AATC’s dynamic micro-speakers comes with its Thiele/Small Parameters (below abbreviated as T/S parameters, which is essentially an ID card. From T/S parameters we can easily derive its impedance curve, which is crucial to evaluate its performance. T/S parameters can also be used to predict low-to-mid frequencies where the lumped analogy holds valid. Now we start from the impedance curve which can be described by the following time-harmonic equation.

is the DC resistance of the coil; is the eddy current loss resulting from electromagnetic effect; is the coil’s inductance; , Q_MS is mechanical Q factor, while Q_­ES­ is electrical Q factor; represents the vibration system’s mass in terms of capacitive component (admittance analogy reflected from the mechanical domain to electrical domain by BL coupling);  represents the suspension system’s compliance in terms of inductive component (admittance analogy reflected from the mechanical domain to electrical domain by BL coupling). The above circuit is described in equivalent circuit form using analogy. There aren’t actually those electrical elements in this dynamic speaker.

In practical we let approach positive infinity, so we reduce to , rendering  in series with .

R_E is DC resistance and it forms the minimum of the impedance curve, sort of like the absolute value of the foundation. At 0hz, i.e. DC, this value can be measured.

In low frequencies especially below the impedance peak, we can safely ignore the inductive effects of L_E(ω). Now the dominating factor is the motional impedance, valued at . This rationalizes the peak around the resonance frequency in the impedance curve. If we clamp the diaphragm from moving, we’ll instead measure a blocked impedance without such peak. This motional impedance arises from the interaction between a capacitive and inductance component, and in reality an extra resistance component to form a damped response. Before the resonance point, the inductance increases and dominates, right after the resonance the dominance quickly shifts to the capacitance with capacitance reaching its maximum and gradually decreasing. At the peak, the inductance equals capacitance and they both cancel due to opposite phase. That is,  and  disappears, leaving only R_ES in effect, rendering the motional impedance .

This is well related to the concept of resonance. At resonance point, energy bounces back and forth between capacitive and inductive elements, without actually being consumed. By doing virtual work, the energy is consumed only by any resistance component such as the damper in the vibration system.

At f­_s­, the total impedance includes the DC resistance along with the motion impedance, i.e. R_E + R_ES, ignoring L_E(ω).

 creates a peak around f_s, similar to what a bandpass system looks like. ;  ; , the equation can be rewritten as below.

It’s obvious that  is a transfer function that performs bandpass function according to the angular frequency, s. Apart from this, as the frequency rises we can no longer omit the effect of . As we put that into the equation, we can now explain why the response is tilting upside in higher frequency. This is due to the increased inductive impedance of the coil when frequency rises.

In higher frequency, the motional impedance no longer plays a role. Instead, we still have R­_E­ in the bottom and the ever-rising L_E. Now we go back to the total equation and focus L_­E­ instead of R_ES、L_CES、C_MES、R_E、R_E’. In some cases, L_E­­ can be approximated to the following equation.

This equation is quite complicated so the n_e­ is usually assumed 1 to make it simpler. Only when n­_e­ equals 1 we have the power of ω as zero. In this case, L­_E­’s unit is Henry, the usual inductor unit. However, when the n­_e­ isn’t 1, then the unit is a variation of Henry.

To conclude, the ups and downs of the impedance curve mainly stems from  and . While the former is due to the inductance of the coil, the latter as motional impedance is due to the Back Electro Motive Force (BEMF). According to Lenz’s Law, a working motor will induce BEMF. As the conducted coil moves up and down in a magnetic field, its material naturally produces a counteracting force, BEMF, to disobey the force thrusted upon by the magnetic field. The direction of BEMF is opposite to the direction of the original voltage, hence reducing the current.

For a dynamic speaker,

• Let BEMF be V_bemf and input voltaeg V_supply. Considering the effect of BEMF, the current going through the coil, . The measured impedance value is now . The impedance should reach its maximum at the peak, and considering the input voltage is ideally V_supply, this means we’d find a minimized current at the resonance point, and to achieve that we need V_bemf to reach its maximum value.
• BEMF is related to the variability of the magnetic field. The faster the magnetic field changes, the strong the BEMF. It means that the BEMF peaks when the coil and diaphragm move at their fastest speed. Refer to the motor’s equation , N is the rotating speed of the motor.
• Please note that the diaphragm and coil move fastest at f_s­, so BEMF peaks and the total impedance peaks as well. Keeping this in mind we can figure out the f_{­s ­}with prior knowledge of other T/S parameters.