DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0027] [Basic Principle]

[0028] Before a description of the basic principle of the present invention, generally known basic mathematical knowledge will be described. A cosine wave A cos(ωt) and sine wave B sin(ωt), which have the same frequency and different amplitudes, are synthesized into the following cosine wave. A and B are amplitudes, and ω is an angular frequency.

A cos(ω t )+ B sin(ω t )=( A ² ₊ B ² ) ^1/2 cos(ω t −ε) for ε=tan ⁻¹ ( B/A ) (1)

[0029] To analyze the synthesis of equation (1), it is convenient to map the cosine wave A cos(ωt) and sine wave B sin(ωt) onto a complex coordinate plane while plotting the amplitude A of the cosine wave A cos(ωt) along the real axis and the amplitude B of the sine wave B sin(ωt) along the imaginary axis.

[0030] More specifically, on the complex coordinate plane, a distance (A ² +B ² ) ^1/2 from the origin gives the amplitude of the synthetic wave, and an angle ε=tan ⁻¹ (B/A) with respect to the real axis gives the phase difference between the synthetic wave and ωt.

[0031] In addition, on the complex coordinate plane, the following relation holds

Cexp ( j ε)= C cos(ε)+ jC sin(ε) (2)

[0032] Equation (2) is an expression of a complex vector. In equation (2), j is the imaginary number unit, C is the length of the complex vector, and ε is the direction of the complex vector. Hence, to analyze the geometrical relationship on the complex coordinate plane, it is convenient to use conversion to a complex vector.

[0033] In the following description, to explain a behavior that is exhibited by an interelectrode electromotive force and the manner the present invention uses the behavior, mapping to the complex coordinate plane and geometrical analysis using a complex vector are employed.

[0034] First, an interelectrode electromotive force which is irrelevant to the flow rate (flow velocity) of a fluid to be measured per unit time will be described. As shown in FIG. 1 , an electromagnetic flowmeter comprises a measuring pipe 1 through which a fluid to be measured flows and a pair of electrodes 2 a and 2 b which oppose each other in the measuring pipe 1 to be perpendicular to both the magnetic field applied to the fluid to be measured and an axis PAX of the measuring pipe 1 and also come into contact with the fluid to be measured. The electrodes 2 a and 2 b detect an electromotive force generated by the magnetic field and the flow of the fluid to be measured.

[0035] The electromagnetic flowmeter also comprises a first exciting coil 3 a and second exciting coil 3 b . In the electromagnetic flowmeter, a plane PLN which includes the electrodes 2 a and 2 b and is perpendicular to the direction of the measuring pipe axis PAX is defined as a boundary in the measuring pipe 1 . At this time, the first exciting coil 3 a and second exciting coil 3 b apply asymmetrical magnetic fields to the fluid to be measured on both sides of the plane PLN, i.e., the boundary in the measuring pipe 1 .

[0036] Of the magnetic field generated from the first exciting coil 3 a , a magnetic field component (magnetic flux density) B 1 which is perpendicular to both an electrode axis EAX that connects the electrodes 2 a and 2 b and the measuring pipe axis PAX on the electrode axis EAX, and of the magnetic field generated from the second exciting coil 3 b , a magnetic field component (magnetic flux density) B 2 which is perpendicular to both the electrode axis EAX and the measuring pipe axis PAX on the electrode axis EAX are given by

B 1 = b 1 cos(ω0 t−θ 1) (3)

B 2 = b 2 cos(ω0 t−θ 2) (4)

[0037] In equations (3) and (4), b1 and b2 are the amplitudes, ω0 is the angular frequency, and θ1 and θ2 are the phase differences (phase delays) from ω0t. The magnetic flux density B 1 will be referred to as the magnetic field B 1 , and the magnetic flux density B 2 will be referred to as the magnetic field B 2 .

