Transmission line

Circuit description

In electrical engineering, a transmission line is a specialized cable or other structure designed to conduct electromagnetic waves in a contained manner. The term applies when the conductors are long enough that the wave nature of the transmission must be taken into account.

opamp

Goals

Calculate current of L1
Calculate voltage across L1

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([Inductor, "L1", 0, 2], circuit)
add_element([Voltage, "Ug", 3, 0], circuit)
add_element([TransmissionLine, "T", [1, 0], [2,0], ["Zc", "t"] ], circuit)

Simulation

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

init_circuit(circuit)
result = simulate(circuit, false)

println(result)

AssertionError error

Again, due to simplifying errors in JuliaSymbolics, we cannot shorten the resulting expression.

We passed false as a second argument to simulate function which indicates a flag to not use simplify. As a result we have long expressions.

As it so happens, in this example, simplify only fails for current I_T_3 (current of transmission line on node 3), and runs successfully for others.

Dict{Any, Any} with 6 entries:
  "I_T_2" => ((-(Ug*exp(-s*t) + (Ug*(R1*exp(-s*t) - Zc*exp(-s*t))*(1 + exp(-s*t))) / (-Zc - R1*(1 + exp(-s*t))))*(L1*s*…
  "V3"    => -((L1*s*(Ug*exp(-s*t) + (Ug*(R1*exp(-s*t) - Zc*exp(-s*t))*(1 + exp(-s*t))) / (-Zc - R1*(1 + exp(-s*t))))) …
  "V2"    => Ug + (-R1*((-(Ug*exp(-s*t) + (Ug*(R1*exp(-s*t) - Zc*exp(-s*t))*(1 + exp(-s*t))) / (-Zc - R1*(1 + exp(-s*t)…
  "I_Ug"  => (-((R1*((-(Ug*exp(-s*t) + (Ug*(R1*exp(-s*t) - Zc*exp(-s*t))*(1 + exp(-s*t))) / (-Zc - R1*(1 + exp(-s*t))))…
  "I_T_3" => (Ug*exp(-s*t) + (Ug*(R1*exp(-s*t) - Zc*exp(-s*t))*(1 + exp(-s*t))) / (-Zc - R1*(1 + exp(-s*t)))) / ((-(R1*…
  "V1"    => Ug

Voltage across L1 is the potential of node 3:

V3 = simplify(result["V3"])
println(V3)

(-L1*s*(-Ug*Zc*(exp(-s*t)^2) - 2Ug*Zc*exp(-s*t))) / (R1*Zc + Zc^2 + L1*R1*s + L1*Zc*s + R1*Zc*(exp(-s*t)^2) + R1*Zc*exp(-s*t) + L1*R1*s*exp(-s*t) + L1*Zc*s*(exp(-s*t)^2) - (Zc^2)*(exp(-s*t)^2) - L1*R1*s*(exp(-s*t)^2))

We can still retrieve the current I_T_3 even though we can't simplify it:

println(result["I_T_3"])

(Ug*exp(-s*t) + (Ug*(R1*exp(-s*t) - Zc*exp(-s*t))*(1 + exp(-s*t))) / (-Zc - R1*(1 + exp(-s*t)))) / ((-(R1*exp(-s*t) - Zc*exp(-s*t))*(L1*s*exp(-s*t) - Zc*exp(-s*t))) / (-Zc - R1*(1 + exp(-s*t))) - Zc - L1*s)