REVIEW OF SCIENTIFIC INSTRUMENTS 86, 024501 (2015)
Constant-current control method of multi-function electromagnetic transmitter Kaichang Xue,1 Fengdao Zhou,1,2 Shuang Wang,1 and Jun Lin1,2,a)
1
College of Instrumentation and Electrical Engineering, Jilin University, Changchun 130021, China Key Laboratory for Geo-Exploration and Instrumentation of Ministry of Education, Jilin University, Changchun 130021, China 2
(Received 24 June 2014; accepted 4 February 2015; published online 24 February 2015) Based on the requirements of controlled source audio-frequency magnetotelluric, DC resistivity, and induced polarization, a constant-current control method is proposed. Using the required current waveforms in prospecting as a standard, the causes of current waveform distortion and current waveform distortion’s effects on prospecting are analyzed. A cascaded topology is adopted to achieve 40 kW constant-current transmitter. The responsive speed and precision are analyzed. According to the power circuit of the transmitting system, the circuit structure of the pulse width modulation (PWM) constant-current controller is designed. After establishing the power circuit model of the transmitting system and the PWM constant-current controller model, analyzing the influence of ripple current, and designing an open-loop transfer function according to the amplitude-frequency characteristic curves, the parameters of the PWM constant-current controller are determined. The open-loop transfer function indicates that the loop gain is no less than 28 dB below 160 Hz, which assures the responsive speed of the transmitting system; the phase margin is 45◦, which assures the stabilization of the transmitting system. Experimental results verify that the proposed constant-current control method can keep the control error below 4% and can effectively suppress load change caused by the capacitance of earth load. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4908191]
I. INTRODUCTION
Electromagnetic prospecting methods are the methods that obtain underground information by using electromagnetic theory with an excitation source. An excitation source includes a natural source and an artificial source. Since power is controllable and the induction signal is stronger, the artificial source provided by the electromagnetic transmitter is widely used in electromagnetic prospecting. To improve mineral exploration’s depth and precision, it is necessary to improve the transmitter’s power and precision of constant-current control, and it is beneficial for practical application.1–3 The higher the transmitter’s power, the deeper the effective prospecting depth will be. Meanwhile, the higher the precision of the constant-current control, the higher the prospecting accuracy will be. To improve the power of the transmitter, it is necessary to improve the rated power of the power components. However, with the improvement of the rated power of the power components, the operating frequency decreases, the responsive speed decreases, and then the precision of the transmitter’s constant-current control decreases. Therefore, there are certain constraints between the transmitter’s power and its control precision. The transmitter, developed by Central South University of China and used for widearea electromagnetic sounding, achieves the output power of 200 kW by using a constant-voltage generator and rectifying diodes to acquire the fixed bus voltage.4 However, it cannot achieve the function of constant-current control for open-loop a)Author to whom correspondence should be addressed. Electronic mail:
lin_jun@ jlu.edu.cn. 0034-6748/2015/86(2)/024501/10/$30.00
control. The Institute of Geology and Geophysics under the Chinese Academy of Sciences has developed the transmitter for the Surface Electromagnetic Prospecting (SEP) system. The transmitter achieves constant-voltage control by adjusting the generator excitation.5 However, it has not been shown to have the function of constant-current control. The Institute of Geophysical and Geochemical Exploration under the Chinese Academy of Geological Sciences has developed a DEM-T70 transmitter, which is part of the array and high-power electromagnetic prospecting system.6,7 The DEM-T70 transmitter achieves the output power of 70 kW by using a variablevoltage generator and rectifying diodes to acquire the variable bus voltage. It achieves constant-current control by controlling the generator excitation current. Since the responsive speed of the generator is generally slow, the response time is longer than 0.1 s. The American company, Zonge International, has developed a multi-function electromagnetic instrument, GDP32II, and the GGT-30 transmitter is part of GDP32II. The GGT-30 transmitter achieves the output power of 30 kW by using a constant-voltage generator at 400 Hz, multiple voltagerange transformers, and rectifying thyristors to adjust the bus voltage. It achieves constant-current control by controlling the rectifying thyristors’ conduction angle. The maximum outof-control time is 0.42 ms. However, the GGT-30’s multiple voltage-range transformers operate in 400 Hz, which is a low frequency and leads to heavy weight. As to Canadian Phoenix’s multi-function electromagnetic instrument V8, the TXU-30 transmitter achieves the output power of 20 kW by using the full-controlled insulated-gate bipolar transistor (IGBT) as the main power components to obtain the variable bus voltage. The TXU-30 achieves constant-current control