[0038] An electromotive force caused by a change in magnetic field is obtained by a time differential dB/dt of the magnetic field. The magnetic fields B 1 and B 2 generated from the first exciting coil 3 a and second exciting coil 3 b are differentiated as follows.

dB 1 / dt=−b 1ω0 sin(ω0 t−θ 1) (5)

dB 2 / dt=−b 2ω0 sin(ω0 t−θ 2) (6)

[0039] When the flow rate of the fluid to be measured is 0, eddy currents generated by the magnetic fields B 1 and B 2 contain only components generated by a change in magnetic fields. An eddy current Ia by the magnetic field B 1 and eddy current Ib by the magnetic field B 2 have directions as shown in FIG. 2 .

[0040] Hence, in the plane that includes the electrode axis EAX and measuring pipe axis PAX, an interelectrode electromotive force Ea that is generated by a change in magnetic field B 1 and is irrelevant to the flow rate (flow velocity) and an interelectrode electromotive force Eb that is generated by a change in magnetic field B 2 and is irrelevant to the flow rate (flow velocity) have opposite directions, as shown in FIG. 2 .

[0041] At this time, a total interelectrode electromotive force E obtained by adding the interelectrode electromotive forces Ea and Be corresponds to a value obtained by calculating the difference between the time differentials dB 1 /dt and dB 2 /dt of the magnetic fields and multiplying the difference by a coefficient k (a complex number related to the conductivity and dielectric constant of the fluid to be measured and the structure of the measuring pipe 1 ).

E=k{−b 2ω0 sin(ω0 t−θ 2)+ b 1ω0 sin(ω0 t−θ 1)} (7)

[0042] Equation (7) can be rewritten to 1 $\begin{array}{cc}E=-k\mathrm{b2}\omega 0\sin \left(\omega 0t\right)\cos \left(-{\theta }_2\right)-k\mathrm{b2}\omega 0\cos \left(\omega 0t\right)\sin \left(-{\theta }_2\right)+k\mathrm{b1}\omega 0\sin \left(\omega 0t\right)\cos \left(-{\theta }_1\right)+k\mathrm{b1}\omega 0\cos \left(\omega 0t\right)\sin \left(-{\theta }_1\right)=\left\{-\mathrm{b2}\sin \left(-{\theta }_2\right)+\mathrm{b1}\sin \left(-{\theta }_1\right)\right\}\omega 0k\cos \left(\omega 0t\right)+\left\{-\mathrm{b2}\cos \left(-{\theta }_2\right)+\mathrm{b1}\cos \left(-{\theta }_1\right)\right\}\omega 0k\sin \left(\omega 0t\right)& \left(8\right)\end{array}$

[0043] When equation (8) is mapped onto a complex coordinate plane based on ω0t, a real axis component Ex and imaginary axis component Ey are given by

Ex={−b 2 sin(−θ2)+ b 1 sin(−θ1)}ω0 k (9)

Ey={−b 2 cos(−θ2)+ b 1 cos(−θ1)}ω0 k (10)

[0044] Ex and Ey in equations (9) and (10) are rewritten to 2 $\begin{array}{cc}\mathrm{Ex}=\left\{-\mathrm{b2}\sin \left(-{\theta }_2\right)+\mathrm{b1}\sin \left(-{\theta }_1\right)\right\}\omega 0k=\left\{-\mathrm{b2}\cos \left(\pi /2+{\theta }_2\right)+\mathrm{b1}\cos \left(\pi /2+{\theta }_1\right)\right\}\omega 0k=\left\{\mathrm{b2}\cos \left(-\pi /2+{\theta }_2\right)+\mathrm{b1}\cos \left(\pi /2+{\theta }_1\right)\right\}\omega 0k& \left(11\right)\\ \mathrm{Ey}=\left\{-\mathrm{b2}\cos \left(-{\theta }_2\right)+\mathrm{b1}\cos \left(-{\theta }_1\right)\right\}\omega 0k=\left\{-\mathrm{b2}\sin \left(\pi /2+{\theta }_2\right)+\mathrm{b1}\sin \left(\pi /2+{\theta }_1\right)\right\}\omega 0k=\left\{\mathrm{b2}\sin \left(-\pi /2+{\theta }_2\right)+\mathrm{b1}\sin \left(\pi /2+{\theta }_1\right)\right\}\omega 0k& \left(12\right)\end{array}$

