A Temperature-Compensation Technique for Improving Resolver Accuracy

Abstract

Variation in the ambient temperature deteriorates the accuracy of a resolver. In this paper, a temperature-compensation technique is introduced to improve resolver accuracy. The ambient temperature causes deviations in the resolver signal; therefore, the disturbed signal is investigated through the change in current in the primary winding of the resolver. For the proposed technique, the primary winding of the resolver is driven by a class-AB output stage of an operational amplifier (opamp), where the primary winding current forms part of the supply current of the opamp. The opamp supply-current sensing technique is used to extract the primary winding current. The error of the resolver signal due to temperature variations is directly evaluated from the supply current of the opamp. Therefore, the proposed technique does not require a temperature-sensitive device. Using the proposed technique, the error of the resolver signal when the ambient temperature increases to 70 °C can be minimized from 1.463% without temperature compensation to 0.017% with temperature compensation. The performance of the proposed technique is discussed in detail and is confirmed by experimental implementation using commercial devices. The results show that the proposed circuit can compensate for wide variations in ambient temperature.

1. Introduction

A resolver, which is a kind of inductive transducer, is a useful device in instrumentation and measurement systems. The operation of a resolver is identical to a variable transformer, which consists of a primary winding as a rotor and two secondary windings placed at right angles from each other as stators [1,2,3,4]. Resolvers are widely used for the measurements of the angle, position, and speed in instrumentation and control systems. Resolvers provide high reliability, durability, and resolution. Therefore, resolvers are especially suitable for harsh environments. Resolvers have been applied in industries, military equipment, robots, aerospace, radars, electric vehicles, and medical and scientific equipment [5,6,7]. Practically, the primary winding of a resolver is excited by a sinusoidal signal. The resolver signals from the two secondary windings known as the quadrature signals are proportional to the sine and cosine functions of the rotor shaft angle, which are in the form of amplitude modulation with a suppressed carrier. Traditionally, a synchronous demodulator based on an analog multiplier, low-pass filter, or analog switch and integrator has been used to extract the shaft-angle signals from the resolver signals [1,8,9,10]. However, the response time and accuracy of these demodulators deteriorate due to the large time constant and phase shift caused by the dominant pole of the low-pass filter and integrator. To overcome these disadvantages, an alternative technique using a peak-amplitude finder was introduced [3,11,12]. This technique provides a simple configuration and rapid response. Recently, techniques for determining position from a resolver signal in digital form have been proposed in the literature [5,13,14,15,16,17]. Unfortunately, these approaches require the resolver to operate at a constant ambient temperature to achieve this accurately.

Many parameters can affect the accuracy of the resolver, such as phase shifts between the primary winding and secondary windings, amplitude imbalance and imperfect quadrature between two resolver signals, and variation in the ambient temperature [1,3,18]. Amplitude imbalance and imperfect quadrature cause errors in the determination of the shaft angle. Recently, a technique to minimize the error of the shaft-angle determination caused by amplitude imbalance and imperfect quadrature was proposed [18]. However, this technique requires a high-speed processor. Consequently, a complex measurement system structure is required. An amplitude imbalance or phase shift between the primary and secondary winding can be prevented using a technique proposed in the literature [3]. Variation in the ambient temperature influences the resistance of the winding, the mutual inductance, the magnetizing current, and the core loss current of the resolver, causing amplitude error in the resolver signals. The traditional technique to minimize this temperature effect is based on the use of an inverse tangent of the ratio of two resolver signals [1,4,8]. However, the disadvantage of this technique is that a large error occurs when the resolver signal is close or equal to zero. The response time is also rather slow for this technique. An approach based on the use of a temperature-sensitive device was developed to compensate for the temperature effect of inductive transducers [19,20,21,22]. The temperature effect in the technique mentioned above is compensated for in an open-loop procedure, which causes inaccuracies in the resolver signal. Other techniques are based on the modification of the transducer’s structure or material to obtain low-temperature sensitivity [23,24,25], and the use of reference inductance to compensate for the temperature effect [26]. The authors of [27] constructed a closed-loop technique to compensate for the temperature of the inductive transducer without using a temperature sensor. This technique requires the output signals of the transducer to be in linear form, which is only suitable for transducers used for the measurement of linear displacement, such as linear variable differential transformers (LVDTs). In contrast, the resolver signals are in the sine and cosine functions. Therefore, the techniques mentioned above are not applicable to these resolvers.