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with the pulse width modulation (PWM) technique, and the maximum out-of-control time is about 50 µs. PWM constantcurrent control can achieve higher precision and responsive speed than generator excitation control and controlled rectifier control. However, the output of TXU-30 is slightly lower. In addition, the TXU-30 has another two main limitations. First, it is commercial equipment; its intellectual property rights belong to Canadian Phoenix. Phoenix Company has just published that TXU-30 adopts the PWM technique for constant-current control. However, few introductions show the precision and responsive speed of constant-current control, and the process of constant-current control, which limits the technical exchanges in academia. Second, China is the most widely used area for electromagnetic transmitters, and its mainstream generators are three-phase 380 V generators, while the TXU-30 uses a three-phase 220 V generator. After three-phase 220 V rectifier, the bus voltage is 310 V, and TXU30 can adopt 600 V IGBTs. The upper limiting frequency of 600 V IGBTs can reach 40 kHz, and the corresponding maximum out-of-control time is about 25 µs. However, when the input is three-phase 380 V, the bus voltage is 540 V after rectifier, and the IGBTs’ voltage should be 1200 V. The upper limiting frequency of 1200 V IGBTs is about 20 kHz, and the corresponding maximum out-of-control time is about 50 µs. Hence, the constant-current control is more difficult when 380 V input is adopted rather than 220 V input. The distributed electromagnetic prospecting system was developed in the authors’ institution. Its multi-function transmitter adopts the new technology in the field of power electronics.8–17 The transmitter uses full-controlled components, IGBT, and relative control methods as key technology. Output power is 40 kW and the maximum out-of-control time is 56 µs. It uses three-phase 380 V generator as a power supply. The constant-current control method design will be illustrated in the following sections.
II. CHARACTERISTICS OF CURRENT WAVEFORM A. Requirements of current waveform
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FIG. 2. Equivalent load of transmitter.
100% duty square wave shown in Fig. 1(b), with a frequency ranging from 1 Hz to 10 kHz. B. Causes of current waveform distortion
Equivalent load of transmitter is shown in Fig. 2. ZL represents the equivalent load. The earth load is two holes on grand, and the distance between the two holes is 1-3 km. L L represents the inductance of the wire connecting two holes, and L L is 1-5 mH. Because the earth load includes resistance and capacitance, Fig. 2 adopts Debye model18 to represent the earth load. m = R1/(R1 + R2), and τ = (R1 + R2)CL. Generally, m < 0.3, τ > 0.1, and 10 Ω < R1 < 80 Ω. Because τ is about 0.110, the branch of CL and R2 should be considered to be below several Hz. When the transmitting frequency f is above 10 Hz, the earth load can be equivalent to R1 R2/(R1 + R2). When the transmitting frequency f is below 30 Hz, ωL L, ω = 2π f , is much less than R1, and the wire inductance can be ignored. When transmitting frequency is below several Hz, the output voltage vL and current i L are shown as Fig. 3(a), and ZL is shown as Eq. (1). When transmitting frequency is several Hz to 30 Hz, ZL can be represented by R1 R2/(R1 + R2). When transmitting frequency is above 30 Hz, the output voltage vL and current i L are shown as Fig. 3(b), and ZL is shown as Eq. (2), ZL = R1 1 − me−t /τ , (1) where t = 0 represents the moment of pulse switch and ZL = R1 R2/(R1 + R2) + jωL L,
(2)
The distributed electromagnetic prospecting system was developed by the authors’ institution and adopts DC resistivity method, induced polarization (IP) method, and controlled source audio-frequency magnetotelluric (CSAMT) method to obtain underground information. When DC resistivity and IP methods are adopted, the transmitting current waveform is 1/8 Hz, 50% duty square wave shown in Fig. 1(a). When CSAMT method is adopted, the transmitting waveform is
where ω = 2π f , f is the transmitting frequency. Therefore, when transmitting frequency is below several Hz, the output current waveform distortion is mainly caused by the resistance and capacitance of earth load, and ZL changes within a transmitting frequency, which is related to responsive speed of constant-current control. When transmitting frequency is above 30 Hz, the output current waveform distortion is mainly caused by wire inductance, ZL does not change
FIG. 1. Current waveforms. (a) 50% duty square wave. (b) 100% duty square wave.
FIG. 3. Output voltage and current waveforms. (a) Below several Hz. (b) Above 30 Hz.
