} + 4 \pi^{2} C_{c}^{2} R_{a}^{2} R_{b}^{3} V_{B} f_{s}^{2} + 16 \pi^{2} C_{c}^{2} R_{a}^{2} R_{b}^{2} R_{s} V_{B} f_{s}^{2} + 12 \pi^{2} C_{c}^{2} R_{a}^{2} R_{b} R_{s}^{2} V_{B} f_{s}^{2} + 8 \pi^{2} C_{c}^{2} R_{a} R_{b}^{3} R_{s} V_{B} f_{s}^{2} + 12 \pi^{2} C_{c}^{2} R_{a} R_{b}^{2} R_{s}^{2} V_{B} f_{s}^{2} + 4 \pi^{2} C_{c}^{2} R_{b}^{3} R_{s}^{2} V_{B} f_{s}^{2} + R_{a}^{3} V_{B} + 3 R_{a}^{2} R_{b} V_{B} + 3 R_{a} R_{b}^{2} V_{B} + R_{b}^{3} V_{B}\right)}{4 \pi^{2} C_{c}^{3} R_{a}^{3} R_{b}^{3} V_{B} f_{s}^{2} + 12 \pi^{2} C_{c}^{3} R_{a}^{3} R_{b}^{2} R_{s} V_{B} f_{s}^{2} + 12 \pi^{2} C_{c}^{3} R_{a}^{3} R_{b} R_{s}^{2} V_{B} f_{s}^{2} + 4 \pi^{2} C_{c}^{3} R_{a}^{3} R_{s}^{3} V_{B} f_{s}^{2} + 12 \pi^{2} C_{c}^{3} R_{a}^{2} R_{b}^{3} R_{s} V_{B} f_{s}^{2} + 24 \pi^{2} C_{c}^{3} R_{a}^{2} R_{b}^{2} R_{s}^{2} V_{B} f_{s}^{2} + 12 \pi^{2} C_{c}^{3} R_{a}^{2} R_{b} R_{s}^{3} V_{B} f_{s}^{2} + 12 \pi^{2} C_{c}^{3} R_{a} R_{b}^{3} R_{s}^{2} V_{B} f_{s}^{2} + 12 \pi^{2} C_{c}^{3} R_{a} R_{b}^{2} R_{s}^{3} V_{B} f_{s}^{2} + 4 \pi^{2} C_{c}^{3} R_{b}^{3} R_{s}^{3} V_{B} f_{s}^{2} + C_{c} R_{a}^{3} R_{b} V_{B} + C_{c} R_{a}^{3} R_{s} V_{B} + 2 C_{c} R_{a}^{2} R_{b}^{2} V_{B} + 3 C_{c} R_{a}^{2} R_{b} R_{s} V_{B} + C_{c} R_{a} R_{b}^{3} V_{B} + 3 C_{c} R_{a} R_{b}^{2} R_{s} V_{B} + C_{c} R_{b}^{3} R_{s} V_{B}}}}{4 \pi^{2} C_{c}^{3} R_{a}^{3} R_{b}^{3} V_{B} f_{s}^{2} + 12 \pi^{2} C_{c}^{3} R_{a}^{3} R_{b}^{2} R_{s} V_{B} f_{s}^{2} + 12 \pi^{2} C_{c}^{3} R_{a}^{3} R_{b} R_{s}^{2} V_{B} f_{s}^{2} + 4 \pi^{2} C_{c}^{3} R_{a}^{3} R_{s}^{3} V_{B} f_{s}^{2} + 12 \pi^{2} C_{c}^{3} R_{a}^{2} R_{b}^{3} R_{s} V_{B} f_{s}^{2} + 24 \pi^{2} C_{c}^{3} R_{a}^{2} R_{b}^{2} R_{s}^{2} V_{B} f_{s}^{2} + 12 \pi^{2} C_{c}^{3} R_{a}^{2} R_{b} R_{s}^{3} V_{B} f_{s}^{2} + 12 \pi^{2} C_{c}^{3} R_{a} R_{b}^{3} R_{s}^{2} V_{B} f_{s}^{2} + 12 \pi^{2} C_{c}^{3} R_{a} R_{b}^{2} R_{s}^{3} V_{B} f_{s}^{2} + 4 \pi^{2} C_{c}^{3} R_{b}^{3} R_{s}^{3} V_{B} f_{s}^{2} + C_{c} R_{a}^{3} R_{b} V_{B} + C_{c} R_{a}^{3} R_{s} V_{B} + 2 C_{c} R_{a}^{2} R_{b}^{2} V_{B} + 3 C_{c} R_{a}^{2} R_{b} R_{s} V_{B} + C_{c} R_{a} R_{b}^{3} V_{B} + 3 C_{c} R_{a} R_{b}^{2} R_{s} V_{B} + C_{c} R_{b}^{3} R_{s} V_{B}} + 1\right)}{R_{a} + R_{b}}\]

Note

SLiCAP can handle very large equations!