[0045] to obtain a complex vector Ec given by 3 $\begin{array}{cc}\mathrm{Ec}=\mathrm{Ex}+\mathrm{jEy}=\left\{\mathrm{b2}\cos \left(-\pi /2+{\theta }_2\right)+\mathrm{b1}\cos \left(\pi /2+{\theta }_1\right)\right\}\omega 0k+j\left\{\mathrm{b2}\sin \left(-\pi /2+{\theta }_2\right)+\mathrm{b1}\sin \left(\pi /2+{\theta }_1\right)\right\}\omega 0k=\left\{\mathrm{b1}\cos \left(\pi /2+{\theta }_1\right)+j\mathrm{b1}\sin \left(\pi /2+{\theta }_1\right)\right\}\omega 0k+\left\{\mathrm{b2}\cos \left(-\pi /2+{\theta }_2\right)+j\mathrm{b2}\sin \left(-\pi /2+{\theta }_2\right)\right\}\omega 0k=\mathrm{b1}\omega 0k\exp \left\{j\left(\pi /2+{\theta }_1\right)\right\}+\mathrm{b2}\omega 0k\exp \left\{j\left(-\pi /2+{\theta }_2\right)\right\}& \left(13\right)\end{array}$

[0046] The above-described coefficient k can be converted into a complex vector given by 4 $\begin{array}{cc}k=\mathrm{rk}\cos \left({\theta }_{00}\right)+\mathrm{jrk}\sin \left({\theta }_{00}\right)=\mathrm{rk}\exp \left(j{\theta }_{00}\right)& \left(14\right)\end{array}$

[0047] In equation (14), rk is a proportional coefficient, and θ00 is the angle of the vector k with respect to the real axis. The angle θ00 changes in accordance with a delay of the magnetic field with respect to the exciting current or a change in conductivity of the fluid. The change in angle θ00 is flow rate measurement error.

[0048] When equation (14) is substituted into equation (13), the interelectrode electromotive force Ec (an interelectrode electromotive force which is caused only by a time-rate change in magnetic field and is irrelevant to the flow velocity) converted into the complex vector is given by 5 $\begin{array}{cc}\mathrm{Ec}=\mathrm{b1}\omega 0k\exp \left(j\left(\pi /2+{\theta }_1\right)\right\}+\mathrm{b2}\omega 0k\exp \left(j\left(-\pi /2+{\theta }_2\right)\right\}=\mathrm{b1}\omega 0rk\exp \left(j\left(\pi /2+{\theta }_1+{\theta }_{00}\right)\right\}+\mathrm{b2}\omega 0rk\exp \left(j\left(-\pi /2+{\theta }_2+{\theta }_{00}\right)\right\}& \left(15\right)\end{array}$

[0049] In equation (15), b1ω0rkexp{j(π/2+θ1+θ00)} is a complex vector whose length is b1ω0rk and angle from the real axis is π/2+θ1+θ00, and b2ω0rkexp{j(−π/2+θ2+θ00)} is a complex vector whose length is b2ω0rk and angle from the real axis is −π/2+θ2+θ00.

[0050] The interelectrode electromotive force caused by the flow rate (flow velocity) of the fluid to be measured will be described next. When the flow velocity of the fluid to be measured is V (V≠0), eddy currents by the magnetic fields B 1 and B 2 respectively contain components V×B 1 and V×B 2 caused by the flow velocity in addition to the eddy current components Ia and Ib when the flow velocity is 0. For this reason, an eddy current Ia′ by the magnetic field B 1 and an eddy current Ib′ by the magnetic field B 2 have directions as shown in FIG. 3 .

[0051] Hence, an interelectrode electromotive force Ea′ generated by the flow velocity V of the fluid to be measured and the magnetic field B 1 and an interelectrode electromotive force Eb′ generated by the flow velocity V and the magnetic field B 2 have the same direction.