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within a transmitting frequency, and ZL is just related to the precision of constant-current control. C. Impact of current waveform distortion
The current waveform distortion mainly impacts the following three aspects. (1) Due to the use of artificial sources in CSAMT method and the limitation of the distance between transmitter and receivers, the plane-wave approximation for the electromagnetic field is no longer reasonable with frequency decreasing, and the electromagnetic wave generally begins to enter the transition area and near area below n × 10 Hz. In this case, the full-area expression of the electric field is shown as Eq. (3)19 and adopted to obtain the apparent resistivity ρs. Equation (3) contains current I so that the current waveform distortion will affect the results of the near-field measurement, IdL ρs 3cos2ϕ − 2 + eik r (1 − ikr) , (3) 3 2πr where dL is the length of the electric dipole in x direction, r is the distance from the receiving point to the center of the electric dipole, E x is the value of the electric field, ϕ is the angle between r and x direction, k is the wave number of the electromagnetic wave, and I is the transmitting current. (2) The expression of apparent resistivity ρs in DC resistivity method is shown as Eq. (4). Equation (4) also involves current I. Therefore, the current waveform distortion affects the measuring accuracy of apparent resistivity, Ex =
ρs = K · ∆U/I,
(4)
where K is a coefficient for the device, ∆U is the voltage between two electrodes, and I is the transmitting current. (3) In an IP method, the expression of polarizability η s is deduced with the excitation waveform shown in Fig. 1(a). Therefore, the current waveform distortion does not accord with the prerequisite of IP measurement. From the above, the constant-current control method has practical value in the near-field measurement of CSAMT, in the measurement of apparent resistivity in DC resistivity method, and in keeping the prerequisite of IP measurement. III. DESIGN OF CONSTANT-CURRENT CONTROL
FIG. 4. Cascaded topology for constant-current control.
load are in series, the currents flowing through E1 and E2 are the same as the values of output positive current pulse and negative current pulse. When keeping the bus voltage of 1# output full-bridge a fixed voltage E1 and keeping the bus current flowing through 2# output full-bridge a fixed current I2, the values of output positive current pulse and negative current pulse can be kept to I2. It is the overall design of constantcurrent control that 1# output full-bridge adopts a constantvoltage source, and 2# output full-bridge adopts a constantcurrent source. The current I2 can be kept to the desired value by adjusting the voltage E2. This overall design not only can achieve the constant-current control but also can keep the voltages of E1 and E2 to be half of output voltage. It is easier to achieve 1000 V by two 500 V sources than to achieve a 1000 V source directly. The related hardware of voltage source E1 and current source I2 is shown in Fig. 5. Voltage source E1 is achieved by 1# AC-DC power supply and 1# constant-voltage circuit, and current source I2 is achieved by 2# AC-DC power supply and 2# constant-current circuit. 1# and 2# AC-DC power supply are the same. The AC-DC power supply converts three-phase 380 V AC to 500 V DC. The AC-DC power supply adopts a highfrequency switching power supply technology. U1 is 50 Hz rectifier. U2 is a full-bridge, which works at 18 kHz, and the output of AC-DC power supply can be adjusted by adjusting the duty cycle of U2. U3 is a high-frequency transformer, and it makes the input and output isolated. Because E1 and I2 are in a floating ground in the cascaded topology shown in Fig. 4, the isolation of input and output is necessary. U4 is high-frequency rectifier, and U5 is LC filter. 1# constant-voltage circuit and 2# constant-current circuit adopt the same topology, Buck step-down chopper, and adopt different control strategies. 1#
A. Overall design
The constant-current control adopts a cascaded topology shown in Fig. 4. ZL is the load of transmitter. VT11, VT12, VT13, and VT14 make up 1# output full-bridge, and VT21, VT22, VT23, and VT24 make up 2# output full-bridge. The bus voltages of 1# and 2# output full-bridge are E1 and E2, respectively, and the maximum values of E1 and E2 are 500 V. When VT11, VT14 and VT21, VT24 turn on, the load voltage is vL = E1 + E2, and the output is a positive current pulse. When VT12, VT13 and VT22, VT23 turn on, the load voltage is vL = −(E1 + E2), and the output is a negative current pulse. Because 1# output full-bridge, 2# output full-bridge, and the
FIG. 5. Related hardware of voltage source E 1 and current source I2.
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FIG. 7. Simplified analysis model. (a) Before t = 0. (b) t > 0.