The purpose of this study was to construct a temperature-compensation technique for a resolver without requiring a temperature-sensitive device. From the transformer principle, the parameters of the secondary winding side can be referred to the primary winding side [28]. Therefore, the error of the resolver signal on the secondary winding side because of temperature can be extracted from the primary winding side. In this study, the error in the resolver signal due to the temperature effect was investigated from the current flowing through the primary winding of the resolver using the opamp supply-current-sensing technique [29,30,31]. The amplitude variation in the resolver signal due to the temperature effect was determined and compensated for using subtract-and-sum topology. The performance of the proposed technique was analyzed and confirmed by experimental implementation. All the devices used in the proposed technique are commercially available. Hence, the attractions of the proposed technique are its performance, simple configuration, and low cost.

2. Principle

The principle of the resolver is shown in Figure 1 [1]. A sinusoidal signal v_ex = V_P sin (ω_ext) is applied to the primary winding L_P, where V_P is the peak amplitude voltage, ω_ex = 2πf_ex is the angular frequency, and f_ex is the frequency of the excitation signal. Two secondary windings are terminated by the load resistors R_L1 = R_L2, and the resolver signals v_S1 and v_S2 are produced in form of quadrature sinusoidal signals as

$$v_{S1}=k_{RS}{v}_{ex}\sin ({\theta }_S)=k_{RS}{V}_P\sin ({\omega }_{ex}t)\sin ({\theta }_S)$$

(1)

$$v_{S2}=k_{RS}{v}_{ex}\cos ({\theta }_S)=k_{RS}{V}_P\sin ({\omega }_{ex}t)\cos ({\theta }_S)$$

(2)

where k_RS denotes the transformation ratio of the resolver and θ_S is the shaft angle. Practically, the amplitude of the resolver signals is affected by variations in the ambient temperature because the mutual inductance, the resistance of both the primary and secondary windings, the magnetizing current, and the core loss current of the resolver depend on variations in the ambient temperature.

Therefore, the transformation ratio k_RS is affected by the ambient temperature. The resolver signals from Equations (1) and (2) can be written as

$$v_{S1}=k_{RS}(1-\alpha \Delta T)V_P\sin ({\omega }_{ex}t)\sin ({\theta }_S)$$

(3)

$$v_{S2}=k_{RS}(1-\alpha \Delta T)V_P\sin ({\omega }_{ex}t)\cos ({\theta }_S)$$

(4)

where α and ΔT are the temperature coefficient of the resolver and the temperature deviation from room temperature at 25 °C, respectively. From Equations (3) and (4), sin (ω_ext) can be simply removed by a synchronous demodulator. Thus, the resolver signals can be expressed as

$$v_{S1}=k_{RS}(1-\alpha \Delta T)V_P\sin ({\theta }_S)$$

(5)

$$v_{S2}=k_{RS}(1-\alpha \Delta T)V_P\cos ({\theta }_S)$$

(6)

The amplitude of the resolver signals depends on variation in the ambient temperature. The variation in the ambient temperature affects all the quantities of the resolver and causes deviation in the transformation ratio k_RS. Based on the principle on which transformers function, the secondary winding currents flowing through the load resistors R_L1 and R_L2 can be referred to the primary side [28]. Therefore, the primary winding current i_P can be used to investigate the secondary winding current. The disturbance of the primary winding current i_P (which includes the magnetizing current and the core loss current) by the ambient temperature can be approximated as

$$i_P=\frac{\left(1-\alpha \Delta T\right)v_{ex}}{\left|Z_{PL}\right|}=(1-\alpha \Delta T)k_R{v}_{ex}$$