[0052] At this time, a total interelectrode electromotive force Ev obtain by adding the interelectrode electromotive forces Ea′ and Eb′ generated by the flow velocity corresponds to the sum of a value obtained by multiplying the magnetic field B 1 by a coefficient kv (a complex number related to the flow velocity V, the conductivity and dielectric constant of the fluid to be measured, and the structure of the measuring pipe 1 ) and a value obtained by multiplying the magnetic field B 2 by the coefficient kv.

Ev=kv{b 1 cos(ω0 t−θ 1)+ b 2 cos(ω0 t−θ 2)} (16)

[0053] When the term of sin and the term of cos of equation (16) are expanded, we obtain 6 $\begin{array}{cc}\mathrm{Ev}=kvb1\cos \left(\omega 0t\right)\cos \left(-{\theta }_1\right)-kvb1\sin \left(\omega 0t\right)\sin \left(-{\theta }_1\right)+kvb2\cos \left(\omega 0t\right)\cos \left(-{\theta }_2\right)-kvb2\sin \left(\omega 0t\right)\sin \left(-{\theta }_2\right)=\left\{b1\cos \left(-{\theta }_1\right)+\mathrm{b2}\cos \left(-{\theta }_2\right)\right\}kv\cos \left(\omega 0t\right)+\left\{-b1\sin \left(-{\theta }_1\right)-\mathrm{b2}\sin \left(-{\theta }_2\right)\right\}kv\sin \left(\omega 0t\right)& \left(17\right)\end{array}$

[0054] When equation (17) is mapped onto the complex coordinate plane based on ω0t, a real axis component Evx and imaginary axis component Evy are given by

Evx={b 1 cos(−θ1)+ b 2 cos(−θ2)} kv (18)

Evy={−b 1 sin(−θ1)− b 2 sin(−θ2)} kv (19)

[0055] Equations (18) and (19) are transformed into a complex vector Evc. 7 $\begin{array}{cc}\mathrm{Evx}=\left\{b1\cos \left(-{\theta }_1\right)+\mathrm{b2}\cos \left(-{\theta }_2\right)\right\}\mathrm{kv}=\left\{b1\cos \left({\theta }_1\right)+\mathrm{b2}\cos \left({\theta }_2\right)\right\}\mathrm{kv}& \left(20\right)\\ \mathrm{Evy}=\left\{-b1\sin \left(-{\theta }_1\right)-\mathrm{b2}\sin \left(-{\theta }_2\right)\right\}\mathrm{kv}=\left\{b1\sin \left({\theta }_1\right)+\mathrm{b2}\sin \left({\theta }_2\right)\right\}\mathrm{kv}& \left(21\right)\\ \mathrm{Evc}=\mathrm{Evx}+\mathrm{jEvy}=\left\{\mathrm{b1}\cos \left({\theta }_1\right)+\mathrm{b2}\cos \left({\theta }_2\right)\right\}\mathrm{kv}+j\left\{b1\sin \left({\theta }_1\right)+\mathrm{b2}\sin \left({\theta }_2\right)\right\}\mathrm{kv}=\left\{\mathrm{b1}\cos \left({\theta }_1\right)+jb1\sin \left({\theta }_1\right)\right\}kv+\left\{\mathrm{b2}\cos \left({\theta }_2\right)+jb2\sin \left({\theta }_2\right)\right\}kv=\mathrm{b1k}v\exp \left(j{\theta }_1\right)+\mathrm{b2kv}\exp \left(j{\theta }_2\right)& \left(22\right)\end{array}$

[0056] The above-described coefficient kv is transformed to a complex vector. 8 $\begin{array}{cc}kv=rkv\cos \left({\theta }_{01}\right)+jrkv\sin \left({\theta }_{01}\right)\}=rkv\exp \left(j{\theta }_{01}\right)& \left(23\right)\end{array}$

[0057] In equation (23), rkv is a proportional coefficient, θ01 is the angle of the vector kv with respect to the real axis. In this case, rkv corresponds to a value obtained by multiplying the proportional coefficient rk (equation (14)) by the flow velocity V and a proportional coefficient γ, so v=Vγ. That is,

rkv=rkVγ (24)

