I. Introduction

The LVDT and the RVDT sensors measure the linear and angular positions of moving parts, respectively. The moving part is attached to the movable core of a transformer as shown in Fig. 1. The core position changes the mutual inductance between the primary and the secondary windings in proportion to the position of the core.

A sine wave, which is a primary excitation, is applied to the primary coil of a RVDT/LVDT sensor to produce a double sideband suppressed carrier amplitude modulation (DSBSC-AM) waveform with the amplitude proportional to the position of the core at the secondary coil as follows:

where $r(t)$, $c(t)=A \cos \left(\omega_{c} t\right)$ and $m(t)$ are the output of the RVDT/LVDT sensor, the primary excitation signal, and the angular or the linear position information of the core, respectively. A signal conditioner is required to extract the position information of the core, $m(t)$, from the output signal of the RVDT/LVDT sensor, $r(t)$, which may be a form of DSBSC-AM demodulator ⁽¹⁾.

Signal conditioners for RVDT/LVDT sensors can be categorized according to whether synchronous demodulation is exploited or not. The methods without the synchronous demodulation include the amplitude detection method ⁽²⁾, the spectral estimation method ⁽³⁾, the oscillator-based method ^(4,5), the dual slope conversion method ⁽⁶⁾, and the ratio-based method ^(7,8). In ⁽²⁾ a LVDT signal conditioner that exploits a peak amplitude detector to generate the control signal for a sample and hold circuit to detect the core position. Since the positional information of the core in a LVDT sensor can be found by calculating the ratio between the output of the LVDT sensor and the primary excitation as in (1) where $m(t)=r(t) / c(t)$, the spectral estimation method calculates the ratio by using the fast Fourier transform. The oscillator-based method treats the secondary coils as an unknown differential inductance that is then measured using an oscillator-based read-out circuitry. In the dual-slope conversion method, the primary coil is excited with a triangular current waveform derived from a couple of DC reference voltages producing the square shaped voltages in the secondary coils that are converted into a digital output by performing the integration and de-integration on the sum of the secondary voltages.

The figure-of-merit representing the linearity performance of a RVDT/LVDT signal conditioner is the maximum linearity error that is the maximum deviation of a measured value from a straight line between the maximum and the minimum values of a measurement. The linearity performances of the amplitude detection, the spectral estimation, the oscillator-based, the dual slope conversion methods are moderate. For example, the maximum linearity errors of the oscillator-based and the dual slope conversion methods are 0.14% and 0.15% of full scale output (%FSO), respectively.

In the ratio-based method either the ratio of the difference and the sum of the secondary signals or that of the difference of the secondary signals and the primary signal is calculated.

Since there is the phase shift between the primary and the secondary signals, additional phase shifting circuits must be used to correctly extract the position information of the core. The synchronous demodulation methods are based on DSBSC-AM demodulation, where the product of the RVDT/LVDT sensor output signal and the locally generated carrier is low-pass filtered. To get the accurate position information of the core the locally generated carrier must be in phase with the sensor output signal, which requires the phase compensation. In previous works, the synchronous demodulation methods are implemented by using blocks such as a peak sensitive demodulator ⁽⁹⁾, a multi-channel digital demodulator ⁽¹⁰⁾ and a Costas-loop based demodulator ⁽¹¹⁾. The peak sensitive demodulator is a modified homodyne detector that carries out a phase compensated synchronous demodulation of the differential output of a LVDT sensor, where a direct digitization of the DSBSC-AM output at the carrier peaks is performed. In ⁽¹⁰⁾ the digital adaptive synchronous AM demodulation is performed, where a prediction technique is employed to recover the position information from the output signal of a RVDT/LVDT sensor.

The signal conditioner in ⁽¹¹⁾ performs DSBSC-AM demodulation by using a Costas loop, where both the primary excitation signal and the sensor output signal are used and the phase error between the sensor output signal and the locally generated carrier signal is corrected by using a feedback mechanism and hence, the need for the phase matching circuitry is eliminated. Even though the signal conditioner shows a better linearity performance than other methods by exploiting digital signal processing techniques and can be used for more demanding applications, the conditioner has disadvantages in that it requires large computational efforts and high implementation costs.

This paper proposes a new RVDT/LVDT signal conditioner structure that achieves a smaller area by removing some blocks in the signal conditioner of ⁽¹¹⁾ which consists of five multipliers, three low-pass filters, one narrowband loop filter, a threshold block and a numerically controlled oscillator (NCO) ⁽¹²⁾. The proposed signal conditioner achieves a small area by replacing three multipliers, one low-pass filter, one narrowband loop filter and a threshold block with a CORDIC block. The linearity performance of the proposed signal conditioner where the loop error is calculated directly by evaluating an arctangent function is better than that of the signal conditioner in ⁽¹¹⁾ where the loop error is calculated indirectly as a sine function of the phase error ⁽¹³⁾.

The organization of this paper is as follows. Sections II and III describe the operations of the Costas-loop based signal conditioner and a general CORDIC algorithm, respectively. The proposed area-optimized RVDT/LVDT signal conditioner structure and its implementation are described in Section IV. After the experimental results are explained in Section V, finally the conclusion is presented in Section VI.