FIG. 6. Related equivalent hardware of constant-current design.
constant-current circuit adopts constant-voltage control, and the feedback voltage is output E1. 2# constant-current circuit adopts constant-current control, and the feedback current is the current flowing through the inductance in Buck topology. When the average current ILo in Buck inductance is I2, the average current of output is also I2. Because E1 is just used for increasing the output power and the constant-current control in 2# constant-current circuit is the key, it is reasonable that constant-current design adopts the equivalent hardware in Fig. 6, in which only 2# AC-DC power, 2# constant-current circuit, and 2# output full-bridge are considered. Before analysis, several definitions are made as follows. Vi represents the equivalent output voltage of AC-DC power supply, and vd represents the voltage of switching node. i Lo and ILo represent instantaneous value and average value of the current flowing through inductance L. i o and Io represent the instantaneous value and average value of the current flowing through the bus of output full-bridge, respectively. E2 represents the bus voltage of Output full-bridge. vL represents the output voltage on ZL. i L represents instantaneous value of the output current flowing through ZL. PWM control receives the reference current value Iref from transmitter controller and keeps ILo = ILref by adjusting the duty cycle of V25; where ILref is corresponding to Iref and is the desired current flowing through L. Because i L is the aim of constant-current control, it is widely believed that PWM controller should adopt i L as feedback signal. However, i L does not flow through IGBT V25, D1, and L directly, and it is more reliable that i Lo is used as feedback signal rather than i L. The error between i Lo and i L will be analyzed in the following. B. Responsive speed and precision
The responsive speed should be analyzed when the equivalent load ZL changes within a transmitting frequency. When the corresponding transmitting frequency is below several Hz, ZL is shown as Eq. (1). A simplified analysis will be done in the following. Because the analysis result is worse than practical situation, the simplified analysis is reasonable. Figure 7 is the simplified model. ILo represents the current flowing through L, and PWM controller always keep ILo as a constant. e2 represents the bus voltage, and it is not a constant. When V21 and V24 turn on for a long time as Fig. 7(a), it is reasonable that ZL = R1, ILo = i o = −i L, and e2 = ILo R1. When V22 and V23 turn on at t = 0 as Fig. 7(b), i o = i L, and ZL is always higher than (1 − m)R1. When ZL takes (1 − m)R1, the analysis is more stringent, and Eq. (5) can be listed. i L(t) is shown as Eq. (6),
de2(t) e2(t) , e2(0) = ILo R1, (5) + i L(t),i L(t) = (1 − m) R1 dt m −t /[(1−m)R 1C] i L(t) = ILo + ILo e . (6) 1−m The second item on the right of Eq. (6) is the error. The maximum amplitude of error is decided by m, and m is decided by earth load. However, the decay time constant is (1 − m)R1C, and C can be changed. If C takes 200 µF, (1 − m)R1C = 11 ms on the worst conditions that R1 = 80 Ω and m = 0.3. The decay time constant is much less than the period corresponding to several Hz’s transmitting frequency. Therefore, the responsive speed can achieve the requirement. However, this analysis is related to a premise that PWM controller always keeps ILo as a constant. This premise needs that the PWM controller has the corresponding responsive speed, which will be considered in Secs. III C and III F. The precision should be analyzed when ωL L is not much less than earth load R. On this condition, transmitting frequency is above 30 Hz, earth load R = R1 R2/(R1 + R2), and the bus voltage of output full-bridge is a constant E2 within a transmitting frequency. Within a switching cycle, the inflow charge of capacitance C is equal to the outflow charge, namely, ILo = Io. The main waveforms are shown as Fig. 8. When vL changes from −E2 to +E2 at t 0, and t 0 = 0, Eq. (7) can be listed before t = t 4. i L(t) is shown as Eq. (8), ILo = C
T di L(t) + i L(t)R, i L( ) = −i L(0), dt 2 where T is the period, and T = 1/ f , ( ) E2 E2 − LR t i L(t) = + i L(0) − e L , R R E2 = L L
i L(0) =
E2 1 − e−RT /(2L L) · . R −1 − e−RT /(2L L)
FIG. 8. Main waveforms of transmitting system.
