Application Examples for Analog Computers — Example 5: Transformer

This document is an English translation of the original German-language text (de).

Example 5 — 15 October 1963

TRANSFORMER

1. Problem Statement

The time-domain behavior of the primary and secondary currents of a transformer under ohmic-inductive and ohmic-capacitive loads is to be determined following switch-on of the mains voltage u. The magnetization characteristic of the iron core is to be taken into account.

Figure 1 — Equivalent circuit of the transformer

Figure 1 shows the equivalent circuit. The symbols mean:

R₁, R₂′ — winding resistances
L₁, L₂′ — leakage inductances
L — main (magnetizing) inductance
Z — load

Where:

Z = R₀ + jωL₀ — ohmic-inductive
Z = R₀ + 1/(jωC₀) — ohmic-capacitive

2. Equations and Transformer Data

Circuit equations:

R₁·i₁ + L₁·(di₁/dt) + L·(di₁/dt) − L·(di₂/dt) − u₀ sin ωt = 0

(R₀ + R₂′)·i₂ + (L₂′ + L₀)·(di₂/dt) + L·(di₂/dt) − L·(di₁/dt) = 0 [ohmic-inductive]

(R + R₂′)·i₂ + L₂′·(di₂/dt) + L·(di₂/dt) − L·(di₁/dt) + (1/C₀)·∫i₂ dt = 0 [ohmic-capacitive]

i_magn = i₁ − i₂

Constants:

Parameter	Value
u₀	√2 · 220 V
R₁	5 Ω
R₂	6.5 Ω
L(I_magn)	see Figure 2
f	50 Hz
L₁	0.0191 H
L₂′	0.0221 H
i_max	23 A
L_max	1.6 H
L₀	0.204 H
R₀	60 Ω
C₀	6.65 μF

Figure 2 — Profile of the function M(I_magn)

(A bell-shaped normalized magnetization curve M vs. I_magn, peak at M = 1.)

3. Normalization

Using the normalized variables:

I = i / i_max
M = L / L_max
M₁ = L₁ / L_max
M₂ = L₂′ / L_max
τ = λ·t

The normalized equations become:

M·İ₁ + M₁·İ₁ + (R₁ / (L_max·λ²))·I₁ − M·İ₂ − (u₀ / (L_max·i_max·λ))·sin(ω/λ · τ) = 0

M·İ₂ + (M₂ + M₀)·İ₂ + ((R₀ + R₂′) / (L_max·λ))·I₂ − M·İ₁ = 0 [ohmic-inductive]

M·İ₂ + M₂·İ₂ + ((R₀ + R₂′) / (L_max·λ))·I₂ − M·İ₁ + (1 / (C₀·L_max·λ²))·∫I₂ dτ = 0 [ohmic-capacitive]

I_magn = (I₁ − I₂) · (i_max / i_magn,max)

These equations are conveniently solved in implicit form using the “open amplifier” technique [2]. Figure 3 shows the computing circuit.

4. Computing Circuit

Figure 3 — Computing circuit

(Block diagram showing integrators, summers, a function generator (FG) for M(I_magn), and multipliers implementing the normalized transformer equations for both load cases. The dashed block at the lower right implements the capacitive-load term 1/(C₀·L_m·λ²).)

5. Results

Figures 4 through 7 show the time-domain behavior of the primary and secondary currents for ohmic-inductive and ohmic-capacitive loads.

Figure 4 — Primary current under ohmic-inductive load (Y₁ vs. t): shows a large transient first peak, then decays to a quasi-steady sinusoidal waveform.
Figure 5 — Secondary current under ohmic-inductive load (Y₂ vs. t): smaller amplitude, settles quickly to steady-state sine wave.
Figure 6 — Primary current under ohmic-capacitive load (Y₁ vs. t): large initial distorted peak followed by a lower-amplitude irregular oscillation settling to a periodic pattern.
Figure 7 — Secondary current under ohmic-capacitive load (Y₂ vs. t): settles to a smooth sinusoidal waveform of moderate amplitude.

References

[1] Richter, Elektrische Maschinen (Electric Machines)

[2] Giloi, W. and Herschel, R., Rechenanleitung für Analogrechner (Computing Guide for Analog Computers), TELEFUNKEN-Fachbuch 1961, Konstanz