(7)

where k_R = 1/|Z_PL|, and Z_PL is the combination of the primary-side impedance and the secondary-side impedance referred to the primary side of the resolver. It should be noted that the changes in the magnetizing current and the core loss current due to temperature are very small compared to the change in the secondary winding current, which can be considered as negligible [25,32]. If the temperature effect on the primary winding current is compensated for, then the secondary winding current is compensated simultaneously. Practically, the primary winding L_P of the resolver is driven by the opamp, as shown in Figure 2a. From Figure 2a, the output state of the opamp is generally a class-AB configuration where the output current of the opamp exists within its supply current. In this study, a bipolar opamp was used for experimental implementation. From Figure 2a, the supply currents i_SP and i_SN of the opamp can be expressed as [29,30,31]

$$i_{SP}=I_B+\frac{\sqrt{\left(i_P^2+4I_S^2\right)}+i_P}{2}\hspace{1em}\hspace{1em}for\hspace{0.17em}\left|i_P\right|<2I_S$$

(8)

and

$$i_{SN}=I_B+\frac{\sqrt{\left(i_P^2+4I_S^2\right)}-i_P}{2}\hspace{1em}\hspace{1em}for\hspace{0.17em}\left|i_P\right|<2I_S$$

(9)

where I_B and I_S denote the quiescent current and the bias current of the class-AB output state of the opamp, respectively.

For the large-amplitude primary winding current i_P or |i_P| > 2I_S, the currents i_SP and i_SN can be described by

$$i_{SP}=\{\begin{array}{l}{I}_B+i_P\hspace{1em}\hspace{1em}\hspace{1em}for\hspace{0.17em}{i}_P>2I_S\\ \\ I_B\hspace{1em}\hspace{0.17em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}for\hspace{0.17em}{i}_P<-2I_S\end{array}$$

(10)

and

$$i_{SN}=\{\begin{array}{l}{I}_B\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}for\hspace{0.17em}{i}_P>2I_S\\ \\ I_B+i_P\hspace{1em}\hspace{0.17em}\hspace{1em}for\hspace{0.17em}{i}_P<-2I_S\end{array}$$

(11)

The primary winding current i_P of a resolver affected by temperature can be observed from the supply current of the opamp. The schematic diagram used to achieve the supply current of the opamp is shown in Figure 2b. From the circuit in Figure 2b, if the input signal v_ex is applied, then the primary winding current i_P can be described by v_ex/Z_PL. The resistors R_P and R_N convert the current signals i_SP and i_SN to the voltage signals v_SP and v_SN, respectively. Opamp A₂ and the resistors R₁, R₂, and R₃ function as a summing amplifier to sum the voltage signals v_SP and v_SN [30,31]. Thus, the primary winding current i_P of the resolver is determined in the form of the voltage signal v_RC. The voltage signal v_RC can be stated as

$$v_{RC}=\frac{R_P{R}_3}{R_1}{i}_{SP}+\frac{R_N{R}_3}{R_2}{i}_{SN}$$

(12)

Practically, the resistances R₁ = R₂ and R_P = R_N are assigned. If the peak amplitude of the current signal i_P is greater than 2I_S or |i_P| > 2I_S, then Equation (12) can be written as

$$v_{RC}=\frac{R_P{R}_3}{R_1}{i}_P=(1-\alpha \Delta T)k_C{k}_R{v}_{in}$$

(13)

The primary winding current of the resolver can be determined from the voltage signal v_RC. Practically, the parameter k_C is normally assigned as 1/k_R. The simulation result of the circuit depicted in Figure 2b obtained using the PSPICE analog simulation program is shown in Figure 3, where a sinusoidal signal of 3 kHz with a peak amplitude of 2V is applied as the input signal v_in. The parameters of the primary winding L_P were measured and modeled from the resolver used in this study. From Figure 3, the phase lag θ_P occurs between the excitation signal v_in and the signal v_RC due to the behavior of the resolver. Therefore, the resolver behavior can be approximated by a single-pole expression as

$$\frac{v_{RC}\left(s\right)}{v_{in}\left(s\right)}=\frac{(1-\alpha \Delta T)k_C{k}_R}{\left(T_{R}s+1\right)}$$

(14)

for

$$T_R=\frac{\tan {\theta }_P}{{\omega }_{ex}}$$

where T_R is the time constant of the resolver. From Equation (14), the circuit in Figure 2b affected by temperature can be represented by the block diagram shown in Figure 2c, where α_R = αk_R.