[0058] When equation (23) is substituted into equation (22), the interelectrode electromotive force Evc converted into complex coordinates is obtained as 9 $\begin{array}{cc}\mathrm{Evc}=\mathrm{b1k}v\exp \left(j{\theta }_1\right)+\mathrm{b2kv}\exp \left(j{\theta }_2\right)=\mathrm{b1rk}v\exp \{\left(j\left({\theta }_1+{\theta }_{01}\right)\right\}+\mathrm{b2rk}v\exp \{\left(j\left({\theta }_2+{\theta }_{01}\right)\right\}& \left(25\right)\end{array}$

[0059] In equation (25), b1rkvexp{j(θ1+θ01)} is a complex vector whose length is b1rkv and angle from the real axis is θ1+θ01, and b2rkvexp{j(θ2+θ01)} is a complex vector whose length is b2rkv and angle from the real axis is θ2+θ01.

[0060] From equations (15) and (25), a total interelectrode electromotive force Eac obtained by adding the interelectrode electromotive force Ec generated by a time-rate change in magnetic field and the interelectrode electromotive force Evc generated by the flow velocity of the fluid is given by 10 $\begin{array}{cc}\mathrm{Eac}=\mathrm{Ec}+\mathrm{Evc}=\mathrm{b1}\omega 0rk\exp \left\{j\left(\pi /2+{\theta }_1+{\theta }_{00}\right)\right\}+\mathrm{b2}\mathrm{\omega 0}\mathrm{rk}\exp \left\{j\left(-\pi /2+{\theta }_2+{\theta }_{00}\right)\right\}+\mathrm{b1rkv}\exp \left\{j\left({\theta }_1+{\theta }_{01}\right)\right\}+\mathrm{b2rk}v\exp \left\{j\left({\theta }_2+{\theta }_{01}\right)\right\}& \left(26\right)\end{array}$

[0061] As is apparent from equation (26), the interelectrode electromotive force Eac is described by the four complex vectors b1ω0rkexp{j(π/2+θ1+θ00)}, b2ω0rkexp{j(−π/2+θ2+θ00)}, b1rkvexp{j(θ1+θ01)}, and b2rkvexp{j(θ2+θ01)}.

[0062] The length of a synthetic vector obtained by synthesizing the four complex vectors represents the amplitude of the output (interelectrode electromotive force Eac), and an angle φ of the synthetic vector represents the phase difference (phase delay) of the interelectrode electromotive force Eac from the phase ω 0t of the input (exciting current).

[0063] In the present invention, the carrier of an angular frequency ω0 is amplitude-, phase-, or frequency-modulated by the modulated wave of an angular frequency ω2 to obtain an exciting current. The exciting current is supplied to the first and second exciting coils 3 a and 3 b to apply asymmetrical magnetic fields to the fluid to be measured on both sides of the plane PLN, i.e., the boundary in the measuring pipe 1 .

[0064] Accordingly, a plurality of frequency components ω0, ω0+ξω2, and ω0−ξω2 (ξ is an integer (ξ≧1); for amplitude modulation, only ξ=1) are generated in the interelectrode electromotive force Eac. From two of these frequency components, an asymmetrical exciting characteristic parameter (amplitude ratio or phase difference) is obtained, which depends on the flow rate of the fluid and does not depend on the variation in delay (θ00) of the magnetic field with respect to the exciting current or the shift of the amplitude of the magnetic field. On the basis of the asymmetrical exciting characteristic parameter, a flow rate measurement error due to the variation in delay of the magnetic field with respect to the exciting current or the shift of the amplitude of the magnetic field is automatically corrected. This is the basic technical idea of the present invention.

[0065] Accordingly, in-phase component noise is removed so that the rectangular wave exciting method need not be used, and the sine wave exciting method can be used.

[0066] [First Embodiment]

[0067] An embodiment of the present invention will be described below in detail. FIG. 4 shows the arrangement of an electromagnetic flowmeter according to the first embodiment of the present invention. The same reference numerals as in FIG. 1 denote the same components in FIG. 4 .