(7)
(8)
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i L(0) = −0.9999E2/R on the worst conditions that L L = 5 mH, R = 10 Ω, and T = 1/100 s. Therefore, i L(t) can be approximately shown as Eq. (9). ILo is shown as Eq. (10), E2 (9) 1 − 2e−Rt /L L , R T /2 T /2 1 1 = Io = i o(t)dt = i L(t)dt T/2 T/2 0 0 ( ) ( ) RT E2 4L L − 2L E2 4L L 4L L = 1+ ≈ 1− . e L− R RT RT R RT (10) i L(t) ≈
ILo
Based on Fourier series, the fundamental frequency component vLb of vL(t) is shown as Eq. (11). Therefore, the fundamental frequency component i Lb of i L(t) can be shown as Eq. (12), vLb = i Lb =
4 · π
E2 R2 + (ωL L)
4 E2 sin (ωt) , π
(11) ) ωL L sin (ωt − θ) , θ = arctan . R 2 (
(12) The relationship between ILo and |i Lb| is shown as Eq. (13) by combining Eq. (10) and Eq. (12). k( f ) is the error about frequency, |i Lb| =
4 ILo k( f ), k( f ) = π
1 1 + (2π f L L/R)
· 2
1 . 1 − 4 f L L/R (13)
k( f ) = 1.19 on the worst conditions that L L = 5 mH, R = 10 Ω, and f = 100 Hz. k( f ) cannot be neglected. However, it is easy to be calibrated. First, the voltage E2 and ILo should be read from meters when transmitting frequencies are about 30 Hz and 100 Hz. Second, L L and R should be calculated by combining Eq. (10) and the two sets of E2 and ILo. This calculation can be achieved in transmitter controller. Finally, k( f ) can be calculated by Eq. (13) in transmitter controller, and the error caused by wire inductance L L can be calibrated. Because the expression for calculating the apparent resistivity ρs in CSAMT is not related to transmitting current when transmitting frequency is above n × 10 Hz, the upper limitation transmitting frequency 100 Hz is chosen in the precision analysis.
FIG. 9. PWM constant-current controller.
fluctuation in the duty cycle of the PWM signal, so special consideration needs to be taken for the ripple current. (2) The limitation between responsive speed and switching losses. The responsive speed of constant-current control is related to whether the circuit before C in Fig. 6 can be represented as a current source ILo or not. The responsive speed is fast when the switching frequency is high; however, high switching frequency leads to great switching losses. Therefore, the switching frequency should not be set too high. The PWM constant-current controller needs to meet the requirements of the responsive speed with a certain switching frequency. The structure of the designed PWM constant-current controller is shown in Fig. 9. Regulator 1 is an active low pass filter for suppressing the ripple current ∆ILo of inductance L. Regulator 2 is a Proportional-Integral (PI) regulator with two poles and a zero and is used to meet the requirements of responsive speed and the stability of the loop-circuit. The PWM generator generates the controlling signal d for the switching component V25. Vp-p is the peak-to-peak value of the sawtooth waveform Vtri. i Lo1 represents the output of Hall current sensor which is used for inducing the current i Lo, and Iref represents the preset current signal. D. Model of the system
There are two aspects that need special consideration, when the PWM constant-current controller is designed.
Based on the hardware of the transmitter and the PWM constant-current controller, the system block diagram shown in Fig. 10 is built. The current i Lo is converted to the voltage signal i Lo1 through the Hall current sensor, and Kio is the gain of the Hall current sensor. The error signal er can be acquired by processing i Lo1 with regulator 1 and regulator 2, as shown in Fig. 9. Fc1(s) and Fc(s) are the transfer functions of regulator 1 and regulator 2. The controlling signal d can be acquired by processing the error signal er with the PWM generator. d, the
(1) The impact of inductance L’s ripple current. The value of inductance L is limited by the weight and current ripple; the current ripple decreases when the value increases, but the weight increases when the value increases, both of which are mutually restricted. Generally, the trade-off inductance value adopts the value that makes ∆ILo/ILo = 0.4 in engineering. ∆ILo is the peak-to-peak value of the inductance’s ripple current. ∆ILo can cause significant
FIG. 10. System block diagram.
C. Design of the PWM controller
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FIG. 11. Equivalent circuit for acquiring Gid(s).