II. Signal Conditioner Based-on Costas Loop

A signal conditioner for a RVDT/LVDT sensor such as the one in Fig. 1 converts the output of the sensor, $r(t)$, to a DC output voltage that represents $m(t)$. One of the effective structures for RVDT/LVDT signal conditioning is based on synchronous AM demodulation which exploits Costas loop as shown in Fig. 2.

In Fig. 2 the frequency of the primary excitation signal is fixed and the phase error, $\phi$, between the output of a RVDT/LVDT sensor and that of the NCO is made zero by using a feedback mechanism where the loop error, $e_{C O S T A S}(t)$, which indirectly represents the phase error, is determined by evaluating the sine function that has $\phi$ as its argument and is proportional to the phase error as follows:

since $\sin (\phi)$ is approximately equal to $\phi$ for small values of $\phi$, which may cause the linearity error that is large in RVDT/LVDT applications where the linearity requirement is very stringent. $\phi$ is typically non-zero upon startup and both the in-phase and the quadrature signals, $i(t)$ and $q(t)$, are nonzero, which leads to a non-zero $e_{COS T A S}(t)$ that is input into the phase increment input port of the NCO and alters the phase of the NCO output waveform to make $\phi$ zero.

When the Costas loop in Fig. 2 is in the locked state, the output signals of the NCO become either $\cos \left(\omega_{c} t\right)$ and $\sin \left(\omega_{c} t\right)$ or $-\cos \left(\omega_{c} t\right)$ and $-\sin \left(\omega_{c} t\right)$ because the loop error, $e_{COS T A S}(t)$, is zero in both the cases. Therefore, the NCO output signals in the locked state can be represented as $\cos \left(\omega_{c} t+\delta\right)$ and $\sin \left(\omega_{c} t+\delta\right)$ where is either zero or $\pi$ radians. After the loop is locked, the demodulated output signal can be found by multiplying a constant gain,$\frac{A}{2}$, with $i(t)$ that has two possible values $\frac{1}{2} \times A \times m(t)$ and $-\frac{1}{2} \times A \times m(t)$ when$\delta$ is zero and $\pi$ radians, respectively, which is called a sign ambiguity.

The sign ambiguity is removed and the sign of the demodulated output is determined by low-pass filtering the product in (3), which is done by using MUL3 and LPF4 in Fig. 2.

where $\cos \left(\omega_{c} t+\phi+\delta\right)$, $\theta$, and $A \cos \left(\omega_{c} t+\theta\right)$ are the in-phase local carrier signal, the phase shift induced between the primary and the secondary signals, and the primary excitation signal, respectively, and $\phi$ is zero radians in the locked state. Low-pass filtering $s(t)$ removes the high frequency component to produce $\frac{1}{2} A \cos (\theta-\delta)$, which is positive or negative when $\delta$ is zero or $\pi$ radians, respectively, in the locked state under the assumption that $|\theta| \leq \pi / 2$, which is valid for a RVDT/LVDT sensor, and $A>0$. The output of Threshold block, which is $-1$ or $1$ for a negative or a positive input, respectively, is multiplied with the scaled $i(t)$ to produce the sign-corrected demodulated output.

For the proper operation of the signal conditioner, the narrowband low-pass filter whose the output is the approximate value of the phase error must have a small cutoff frequency. The signal conditioner in ⁽¹¹⁾ has a cutoff frequency of 10 Hz, which leads to a 257-tap digital low-pass filter.

III. Operation of CORDIC

A CORDIC is known to be a simple and efficient algorithm to calculate trigonometric functions typically converging with one bit per an iteration ⁽¹⁴⁾. A CORDIC block can operate either in a rotation mode or a vectoring mode where an arctangent function can be computed. As shown in Fig. 3, the CORDIC in a vectoring mode rotates the input vector until the resulting vector is aligned with $\mathcal{X}$ axis by minimizing the $y$ component of the residual vector at each iteration as follows:

where $d_{i}=1$ if $y_{i}<0$, $d_{i}=-1$ otherwise. When $y_{n}$ becomes zero, the iteration stops and the residual vectors are as follows:

where $\left(x_{0}, y_{0}\right)$ is an input vector and $z_{n}$ becomes $\tan ^{-1}\left(\frac{y_{0}}{x_{0}}\right)$ if $z_{0}$ is set to zero.

IV. Proposed Signal Conditioner

In this section, the structure of the proposed signal conditioner and its implementation are described. In the proposed signal conditioner, a CORDIC block is used to find the phase error instead of a narrowband low-pass filter and a multiplier. By using the CORDIC block the need for a sign ambiguity removal is also eliminated.

V. Experimental Results

VI. Conclusion

In this paper, a new area-optimized RVDT/LVDT signal conditioner structure is proposed. In the proposed signal conditioner a CORDIC block replaces two low-pass filters, three multipliers and one threshold block that are used to generate the loop error signal and to determine the sign of the signal conditioner output. By employing the proposed structure, the total area of the signal conditioner that is synthesized by using a standard CMOS technology is reduced by 44.1% and the linearity performance comparable to that of a state-of-the-art commercial signal conditioner is achieved.