[0068] The electromagnetic flowmeter according to this embodiment comprises a measuring pipe 1 , electrodes 2 a and 2 b , first and second exciting coils 3 a and 3 b , and a power supply section 4 which supplies a first exciting current to the first exciting coil 3 a and a second exciting current to the second exciting coil 3 b.

[0069] The electromagnetic flowmeter also comprises a signal conversion section 5 and flow rate output section 6 . The signal conversion section 5 obtains an amplitude by separating a component having an angular frequency ω0 from an electromotive force detected by the electrodes 2 a and 2 b , obtains an amplitude by separating a component having a sum frequency (ω0+ω2) or a difference frequency (ω0−ω2) of the angular frequency ω0 and an angular frequency ω2 from the electromotive force, and obtains the ratio of these amplitudes. The flow rate output section 6 calculates the flow rate of a fluid to be measured on the basis of the amplitude ratio obtained by the signal conversion section 5 .

[0070] The first exciting coil 3 a is arranged downstream of a plane PLN at a position separated from it by an offset distance d1. The second exciting coil 3 b is arranged upstream of the plane PLN at a position separated from it by an offset distance d2. That is, the second exciting coil 3 b is arranged on the opposite side of the first exciting coil 3 a with respect to the plane PLN

[0071] The power supply section 4 supplies a first sine wave exciting current having the first angular frequency ω0 to the first exciting coil 3 a . In this embodiment, b1=b and θ1=0 in equation (3). Of the magnetic field generated from the first exciting coil 3 a when the first exciting current is supplied from the power supply section 4 , a magnetic field component B 1 that is perpendicular to both an electrode axis EAX and a measuring pipe axis PAX on the electrode axis EAX is given by

B 1 = b cos(ω0 t ) (27)

[0072] The power supply section 4 also supplies a second exciting current to the second exciting coil 3 b . The second exciting current is obtained by amplitude-modulating a sine wave carrier having the same angular frequency ω0 as that of the carrier component of the first exciting current and a predetermined phase difference θ2 by using a modulated sine wave having the second angular frequency ω2.

[0073] Of the magnetic field generated from the second exciting coil 3 b when the second exciting current is supplied from the power supply section 4 , an amplitude b2 of a magnetic field component B 2 that is perpendicular to both the electrode axis EAX and the measuring pipe axis PAX on the electrode axis EAX is given by

b 2= b{ 1+ m _a cos(ω2 t )} (28)

[0074] In equation ( 28 ), m _a is an amplitude modulation index. From equations (4) and (28), the magnetic field B 2 is given by

B 2 = b{ 1+ m _a cos(ω2 t )} cos(ω0 t−θ 2) (29)

[0075] In equation (26), b1=b, θ1=0, and θ01=θ00. When the magnetic fields B 1 and B 2 are given by equations (27) and (29), we obtain 11 $\begin{array}{cc}\mathrm{Eac}=\mathrm{Ec}+\mathrm{Evc}=b\mathrm{\omega 0}rk\exp \left\{j\left(\pi /2+\mathrm{\theta 00}\right)\right\}+b\left\{1+m_a\cos (\mathrm{\omega 2}t)\right\}\mathrm{\omega 0}rk\exp \left\{j\left(-\pi /2+\mathrm{\theta 2}+\mathrm{\theta 00}\right)\right\}+brkv\exp \left\{j\left(\mathrm{\theta 00}\right)\right\}+b\left\{1+m_a\cos (\mathrm{\omega 2}t)\right\}rkv\exp \left\{j\left(\mathrm{\theta 2}+\mathrm{\theta 00}\right)\right\}=b\omega 0rk\exp \left\{j\left(\pi /2+\mathrm{\theta 00}\right)\right\}+b\mathrm{\omega 0}rk\exp \left\{j\left(-\pi /2+\mathrm{\theta 2}+\mathrm{\theta 00}\right)\right\}+brkv\exp \left\{j\left(\mathrm{\theta 00}\right)\right\}+brkv\exp \left\{j\left(\mathrm{\theta 2}+\mathrm{\theta 00}\right)\right\}+m_a\cos (\mathrm{\omega 2}t)b\mathrm{\omega 0}rk\exp \left\{j\left(-\pi /2+\mathrm{\theta 2}+\mathrm{\theta 00}\right)\right\}+m_a\cos (\mathrm{\omega 2}t)brkv\exp \left\{j\left(\mathrm{\theta 2}+\mathrm{\theta 00}\right)\right\}& \left(30\right)\end{array}$