DC component of d, is proportional to er. Fm is the scaling factor of the PWM generator, and Fm = 1/Vp-p. The controlling signal of d operates on the switching component V25 in Fig. 6, causing the noise voltage vn, which produces a ripple current on inductance L. The switching node voltage vd represents the composite result of the AC and DC components. Gid(s) represents the relationship between the switching voltage vd and the current i Lo. Fig. 11 shows the equivalent circuit for calculating Gid(s). Rc, as shown in Fig. 11, is the equivalent series resistance of capacitance C. The meanings of the other letters in Fig. 11 are the same as those shown in Fig. 6. Gid(s) is shown as Eq. (14), Gid(s) =
i Lo [1 + s (R + Rc) C]/[LC (R + Rc)] . (14) = R R cC+L vd s2 + s LC(R+R + (R+RRc)LC c)
Fc1(s) and Fc(s) are shown as Eq. (15) and Eq. (16), respectively. The meanings of the symbols shown in Eqs. (15) and (16) are identified in Fig. 9, Fc1(s) = ωp1 =
Kc1(1 + s/ωz1) R22 + R23 , ωz1 = , (1 + s/ωp1) R22 R23C21 R12 (R22 + R23) 1 , Kc1 = , R23C21 R11 R21
(15)
Fig. 12 shows the main waveforms for analysis of the ripple current. First, the noise component vn produces the ripple current ∆i Ln, which is the AC components of the i Lo. Second, the ∆i Ln produces the AC components of er. Third, the AC components of er lead that practical d is not the desired value. The desired value d ref is determined by the reference Iref and is corresponding to the average value of er. The average value of er is represented by dotted line in Fig. 12(c). Therefore, the practical value d = d ref + d n, where d n represents the error caused by the AC components of er. Finally, d n leads that ILo has an error. Based on Fourier series, the noise component vn of vd is shown as Eq. (18). Because the voltage of C can be considered as a constant, the ripple current ∆i Ln can be shown as Eq. (19), ) ( ∞ 2Vi sin jdπ , (18) vn = vn j cos ( jωst) , vn j = jπ j=1 ∆i Ln =
∞ ( vn j π ) cos jωst − j . jωs L 2 j=1
(19)
Because the fundamental frequency is much higher than others, the ripple current ∆i Ln can be shown as Eq. (20), approximately, ( vn1 π) ∆i Ln ≈ cos ωst − . (20) ωs L 2 The error component d n can be shown as Eq. (21) from Fig. 10. The error of ILo can be shown as Eq. (22), d n = kdd n, d n = ∆i Ln KioFmFc1(s)Fc(s),
(21)
where ωz1, ωp1, and Kc1 are zero, pole, and gain of Fc1(s), respectively;
where kd is between −1 and +1 because the intersection of vtri and er is not corresponding to a fixed angle of er,
Kc (1 + s/ωz2) 1 , ωz2 = , R s 1 + s/ωp2 32C31 1 C31 + C32 , Kc = , ωp2 = R32C31C32 R31 (C31 + C32)
ILref − ILo Vid ref/R − Vid/R −d n = = , ILref Vid ref/R d − dn
Fc(s) =
(16)
where ωz2, ωp2, and Kc are zero, high-frequency pole, and gain of Fc(s), respectively. The expression of the open-loop transfer function G(s) is as follows: G(s) = KioFc1(s)Fc(s)FmViGid(s).
(22)
where ILref is the desired value of Iref . There are two conclusions can be acquired from Eqs. (20)–(22). First, the error of ILo can be decreased by decreasing
(17)
Other confirmed parameters of the hardware are displayed in Table I. f s is the switching frequency, and ωs = 2π f s. E. Analysis of the ripple current
The ripple current flowing through L has an effect on precise of constant-current control, and it should be analyzed. TABLE I. Known parameters. Parameters Value Parameters Value
Vi 500 V C 2 ×10−3 F
Fm 0.42 V−1 Rc 0.13 Ω
Kio 0.05 V/A f s(ω s) 18 kHz(113 krad/s)
L 0.5 × 10−3 H R 5 Ω-5 kΩ
FIG. 12. Main waveforms of ripple current analysis. (a) vd. (b) i Lo. (c) PWM generator.
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the gain of Fc1(s) · F(s) at switching frequency. Second, the error is high at low duty cycle, and is small at high duty cycle. Therefore, when the output voltage is between 500 V and 1000 V, it is a reasonable method to increase the duty cycle of 2# constant-current circuit by adjusting the output voltage E1 of 1# constant-voltage circuit. When the output voltage is below 500 V, it is a reasonable method to increase the duty cycle that ZL is only connected to the output of 2# output full-bridge, and adjusting the output voltage Vi of 2# AC-DC power supply. F. Design of open-loop transfer function
Figure 13 is the schematic diagram for designing the open-loop transfer function, which can be achieved using the following steps. (1) Draw the amplitude-frequency characteristic diagram of Go(s) = KioFmViGid(s). According to the parameters in Table I, Gid(s) can be approximately simplified as 1 Kid (1 + s/ωzo) ,ωzo ≈ , 2 RC + sωpo/Q + ωpo 1 L 1 1 , Kid ≈ ,Q ≈ , ωpo ≈ LC LC R Rc C
Gid(s) =
s2
(23)
where ωzo, ωpo, Kid, and Q are zero, second-order pole, gain, and quality factor of Gid(s), respectively. Go(s) is drawn with a different load R in MATLAB by using the simplified expression of Gid(s). After analyzing, it is proven that Go(s) has little to do with load R when the angular frequency is above ωpo. The gain can be considered to be approximately 28 dB at ωpo. The amplitude-frequency characteristic of Go(s) can be expressed as shown in Fig. 13(a). (2) Determine the positions of zero and pole of Fc1(s). The amplitude-frequency characteristic of Fc1(s) is shown in Fig. 13(b). The changing rate of Fc1(s) is −20 dB/dec between the pole ωp1 and the zero ωz1. The following five aspects should be considered when determining the zero and pole of Fc1(s). (a) Fc1(s) is used for suppressing the ripple current of inductance L. To have a favorable effect
FIG. 13. Design of open-loop transfer function. (a) Go(s). (b) Fc1(s). (c) Fc(s). (d) G(s).
