How to use hysteresis in SPICE to reduce distortion in tape recordings

Analog tape distortion

The model we discussed in the previous article can be used to analyze the reduction in distortion of an analog tape recorder when a high-frequency sine wave bias signal is added to the analog signal being recorded.

For the first order, consider that the tape does not go into saturation, so the model does form a reasonable first order model for a real core. That is, the deadband reflects the remanence of the core, and this hysteresis causes the sinusoidal simulation signal to be clipped in the manner shown in Figures 1 and 2, and the hysteresis is shown in Figures 3 and 4.

Figure 1. Clockwise from top left, Figures 1 through 4 in Part 1 of this series.

As you can see, these hysteresis characteristics obviously lead to significant distortion.

High Frequency (HF) Signal Example

Consider a large, high frequency, sinusoidal magnetizing signal. Even with hysteresis, the average value of the recorded signal will be zero. If this signal is now DC offset from another signal, the HF (high frequency) signal will now essentially swing around this offset signal, regardless of the dead zone, because the large signal always drives the magnetization through the dead zone. This is true despite the waveform distortion of the HF signal.

Therefore, the average value of the HF bias signal will be equal to the offset signal, and it is this average value that forms the recorded signal for playback. In this way, the offset signal does not experience dead-band distortion as if it were the only signal applied. The waveform of the HF bias does not matter, as long as its frequency is high enough that any spurious signals are outside the bandwidth of the desired signal, as they can be filtered out.

A schematic illustrating this is shown below. The diode and capacitor simulate the DC hysteresis and nonlinear dead zone. It is the dead zone that causes the distortion.

You will see how HF oscillator biasing reduces tapped recording distortion. Without HF biasing, hysteresis would severely distort the output signal. The output DC magnetization lags the input as the signal changes direction. However, forcing the input to swing positive or negative overrides the hysteresis and allows the output to depend on the average of the hysteresis curve. The input signal is constructed with two frequencies to show that both THD (Total Harmonic Distortion) and IMD (Intermodulation Distortion) are cancelled.

Figure 2. Schematic diagram of tape distortion reduction

The schematic shows the sum of three sinusoidal voltages. Two of the signals represent a multi-tone input and the other is a high frequency offset signal. These two signals illustrate the effects of intermodulation distortion. Nonlinear systems will show sum and difference frequencies.

Original input/output signal

A typical raw single input/output signal looks like this:

Figure 3. Single frequency signal, VIN=1V

Figure 4. Single frequency signal, VIN=12V

Figures 3 and 4 show that the effective magnetizing signal “voltage lags” its input and is subsequently severely distorted due to the hysteresis that occurs when the signal changes direction. As we discussed in the previous article, standard SPICE techniques for generating voltage hysteresis do not model this distortion of the waveform peaks.

Mixing the original signal

The mixed original signal looks like this:

Figure 5. Mixing signal, VINA = 1V, VINB = 1V

Figure 5 shows that the input signal is significantly distorted.

The FFT (Fast Fourier Transform) of the mixed unbiased signal is shown below:

Figure 6. Unbiased mixed signal FFT VINA=1V, VINB=1V

Figure 7. Unbiased mixed-signal FFT, VINA = 6 V, VINB = 6 V

Under unbiased conditions, these show significant intermodulation distortion at 500 Hz, 1kHz, and 1k5.

Mixed high frequency signals

The mixed HF bias signal is shown below:

Figure 8. HF bias mixed signal

The FFT of the mixed HF bias signal is shown below:

Figure 9. HF bias mixed signal FFT

Therefore, adding HF bias shows a significant reduction in intermodulation products.

Summary

This article has demonstrated a technique that allows hysteresis to be modeled within the capabilities of standard SPICE. This technique can be extended to build inductors with hysteresis and will be described in a subsequent article.