Integrator

Circuit description

An integrator in measurement and control applications is an element whose output signal is the time integral of its input signal. It accumulates the input quantity over a defined time to produce a representative output.

Goals

• Calculate integrator response
• Calculate integrator response for R1=R2=R3=R

Modeling the circuit

First we include Symbolics.jl and CircuitS.

using Symbolics
include("../CircuitS.jl")

The we create the circuit and add all of the elements as shown in the picture above:

circuit = create_circuit()
add_element([Resistor, "R1", 1, 3], circuit)
add_element([Resistor, "R2", 4, 5], circuit)
add_element([Resistor, "R3", 5, 2], circuit)
add_element([Capacitor, "C", 3, 4, "U0"], circuit)
add_element([Voltage, "Ug", 1, 0], circuit)
add_element([OpAmp, "OpAmp1", [3, 0], 2], circuit)
add_element([OpAmp, "OpAmp2", [0, 5], 4], circuit)

Simulation

After we've built everything, we initialise and simulate the circuit:

init_circuit(circuit)
result = simulate(circuit)

println(result)

Dict{Any, Any} with 8 entries:
  "V5"       => 0
  "V3"       => 0
  "V2"       => (R3*Ug + C*R1*R3*U0) / (C*R1*R2*s)
  "I_OpAmp2" => (-Ug - C*R1*U0 - C*R2*Ug*s) / (-C*R1*R2*s)
  "V4"       => (-Ug - C*R1*U0) / (C*R1*s)
  "I_Ug"     => Ug / (-R1)
  "I_OpAmp1" => (-Ug - C*R1*U0) / (C*R1*R2*s)
  "V1"       => Ug

Integrator response is the voltage of node 2:

println(result["V2"])

(R3*Ug + C*R1*R3*U0) / (C*R1*R2*s)

We can replace replace R1=R2=R3=R the classic way, before the simulation:

@variables R1 R2 R3 R
init_circuit(circuit, Dict([R1=>R, R2=>R, R3=>R]))
result = simulate(circuit)

println(result["V2"])

(Ug + C*R*U0) / (C*R*s)

Or we can do it post simulation with JuliaSymbolics substitute function:

init_circuit(circuit)
result = simulate(circuit)

@variables R1 R2 R3 R
V2 = result["V2"]
V2 = SymbolicUtils.substitute(V2~0,Dict([R1=>R, R2=>R, R3=>R]))
V2 = simplify(V2, expand=true)
println(V2)