of suppression, the attenuation should not be less than 5 times (i.e., 14 dB) at the switching angular frequency ωs. Therefore, ωz1 should be larger than 6 times of ωp1. (b) When the distance between zero and pole is more than 10 times, the phase-frequency characteristic will deteriorate rapidly between pole and zero. Therefore, ωz1 should not exceed ωp1 for 10 times. (c) The gain in low frequency will decrease when the zero and pole of Fc1(s) are set before ωpo, which results in a low responsive speed. Therefore, the zero and pole should be set after ωpo. (d) Go(s) has a second-order pole at ωpo. Adding pole within the angular frequency ranging below 10 times of ωpo will cause the rapid deterioration of G(s)’s phase-frequency characteristic. Therefore, ωp1 should be larger than 10 times of ωpo. (e) The current ripple of i Lo is mainly at the switching angular frequency, namely, ωs = 113 krad/s. ωz1 needs to be less than ωs so that Fc1(s) can achieve a favorable effect on the ripple suppression. From the above, ωp1 = 1.5 × 104 rad/s and ωz1 = 9 × 104 rad/s are taken. (3) Determine the positions of zero and poles of Fc(s). The amplitude-frequency characteristic of Fc(s) is shown in Fig. 13(c). Since the changing rate of Go(s) is +20 dB/dec between ωzo and ωpo, the gain decreases at low frequency. Therefore, the zero of Fc(s) is set at ωpo to increase the gain below ωpo, namely, taking ωz2 = 103 rad/s. The highfrequency pole ωp2 can be used to suppress the highfrequency interference of the system. When pole is close to ωz1, it may cause a premature negative phase shift. Meanwhile, when pole is far away from ωz1 and ωs, it may cause an insufficient suppressive range of highfrequency interference. Therefore, ωp2 = 3 × 105 rad/s is taken. (4) Determine the gain of Fc1(s) and Fc(s). When G(s) is drawn with 0 dB of Fc1(s) at low frequency (ω < ωp1) and 0 dB of Fc(s) at middle frequency (ωz2 < ω < ωp2), the amplitude-frequency characteristic of G(s) is shown as in Fig. 13(d). The gain of G(s) is no less than 28 dB below 103 rad/s (i.e., 160 Hz), which makes the responsive speed meet the requirements. The crossing angular frequency ωcross = 2 × 104 rad/s. ωcross is one fifth of the switching angular frequency ωs, which keeps the loop stable to a certain extent. Therefore, it is reasonable to
FIG. 14. Bode plot of open-loop transfer function.
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TABLE II. Parameters of PWM constant-current controller.
TABLE III. Fixed offset of constant-current control.
Parameters Value (kΩ)
R 11 10
R 12 10
R 21 20
R 22 4
R 23 20
Parameters Value
C 21 3.3 nF
R 31 33 kΩ
R 32 36 kΩ
C 31 33 nF
C 32 0.1 nF
determine Kc1 and Kc when the gain of Fc1(s) in low frequency is 0 dB and the gain of Fc(s) in middle frequency is 0 dB. (5) Check the stability of the loop. According to the analysis above, Bode plot of G(s) is drawn in MATLAB, as shown Fig. 14. It can be seen from Fig. 14 that when the amplitude-frequency characteristic is 0 dB, the corresponding phase margin is 45◦, which meets the requirement of loop stability. When R takes 5 Ω, 30 Ω and 5 kΩ, the gain is higher than 28 dB below 103 rad/s (i.e., 160 Hz). Therefore, it achieves a favorable response rate with R between 5 Ω and 5 kΩ. (6) Determine the parameters of the PWM constant-current controller. According to the zeros, poles, and gains of Fc1(s) and Fc(s), the parameter values shown in Table II can be acquired.