However: In order to provide useful design information:

“Models should be as simple as possible, but not simpler”

Numeric time-domain response

# Numeric time-domain response
timeResult_num  = sl.doTime(ACcoupling, pardefs=params)

# Plot the numeric response
sl.plotSweep("AC_time", "Time-domain response", timeResult_num, 0, 2, 500, sweepScale="u") 

power_impulse = sl.doImpulse(ACcoupling, source="V1")
M             = power_impulse.M
Iv            = power_impulse.Iv
Dv            = power_impulse.Dv
denom         = power_impulse.denom
numer         = power_impulse.numer
transfer      = power_impulse.laplace

The plot is hown below.

Return attributes

Important

Calculation of the unit impulse response requires execution of:

• doNumer()

• doDenom()

• doLaplace()

This returns the following result attributes:

power_impulse = sl.doImpulse(ACcoupling, source="V1")
M             = power_impulse.M
Iv            = power_impulse.Iv
Dv            = power_impulse.Dv
denom         = power_impulse.denom
numer         = power_impulse.numer
transfer      = power_impulse.laplace

Related math functions

Convert a step response into a periodic pulse response

The function step2PeriodicPulse converts a step response into a periodic pulse response.

# Convert a step response into a periodic pulse response
# Example step response
step_resp  = sp.sympify("A*(1-exp(-t/tau_s))")
# Define pulse time 'tau_p' and pulse period 'T_p'
tau_p, T_p = sp.symbols("tau_p, T_p")
# Generate the response to 3 pulses
three_pulse_resp = sl.step2PeriodicPulse(step_resp, tau_p, T_p, 3)

Formatted result:

\[\begin{split}\begin{align} H_{3p} = & A \left(1 - e^{- \frac{t}{\tau_{s}}}\right) \theta\left(t, 1\right) + A \theta\left(- 2 T_{p} + t, 1\right) \left(1 - e^{- \frac{- 2 T_{p} + t}{\tau_{s}}}\right) \nonumber \\ & + A \theta\left(- T_{p} + t, 1\right) \left(1 - e^{- \frac{- T_{p} + t}{\tau_{s}}}\right) - A \theta\left(t - \tau_{p}, 1\right) \left(1 - e^{- \frac{t - \tau_{p}}{\tau_{s}}}\right) \nonumber \\ & - A \theta\left(- 2 T_{p} + t - \tau_{p}, 1\right) \left(1 - e^{- \frac{- 2 T_{p} + t - \tau_{p}}{\tau_{s}}}\right) - A \theta\left(- T_{p} + t - \tau_{p}, 1\right) \left(1 - e^{- \frac{- T_{p} + t - \tau_{p}}{\tau_{s}}}\right) \end{align}\end{split}\]

where \(\theta(t)\) is Sympy’s Heaviside Function, occuring at \(t=0\) (see below).

Warning

Numeric evaluation of periodic periodic pulse responses created in this way, may take a while!

Sympy’s Heaviside function

sympy.Heaviside(t, 1): \(\theta(t,1)=0\) for \(t<0\) and \(\theta(t,1)=1\) for \(t \ge 0\).

>>> import sympy as sp
>>> sp.Heaviside(-1,1)
0
>>> sp.Heaviside(0,1)
1
>>> sp.Heaviside(1,1)
1

Display equations on HTML pages and in LaTeX documents

The report module Create reports, discusses how HTML snippets and LaTeX snippets can be created for variables, expressions, equations and tables.

As a matter of fact, all equations, tables and figures on this page are created with this module:

# Generate RST snippets for the Help file
rst = sl.RSTformatter()

rst.eqn("h_p(t)", power_impulse_sym).save("eqn-ACcoupling-power-impulse")
rst.eqn("a_s(t)", signal_step_sym).save("eqn-ACcoupling-signal-step")
rst.eqn("V_out", timeResult_sym).save("eqn-ACcoupling-time")
rst.eqn("H_3p", three_pulse_resp, multiline=2).save("eqn-H_3p-time")

# Generate LaTeX snippets for the Help file
ltx = sl.LaTeXformatter()

ltx.eqn("h_p(t)", power_impulse_sym).save("eqn-ACcoupling-power-impulse")
ltx.eqn("a_s(t)", signal_step_sym).save("eqn-ACcoupling-signal-step")
ltx.eqn("V_out", timeResult_sym).save("eqn-ACcoupling-time")
ltx.eqn("H_3p", three_pulse_resp, multiline=1).save("eqn-H_3p-time")

Plot frequency characteristics

SLiCAP plot functions are discussed in Create plots