[0076] Four vectors on the right-hand side of equation (30), i.e., bω0rkexp{j(−π/2+θ00)} as the first term, bω0rkexp{j(−π/2+θ2+θ00)} as the second term, brkvexp{j(θ00)} as the third term, and brkvexp{j(θ2+θ00)} as the fourth term correspond to fundamental vectors obtained when no amplitude modulation is done.

[0077] The vector as the fifth term on the right-hand side of equation (30), i.e., m _a cos(ω2t)bωrkexp{j(−π/2+θ2+θ00)} can be rewritten to bω0rkm _a cos {ω0t−(−π/2+θ2+θ00)} cos(ω2t) as time expression. This time expression can further be rewritten to 12 $\begin{array}{cc}b\omega 0rkm_a\cos \left\{\omega 0t-\left(-\pi /2+\mathrm{\theta 2}+\mathrm{\theta 00}\right)\right\}\cos (\mathrm{\omega 2}t)=\left(1/2\right)b\mathrm{\omega 0}{\mathrm{rkm}}_a\left[\cos \left\{\omega 0t-\left(-\pi /2+\mathrm{\theta 2}+\mathrm{\theta 00}\right)+\mathrm{\omega 2}t\right\}+\cos \left\{\omega 0t-\left(-\pi /2+\mathrm{\theta 2}+\mathrm{\theta 00}\right)-\mathrm{\omega 2}t\right\}\right]=\left(1/2\right)b\mathrm{\omega 0}{\mathrm{rkm}}_a\cos \left\{\left(\mathrm{\omega 0}+\mathrm{\omega 2}\right)t-\left(-\pi /2+\mathrm{\theta 2}+\mathrm{\theta 00}\right)\right\}+\left(1/2\right)b\mathrm{\omega 0}{\mathrm{rkm}}_a\cos \left\{\left(\mathrm{\omega 0}+\mathrm{\omega 2}\right)t-\left(-\pi /2+\mathrm{\theta 2}+\mathrm{\theta 00}\right)\right\}& \left(31\right)\end{array}$

[0078] As is apparent from equation (31), the fifth term on the right-hand side of equation (30) forms a vector (1/2)bω0rkm _a exp{j(−π/2+θ2+θ00)} on each of a complex plane based on the angular frequency (ω0+ω2) and a complex plane based on the angular frequency (ω0−ω2).

[0079] The vector as the sixth term on the right-hand side of equation (30), i.e., m _a cos(ω2t)brkvexp{j(θ2+θ00)} can be rewritten to brkvm _a cos {ω0t−ω2+θ00)} cos(ω2t) as time expression. This time expression can further be rewritten to 13 $\begin{array}{c}{\mathrm{brkvm}}_a\cos \left\{\mathrm{\omega 0}t-\left(\mathrm{\theta 2}+\mathrm{\theta 00}\right)\right\}\cos \left(\mathrm{\omega 2}t\right)=\left(1/2\right){\mathrm{brkvm}}_a\left[\cos \left\{\mathrm{\omega 0}t-\left(\mathrm{\theta 2}+\mathrm{\theta 00}\right)+\mathrm{\omega 2}t\right\}+\cos \left\{\mathrm{\omega 0}t-\left(\mathrm{\theta 2}+\mathrm{\theta 00}\right)-\mathrm{\omega 2}t\right\}\right]=\left(1/2\right){\mathrm{brkvm}}_a\cos \{\left(\mathrm{\omega 0}+\mathrm{\omega 2}\right)\end{array}$