3. Temperature-Compensation Technique

A simple closed-loop configuration to compensate for the temperature effect is shown in Figure 4a, where the subtract-and-sum topology is depicted within the dashed-line frame. The phase-lead compensator in Figure 4a is required to compensate for the phase lag θ_P of the resolver’s behavior. Therefore, the time constant T_l of the phase-lead network is set to equal the time constant T_R for the in-phase condition between the signals v_F and v_in. From Figure 4a, the closed loop transfer function can be stated as

$$v_F(s)=\frac{\left(1+k_P\right)\gamma k_C{k}_F{k}_R}{\left(1+\gamma k_C{k}_F{k}_P{k}_R\right)\left(1+\frac{\gamma T_{l}s}{\left(1+\gamma k_C{k}_F{k}_P{k}_R\right)}\right)}{v}_{in}(s)-\frac{\gamma {\alpha }_R{k}_C{k}_F{k}_R}{\left(1+\gamma k_C{k}_F{k}_P{k}_R\right)\left(1+\frac{\gamma T_{l}s}{\left(1+\gamma k_C{k}_F{k}_P{k}_R\right)}\right)}\Delta T$$

(15)

where k_P and k_F are the proportional gain and the compensation gain, respectively, and γ is the attenuation factor of the phase-lead compensator. Normally, k_C = 1/k_R and k_F = 1/γ are assigned. If k_P >> 1 and γk_Ck_Fk_Pk_R >> 1, then Equation (15) can be expressed as

$$v_F(s)=\left(v_{in}(s)-\frac{{\alpha }_R}{k_P}\Delta T\right)\frac{1}{\left(1+\frac{\gamma T_{l}s}{k_P}\right)}$$

(16)

From Equation (16), the magnitude of the signal v_F for v_in = V_Psin(ω_ext) can be approximated as

$$\left|v_F(j\omega )\right|=\left(V_P-\frac{{\alpha }_R}{k_P}\Delta T\right)\left[\frac{1}{\sqrt{\left(1+{\left(\frac{\gamma T_l{\omega }_{ex}}{k_P}\right)}^2\right)}}\right]$$

(17)

From Equation (17), the parameters of γ = 0.1, T_l = 67.63 μs, and ω_ex = 18.85 krad/s were achieved from experimental implementation. Thus, the denominator of the terms in the square brackets in Equation (17) can be approximated as 1. It is evident that the temperature effect of the primary winding current i_P on the resolver can be minimized by increasing the gain value k_P. The proposed circuit for the block diagram in Figure 4a is shown in Figure 4b. The block diagram of the sum-and-subtract topology in the dashed-line frame in Figure 4a can be replaced by the dashed-line block SS₁ in Figure 4b to minimize the active device used for the experimental implementation.

The proposed circuit consists of three opamps as active devices. The operation of the proposed circuit can be explained as follows: from Figure 4b, the opamp A₁ and resistors R_f1 and R_f2 function following the sum-and-subtract scheme to obtain the error signal between the input signal v_in and the feedback signal v_F. The capacitance C_C is employed to avoid unstable sum-and-subtract scheme operation [27]. For a small value of the capacitance C_C, the output signal v_ex of the opamp A₁ can be stated as

$$v_{ex}=v_{in}+\frac{R_{f2}}{R_{f1}}\left(v_{in}-v_F\right)=v_{in}+k_P\left(v_{in}-v_F\right)$$

(18)

Notably, the signal v_ex is provided as the excitation signal for the resolver. From Equation (18), the signal v_ex is in sum-and-subtract form, corresponding to the sum-and-subtract topology in Figure 4a. The primary winding current i_P is detected by the resistors R_P and R_N in the form of voltage signals v_SP and v_SN, respectively. Opamp A₂ forms the inverting amplifier to obtain the signal v_RC from the v_SP and v_SN signals. The phase-lead compensator formed by capacitor C_l1 and resistors R_l1 and R_l2 provides the phase lead θ_L to compensate for the phase lag θ_P caused by the behavior of the resolver. Opamp A₃, the resistors R_C1 and R_C2, and the variable resistor R_V provide the compensation gain k_F for the phase-lead compensator. The relationship between the voltage signals v_RC and v_F in Figure 4b for R_C1 = R_C2 can be expressed as

$$\frac{v_F(s)}{v_{RC}(s)}=\frac{2\left(s+\frac{1}{T_l}\right)}{\beta \left(s+\frac{1}{\gamma T_l}\right)}=\frac{\gamma k_F\left(T_{l}s+1\right)}{\left(\gamma T_{l}s+1\right)}$$

