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Created on 01.06.1983 12:01.

Updated 03.01.2020 12:50.

Author: Popov P., Shorov V ..

It is generally accepted that the sound quality of a loudspeaker is almost entirely determined by its frequency response by sound pressure, its unevenness in the range of reproducible frequencies and the harmonic coefficient. However, the subjective (by experts) assessment of the sound of not only amateur, but also industrial sound reproducing equipment shows that loudspeakers with good parameters do not always sound equally good.

A careful study of the operation of the speakers allowed us to assume that one of the reasons for this phenomenon may be the difference in the characteristics of the transient processes of the midrange heads included in the loudspeakers.

## Equivalent head circuit

To analyze transient processes in the area of the piston action of the head (low-frequency range of reproducible frequencies), it is convenient to use its equivalent circuit shown in Fig. 1a. Here R_{e} and L_{e} Are the resistance and inductance of the voice coil of the head, respectively, C = m and L = c are the electrical equivalents, respectively, of the mass m and the flexibility of the suspension from the mobile system, and R_{e} – electrical equivalent of radiation and friction losses of the suspension unit. The numerical values of the equivalents are converted to the electrical input of the head.

In the area of the piston action of the head, the effect of inductance L_{k} its frequency and time characteristics can be neglected. As a result, the equivalent circuit of the head takes the form shown in Fig. 1b.

It is known that the figure of merit of a circuit of a parallel connected resistor, inductor and capacitor is equal to the ratio of the conductance of the reactive (inductive or capacitive) and resistive branches. The quality factor of the circuit shown in Fig. 1b,

Here G_{a}, = 1 / R_{a}, Is the conductivity of the resistive branch, ω_{s}, = √ (mC) – resonant circular frequency of the head moving system. The figure of merit (Q_{a}), is called the acoustic Q-factor of the head, since it takes into account losses only in the mechanical oscillatory system (R_{a}).

If the one shown in fig. 1b connect the circuit to a generator with zero output impedance, then the considered LC circuit will be shunted by the resistance R_{e}… In this case, its quality factor is determined by the formula Q_{e}= ω_{s}mRe and is called the electrical Q-factor of the head (when determining it, the influence of R_{a}).

The quality factor determined taking into account the influence of the resistances R_{a} and R_{e} is called the equivalent Q-factor of the head Q_{t}… With zero internal resistance of the input voltage source, it is equal to:

Q_{t}= Q_{a}* Q_{e}/ (Q_{a}+ Q_{e})

And since for all heads Q_{a}>> Q_{e} quantity Q_{t}, only slightly differs from Q_{e}…

When passing from the characteristics of the electrical equivalent of the head to its acoustic characteristics, it must be borne in mind that the voltage on a parallel circuit consisting of elements m, C and R_{a} (Fig. 1b), is an electrical analogue of the vibrational speed of a moving system. Thus, the greater the value of R_{e}and, therefore, Q_{t}, for a given value of R_{a}, the greater the unevenness of the dependence of the voltage on the circuit on the frequency, which corresponds to the greater unevenness of the sound pressure developed by the head in the area of the piston action.

The resonant frequencies of the low-frequency heads lie within the frequency ranges they reproduce, therefore, when choosing these heads, special attention is paid to the numerical value of the equivalent Q-factor of the head Q_{t} and if it exceeds the required one, measures are taken to reduce it and improve the frequency response.

A different situation arises when choosing a mid-frequency (MF) heads. Their resonant frequencies are, as a rule, below the range of frequencies they reproduce. As a result, if by the traditional method (smoothly changing the frequency of the generator) to remove the frequency response of the loudspeaker by sound pressure, the unevenness of the characteristics of the midrange head near its resonant frequency is practically not detected in the resulting characteristic of the loudspeaker, since the voltage of the resonant frequency supplied to this head will be significantly weakened bandpass filter.

Meanwhile, the real mode of the midrange heads is significantly different from that discussed above. The voltage of the broadcast signal at the output of the loudspeaker bandpass filter can be considered as harmonic (sinusoidal), the amplitude and frequency of which are continuous and, in general, change rather sharply in time. For this reason, the head constantly operates in a transient (dynamic) mode, and not in a steady-state mode of sinusoidal oscillations, which occurs when the frequency response is removed by sound pressure.

## Dynamic mode of the midrange heads

For a quantitative assessment of the transient process, let us again turn to the equivalent circuit of the head (Fig. 1b) and assume that the voltage applied to the input of the device has a sinusoidal shape. Sinusoidal voltage frequency f_{1} lies in the passband of the midrange filter of the heads and exceeds the resonant frequency of the head f_{s}…

The calculation shows that under these conditions, the voltage on a parallel oscillatory circuit (an electrical analogue of the oscillatory speed of a moving system) consists of two components. The first component is a sinusoidal voltage with frequency f_{1} and the amplitude calculated using the methods for calculating the steady-state mode of sinusoidal oscillations^{[1]}… Ego is the so-called forced component, since its frequency coincides with the frequency of the voltage applied to the head.

The second component is called the free component of the transient process, since the law of its change in time is determined exclusively by the values of the parameters of the equivalent circuit of the head and does not depend on the frequency of the voltage applied to it. With an equivalent quality factor of the head Q_{t}> 0.5 free component is a sinusoid with amplitude decreasing in time. The circular frequency of this sinusoid is ω_{s }is called the frequency of free vibrations and is determined by the formula:

This frequency is always less than the resonant frequency of the head in, and approaches it as the quality factor increases. Already at Q_{t} = 1.5, the difference in the numerical values of both frequencies does not exceed 6%.

The amplitude of free oscillations decreases in time according to the law:

e^{–πt}^{/ QtTs},

where T_{s}, = 1 / f_{s} Is the period of the resonant frequency of the head, e is the base of natural logarithms.

The initial (at t = 0) value of the amplitude of free oscillations depends on the initial phase of the input voltage applied to the head and on the parameters of the circuit. In principle, it can reach the amplitude value of the forced component.

## The sound and the calculation of its duration

The free component of vibrations of the moving system of the head (or a group of heads) generates the so-called overtones, which are quite noticeable by ear in poorly designed loudspeakers and degrades the quality of their sound. Since the emergence of a free component is a fundamentally irreparable phenomenon and constantly accompanies the transition process in any …

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