Windingless Inductor1

Inductors are an important part of passive circuit design. Depending on the frequency, inductors can be composed of a few turns of wire wound around a small ferrite core or of thousands of windings surrounded by a weighty iron core. In some applications, they may be replaced with smaller and less expensive synthetic inductors synthesised with operational amplifiers. For inductors, a 90 degree phase lag of sinusoidal current compared to voltage is characteristic. However, ohmic losses in the windings and cores cause a deviation of phase lag from theoretical 90 degree value. In the following simulation, this displacement may be used as a criterion of quality in an extended frequency range.

The circuit

The input file (windingless_inductor_1.cir)

electronic capacitor I

* C series = c1 r2 / r1
* R series = r1

.control
set units = degrees
ac dec 40 0.01Hz 1kHz
plot db(i(v1)) ph(i(v1)) xlabel f[Hz]
  + ylabel 'current magnitude [dB] / phase[degree]' title 'AC analysis'
plot mag(i(v1)) real(i(v1)) imag(i(v1)) xlabel f[Hz] ylabel current[A]
  + title 'AC analysis'
.endc

v1 0 1 dc 0 ac 1

r1 4 1 1k
r2 2 3 100k

c1 3 0 10nF

e1 2 0 1 2 1meg
e2 4 0 3 4 1meg

.end

The results

Windingless Inductor 2

A less expensive alternative is shown in the next circuit. It needs only one amplifier but has higher ohmic losses caused by resistors r1 + r2 lying in series with the inductance c1 * r1 * r2. In the graph, this is indicated by a phase lag in the current.

The circuit

The input file (windingless_inductor_2.cir)

windingless inductor II

* L series = r1 r2 c1
* R series = r1 + r2

.control
set units = degrees
ac dec 40 10Hz 1megHz
plot db(i(v1))+200 ph(i(v1)) xlabel f[Hz]
  + ylabel 'current magnitude [dB] / phase[degree]' title 'AC analysis'
plot mag(i(v1)) real(i(v1)) imag(i(v1)) xlabel f[Hz] ylabel current[A]
  + title 'AC analysis'
.endc

v1 0 1 dc 0 ac 1

r1 1 3 1k
r2 0 2 100k

c1 1 2 10nF

e1 3 0 2 3 1meg

.end

The results