(19)

where γ = R_l2/(R_l1 + R_l2) is an attenuation factor, k_F = 2/β is a compensation factor, and T_l = R_l1C_l1. The compensation factor k_F is set to 1/γ for |v_F(jω)| = |v_RC(jω)|. Notably, the pole at s = −1/γT_l must be far from the left of the zero at s = −1/T_l to prevent phase error at the excitation frequency. Therefore, the factor γ ≤ 0.1 is chosen to assign the location of the pole shifted left from the location of the 0 by at least 1 decade. The phase lead θ_L in Equation (19) can be stated as

$${\theta }_L={\tan }^{-1}\left({\omega }_{ex}{T}_l\right)-{\tan }^{-1}\left(\gamma {\omega }_{ex}{T}_l\right)$$

(20)

If θ_L = |θ_P| is set, then the voltage signal v_F and the input signal v_in are in phase.

4. Performance Analysis

In practical implementation, a deviation from the ideal performance is caused by the non-ideal characteristic of the devices used in the proposed circuit. The tolerances of the resistors and non-identical supply currents of opamp A₁, i_SP, and i_SN contribute to the amplitude error and the DC offset voltage in the v_RC and v_F signals. The feedback signal v_F, which includes error due to the non-ideal characteristic of the devices used in the circuit in Figure 4b, can be written as

$$v_F=\frac{\gamma 2R_3{R}_{C2}{R}_P}{\beta R_1{R}_{C1}}\left(1+{\epsilon }_1\right)i_P+V_{offset}$$

(21)

for

$${\epsilon }_1={\delta }_A+{\delta }_R$$

and

$$V_{offset}=\frac{\gamma 2R_3{R}_{C2}{R}_P\left({\Delta }_B+{\Delta }_S\right)}{\beta R_1{R}_{C1}}$$

where δ_A and δ_R are the tolerances of the resistors R₁, and R_P, respectively; Δ_B and Δ_S are the different currents of the quiescent currents (I_B1 − I_B2) and the bias currents of the class-AB output state (I_S1 − I_S2) of the opamp A₁, respectively; and V_offset is a DC offset voltage. Practically, the resistances in Equation (21) are set to R_C1 = R_C2, R_P = 500 Ω, R₁ = 10 kΩ, and R₃ = 322.7 kΩ, which were chosen for the tolerance of 0.1% or δ_A = δ_R = 0.001. From Equation (21), the error ε₁ can be calculated as 2 × 10⁻³. The effect of the error ε₁ can be minimized by tuning the compensation factor k_F = 2/β. The different currents Δ_B and Δ_S are measured from opamp A₁ as about 3.8 and 2.1 μA, respectively. Therefore, the offset voltage V_offset can be calculated as 95.2 mV for k_F = 1/γ. The offset voltage V_offset can be cancelled by adjusting the voltage V_C in Figure 4b to an appropriate value. The tolerance of the resistors in the proposed circuit also induces harmonic distortion in the feedback signal v_F for the amplitude of the primary winding current i_P varying in the range of ±2I_S. The total harmonic distortion (THD) as a percentage of the feedback signal v_F for −2I_S ≤ i_P ≤ 2I_S can be expressed as

$$\mathrm{THD}=\frac{\left({\delta }_A+{\delta }_R+{\delta }_S\right)i_P}{16\left(1+\frac{{\delta }_A+{\delta }_R}{2}\right)I_S}\times 100\%$$

