Hysteresis Loops and Transfer Curves

Several workers have observed⁽¹⁾ that the transfer curves of Fig. 211 are similar in shape to the left-hand or return trace of the hysteresis loop. There is a connection between the two. In Fig. 21, p. 25, it was shown that in a core with both a-c and d-c magnetization the minor hysteresis loop follows the back trace of the major loop in the negative or decreasing B direction, and proceeds along a line with less slope in the positive direction until it joins the normal permeability curve at B_m. Also, it was pointed out in connection with Fig. 69, p. 94, that, if AB has the maximum value B_m, the result is the banana-shaped figure OB_mD'. Here again the loop representing flux excursion O-B_m follows the left-hand side of the hysteresis loop in the downward or negative direction.

Fig. 215. Minor loops in rectangular hysteresis loop core material.

In a rectangular hysteresis loop material with B-H loop shown in Fig. 215 (a), the path traced over a flux excursion B₀B_S is more irregular in shape but still follows the left-hand trace of the loop. If magnetic amplifier cores are biased to a series of reset flux positions B₀ to B₃ the corresponding flux excursions and minor loops are those shown in Fig. 215(b). Usually, the load current far exceeds the control current necessary to reset the cores, so that these loops actually have a much longer region over which the loop width is practically zero, as shown in Fig. 215 (c). This is true of all the loops regardless of flux excursion.

The foregoing is true of a slowly varying flux excursion, so that the locus of the lower end points of the minor loops is the left-hand trace of the d-c hysteresis loop. Most magnetic materials, including rectangular loop materials, have a wider loop when the hysteresis loop is taken under a-c conditions, because of eddy currents. The difference between loops is as shown in Fig. 216.

Fig. 216. D-c and a-c B-H loops for grain-oriented nickel steel.

The locus of the end points of the minor loops under a-c flux excursions is neither the a-c nor the d-c loop but an intermediate line such as that drawn dot-dash in Fig. 216. The slope of this line is less than that of either the a-c or the d-c loop, and the gain of the magnetic amplifier is accordingly reduced.

An analysis for the self-saturated magnetic amplifier of Fig. 214(a) is given below. Load current is assumed to have the same shape as e_L in Fig. 214(b), and the following assumptions are made:

• Sinusoidal supply voltage and negligible a-c source impedance.
• Negligible reactor and rectifier forward voltage iR drops.
• Negligible rectifier back leakage current.
• Negligible magnetizing current compared to load current.
• Negligible saturated inductance.
• High control circuit impedance.
• E = 4.44fNΦ_s·10^-8.     [113]

Θ₁ = firing angle as in Fig. 214(b).
t₁ = Θ₁/ω.
ω = 2π · supply frequency f.
E = rms supply voltage.
Φ_s = saturation flux = B_SA_C (for B_s see Fig. 215).
A_c = core section in cm².
Φ₀ = reset core flux = B₀A_C (for B₀ see Fig. 215).
R_L = load resistance.
I_av = average load current.
i = instantaneous load current.
N = turns in load winding.

Under the assumptions, equation 1 becomes

[114]

Integrating equation 114 gives

[115]

and

[116]

During the interval Θ₁ < ωt < π, load voltage is

[117]

where R_L is the load resistance. This may be integrated to give

[118]

Combining equations 116 and 118 and substituting equation 113,

[119]

The left side of equation 119 is the average load current over the conducting interval π/ω - t₁. Average load current over the whole cycle is

[120]

Equation 120 has two flux terms: Φ_s, which is a fixed quantity for a given core material; and Φ₀. The relation between Φ₀ and control current is, as indicated in Fig. 215, the return trace of the major hysteresis loop. Thus equation 120 states that the average load current is the sum of a constant term and a term which has the same shape as the return trace of the hysteresis loop. Quantitatively, a self-saturated half-wave magnetic amplifier has a current transfer curve the same as the return trace of the core hysteresis loop, except that ordinates are multiplied by fA_cN/10⁸R_L and are displaced vertically by an amount fB_sA_cN/10⁸R_L.

Comparison with equation 113 reveals that the ordinate multiplier and vertical displacement are E/4,44R_LB_S and E/4,44R_L, respectively. As noted above, the return trace should be modified to mean the dot-dash line of Fig. 216.

(1)	See "Self-Saturation in Magnetic Amplifiers," by W. J. Dornhoefer, Trans. A1EE, 68, 835 (1949).