(22)

where δ_S = Δ_S/I_S. From Equation (22), the maximum percentage of THD occurs when the amplitude of the primary winding current |i_P| is equal to 2I_S. If the class-AB bias current I_S of opamp A₁ is 1.302 mA, then the THD percentage will be about 0.09%. Notably, the effect of THD can be prevented. The second factor is the unstable operation of the proposed circuit caused by intermodulation distortion (IMD) and phase shift due to the behavior of the opamps. IMD occurs on the v_ex signal due to the intrinsic pole of the opamp A₁. To avoid IMD, a capacitance C_C is connected in parallel to the resistance R_f2 to generate the dominant pole for the sum-and-subtract SS₁. The dynamic behavior of the opamps A₁, A₂, and A₃, caused by the gain bandwidth product (GBP) and the gains k_P and k_F, introduces a phase shift in the feedback signal v_F. Figure 4c shows a block diagram of the proposed circuit where the dynamic behaviors of the opamps are depicted. The time constants T₁, T₂, and T₃ of the opamps A₁, A₂, and A₃, respectively, depend on the amplification gains and GBPs of each opamp. From Figure 4b, the time constant T_C = R_f2C_C is obtained. In addition, the corner frequency ω_C of the term (T_Cs + 1)⁻¹ should be assigned in the range of 10ω_ex ≤ ω_C ≤ 0.1GBP₁ to avoid the unstable operation of the proposed scheme. Practically, the time constant T_C = (0.1GBP₁)⁻¹ is assigned. The time constant T_l = T_R is provided to compensate for the phase lag of the primary winding in the resolver structure. The time constants T₁, T₂, and T₃ can be stated as

$$T_1=\frac{\left(1+k_P\right)}{GB{P}_1}$$

(23)

$$T_2=\frac{(1+k_C)}{GB{P}_2}$$

(24)

and

$$T_3=\frac{\left(1+k_F\right)}{GB{P}_3}$$

(25)

where GBP_i is the gain bandwidth product of opamp A_i used in this paper. From Equations (23)–(25), the time constants T₁, T₂, and T₃ are proportional to the value of the gains k_P, k_C, and k_F, respectively. Notably, a high value of k_P is used to minimize the temperature effect. In addition, high values of k_P and k_F cause the proposed scheme to operate unstably. The phase shift caused by the pole at s = −1/T₁ should approach 0° at the excitation frequency ω_ex to avoid unstable operation. Therefore, the time constant T₁ is assigned to be less than at least 1 decade for the period of the excitation frequency ω_ex or T₁ ≤ 1/10ω_ex. From Equation (19), the pole at s = −1/γT_l of the phase-lead compensator should be placed to the left of the 0 at s = −1/T_l by more than 1 decade by adjusting the factor γ to prevent phase error. Therefore, the maximum values of the amplification factors k_P and k_F for maintaining the stable operation of the proposed technique can be expressed as

$$k_P=\frac{GB{P}_1}{10{\omega }_{ex}}-1$$

(26)

and

$$k_F=\frac{GB{P}_3}{10{\omega }_{ex}}-1$$

(27)

From Equations (26) and (27), the maximum value of k_P and k_F for the stability condition is 132.33 for ω_ex = 18.85 krad/s GBP₁ = GBP₂ = GBP₃ = 25.13 Mrad/s. From Equations (23) and (25), the time constants T₁ and T₃ are calculated as 4.02 and 0.44 μs for k_P = 100 and k_F = 10, respectively. The compensated gain k_C is set to 32.27. Therefore, the time constant T₂ can be determined from Equation (24) as 1.32 μs. From Figure 4b, the resistances R_f1 and R_f2 are assigned as 2 and 200 kΩ for the proportional gain k_P = 100, respectively. If the time constant T_C = R_f2C_C is set to 0.1GBP₁, about 0.4 μs, then the capacitance C_C can be determined as 1.99 pF. The phase shifts θ₁, θ₂, and θ₃ of the poles at s = −1/T₁, s = −1/T₂, and s = −1/T₃, respectively, can be described by

$${\theta }_1=-{\tan }^{-1}\left[\frac{{\omega }_{ex}(1+k_P)}{GB{P}_1}\right]$$

(28)

$${\theta }_2=-{\tan }^{-1}\left[\frac{{\omega }_{ex}(1+k_C)}{GB{P}_2}\right]$$

(29)

and

$${\theta }_3=-{\tan }^{-1}\left[\frac{{\omega }_{ex}(1+k_F)}{GB{P}_3}\right]$$

(30)

For k_P = 100, k_F = 10, and k_C = 32.27, the phases θ₁, θ₂, and θ3 are −4.33°, −1.43°, and –0.47°, respectively. These phase shifts deteriorate the stability of the proposed technique but can be compensated by increasing the phase θ_L of the phase-lead compensator. Therefore, the time constant T_l of the phase-lead compensator can be written as

$$T_l=\frac{\tan \left(\left|{\theta }_P\right|+\left|{\theta }_1\right|+\left|{\theta }_2\right|+\left|{\theta }_3\right|\right)}{{\omega }_{ex}}$$

(31)

For θ_P = −51.89°, the time constant T_l can be determined as 85.3 μs.

In addition, the temperature effect causes deviation of the winding inductance, which is exhibited in the phase shift term φ_D in the primary winding current. The phase shift φ_D can be approximated by the first-order pole at s = −1/T_D, where T_D = (tan φ_D)/ω_ex. The effect of phase shift φ_D in the time constant T_D can be included in Equation (17) as

$$\left|v_F(j\omega )\right|=\left(V_P-\frac{{\alpha }_R}{k_P}\Delta T\right)\left[\frac{1}{\sqrt{\left(1+{\left(\frac{\left(\gamma T_l+T_D\right){\omega }_{ex}}{k_P}\right)}^2\right)}}\right]$$

(32)

Practically, the time constant T_D is much less than the time constant γT_l and can be omitted. From Equation (32), the denominator of the term in the square brackets can be approximated as 1, corresponding to Equation (17). Therefore, the performance of the proposed scheme is unaffected by the phase shift φ_D.

5. Experimental Results

The proposed technique was implemented using commercially available devices. The opamps used in this study were LF351 for opamp A₁ and LF353, comprising two opamps in the same package for opamps A₂ and A₃, where LF351 and LF353 provide a GBP of 25.13 Mrad/s. The measured average values of the currents I_B, I_S, Δ_B, and Δ_S for opamp A₁ were 2.39 mA, 1.3 mA, 3.8 μA, and 2.1 μA, respectively. The passive devices were chosen as R₁ = R₂ = 10 kΩ, R_C1 = R_C2 = 30 kΩ, R_f2 = 200 kΩ, R_f1 = 2 kΩ, R_P = R_N = 500 Ω, C_C = 2 pF, and a variable resistor R_v = 1 kΩ. All resistors used in the proposed circuit were selected for having a tolerance of 0.1%. The power supply was set to ±12 V. The resolver used to demonstrate the performance of the proposed technique was a SANYO 101-4100 with a transformation factor k_RS of 0.37. The excitation signal was assigned as a 3 kHz sinusoidal wave with 2 V peak amplitude. The load resistors R_L1 and R_L2 for the resolver were assigned as 100 kΩ. The impedance of the resolver including the impedance of the secondary side referred to the primary side was measured from the primary winding using a HIOKI 3532-50 LCR meter as 16.135 kΩ at room temperature (25 °C). Therefore, the conversion factor k_R was determined as 61.977 × 10⁻⁶ A/V. From Equation (13), the factor k_C = 1/k_R = 16.135 × 10³ V/A was set. Therefore, the resistor R₃ of Equation (13) was determined as 322.7 kΩ for R_N = R_P = 500 Ω and R₁ and R₂ = 10 kΩ. The variable resistor was used for the resistor R₃ to achieve 322.7 kΩ. The circuit in Figure 2b was used to determine the dynamic behavior of the resolver.

Figure 5 shows the measured result of the phase shift between the signal v_in and the primary winding current represented by the signal v_RC. From Figure 5, the phase lag θ_P between signal v_in and signal v_RC was measured as 51.98° at a room temperature of 25 °C. Therefore, the primary winding impedance including the impedance of the secondary side referred from the primary side can be provided in polar form as 16.135 kΩ ∠ −51.98°. The time constant T_R representing the dynamic behavior of the circuit in Figure 2b was calculated from Equation (14) as 67.63 μs. From Figure 4b, the time constant T_l = T_R was assigned to compensate for the phase lag due to the total impedance of the resolver. The attenuation factor γ = 0.1 was chosen in this experiment. Subsequently, γT_l = 676 μs was obtained. The resistances R_l1 and R_l2 were determined as 6.76 kΩ and 676 Ω, respectively, for C_l1 = 0.01 μF. In this experiment, the resistances R_l1 and R_l2 were chosen as 6.8 k and 680 Ω, respectively. Therefore, the attenuation factor γ was recalculated as 0.09. The feedback gain k_F was determined as 11.11 by adjusting the variable resistor R_v. The operation environment of the resolver in this experiment was a chamber capable of varying the temperature from 25 to 70 °C. The shaft angle θ_S of the resolver was set to 45° to obtain equal amplitudes of the secondary winding signals v_S1 and v_S2. The experimental setup and the prototype of the proposed circuit in Figure 4b are shown in Figure 6a and 6b, respectively.

If phase θ_P of the primary winding current at 25 °C is set as a phase reference, the phase shift φ_D of the primary winding current shift from the phase reference θ_P can be seen in Figure 7 for the variations in the ambient temperature from 25 to 70 °C. From Figure 7, the maximum value of the phase shift φ_D at 70 °C is about 1.78°, which corresponds to the time constant T_D = 1.649 μs. The time constant γT_l = 676 μs is much larger than the time constant T_D. Notably, the effect of the phase shift φ_D on the circuit performance is insignificant and can be neglected.

The phase shift φ_D can be inferred from the change in the resolver inductance. From Figure 7, the variation in the ambient temperature had little influence on the winding inductance of the resolver, in agreement with a recently reported approach [25]. The measured result of the primary winding current in Figure 2a, represented by the signal v_RC versus the variation in the ambient temperature, is shown in Figure 8a, which was used to determine the temperature coefficient α of the resolver. The temperature coefficient α of the resolver in the term of α_R was approximated from the signal v_RC as −1.71 mV/°C for a 2 V peak amplitude of the excitation signal. The temperature effect on the secondary winding signals v_S1 and v_S2 was also measured as shown in Figure 8b. The percentage error for this experimental result can be described as

$$percentage\hspace{0.17em}error=\frac{\left|expected\hspace{0.17em}value-measured\hspace{0.17em}value\right|}{expected\hspace{0.17em}value}\times 100\%$$

(33)

As shown in Figure 8a,b, the magnitude of signals v_RC, v_S1, and v_S2 decreased with increasing ambient temperature. The percentage errors at 70 °C of the signals v_RC, v_S1, and v_S2 were 3.728%, 1.463%, and 1.44%, respectively. These errors are too high for precision control systems. Therefore, the resolvers require temperature compensation. From Figure 8b, the effect of temperature variation on the resolver signal v_S2 was the same as on the resolver signal v_S1. Therefore, only the resolver signal v_S1 was used to demonstrate the performance of the proposed technique.

The prototype board in Figure 6b was connected to the resolver for the investigation of the circuit’s performance, where the operation environment of the resolver was the same as in the above-described experiment. The experimental results for both signals v_RC and v_S1 in terms of percentage error are provided in Figure 9a,b, respectively. Figure 9a,b show that the errors of the signal v_RC and the resolver signals were significantly reduced. Figure 9c depicts the waveform of the excitation signal v_ex and the signal v_F for an ambient temperature of 70 °C. The signals v_ex and v_F were in phase. From Equation (32), the time constant T_D was weighted by the proportional gain k_P. Therefore, the performance of the proposed circuit was independent of the time constant T_D or the phase shift φ_D. As shown in Figure 8a and Figure 9a, the percentage error of the primary winding current represented by the signal v_RC could be minimized from 3.728% to 0.043% at 70 °C. Additionally, the percentage errors of the resolver signal v_S1 in Figure 8b and Figure 9b could be reduced from 1.463% to 0.017%. Obviously, the accuracy of the resolver was improved more than 100-fold using the proposed technique. These experimental results confirm that the proposed circuit provides high-quality performance and offers advantages due to a small and simple circuit configuration. In addition, the proposed circuit can be placed between the resolver and a commercial signal conditioner without disturbing the operation of the system.