IV. EXPERIMENTAL RESULTS
The experiment was conducted on the transmitting system shown in Figs. 4 and 5. In Fig. 6, V25 and D1 are IGBT and freewheeling diode in FF200R12KT4 module, respectively; V21 − V24 use 2MBI200U4H-170 module; the Hall current sensor uses CS050B-4 V. The switching frequency f s is 18 kHz, and the constant-current control chip is SG3525. The measured waveforms and the average current are obtained using a TDS1012B oscilloscope and its own measuring function. A. Dynamic characteristics
Figure 15 shows the waveforms of the dynamic characteristics test. Iref is a step signal shown in the under part of Fig. 15. i Lo, shown in the upper, is the corresponding current waveform when R = 11.1 Ω. It can be seen from Fig. 15 that when Iref
ILref (A) ILo (A) Error (%) ILref (A) ILo (A) Error (%)
5 5.1 +2 25 26.1 +4.4
10 10.5 +5 30 30.8 +2.7
15 15.9 +6 35 35.3 +0.8
20 21.1 +5.5 40 39.2 −2
changes from low to high, the delay time t d of the current i Lo is 0.7 ms. The rise time t r is 0.3 ms, the transition time t s is 3 ms, and the overshoot δp is 10%. The current i Lo will drop to zero quickly when Iref changes from high to low. The dynamic experimental results show that the stepsignal’s response time of the constant-current circuit is 3 ms, namely, that when the disturbance is below 330 Hz, it can achieve a favorable effect of suppression. Therefore, it is reasonable that the average current ILo is viewed as a constantcurrent source. B. Fixed offset
ILo is shown in Table III when ILref ranges from 5 to 40 A. Table III is acquired on the condition that ZL = 25 Ω, and the transmitter frequency is 11 Hz. The error ranges from −2% to +6%. Because the error is fixed, the error can be calibrated by using the data in Table III. k(ILref ), shown as Eq. (24), is used for defining the fixed offset of transmitter, k(ILref ) = ILo/ILref .
(24)
C. Load-regulation characteristic
ILo is shown in Table IV when ZL ranges from 11.1 Ω to 40 Ω. Table IV is acquired under the condition that ILref = 20 A, and the transmitter frequency is 11 Hz. The error ranges from +4.5% to +7.5%. The error includes two parts, one is the fixed error +5.5% in Table III, another is the error caused by the load. The error caused by the load ranges from −0.5% to +2%, and is related to load-regulation characteristic. D. Frequency characteristic
The frequency characteristic is measured on the conditions that L L = 4.7 mH and R = 25.6 Ω. The accurate values of L L and R are obtained by Agilent U1733C. Two sets of E2 and ILo are obtained from transmitter meters to obtain the estimated values of L L and R. E2 = 440 V and ILo = 16.2 A when transmitting frequency is 32 Hz. E2 = 475 V and ILo = 16.0 A when transmitting frequency is 128 Hz. Transmitter TABLE IV. Results of the load-regulation characteristic. Z L (Ω) ILo (A) Z L (Ω) ILo (A) FIG. 15. Waveforms of the dynamic characteristics test.
11.1 21.3 20.0 21.5
12.5 21.2 25.0 20.9
14.3 21.3 33.3 21.2
16.7 21.4 40 21.2
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FIG. 16. Waveforms of the experiments on the earth load. (a) Waveforms of constant-voltage. (b) Waveforms of constant-current. (c) Detailed waveforms of constant-voltage. (d) Detailed waveforms of constant-current.
controller figures out that L L = 5.1 mH and R = 26.7 Ω by two sets of data and Eq. (10). Then, transmitter controller can figure out k( f ) with L L = 5.1 mH, R = 26.7 Ω, and Eq. (13). Finally, ILref can be obtained by Eqs. (13) and (24), and ILref = |i Lb|/[(4/π) · k( f ) · k(ILref )], where |i Lb| is the desired fundamental frequency amplitude. Table V shows the main data of the desired |i Lb| is 40 A. |i Lb| is obtained by Fast Fourier Transfer (FTT) of i L. The error of |i Lb| ranges from −3.5% to −2.5%. k( f ) ranges from 1.00 to 1.10 when f ranges from 1 Hz to 128 Hz, which means that the error caused by frequency change can reach 10%. However, the fluctuation of |i Lb| is 1% when the calibration method proposed in Sec. III B is adopted.
TABLE V. Results of the frequency characteristic. f (Hz) k( f ) ILref (A) ILo (A) |i Lb| (A) f (Hz) k( f ) ILref (A) ILo (A) |i Lb| (A)
128 1.10 27.9 28.3 39.1 8 1.01 30.5 30.8 38.6
64 1.05 29.2 29.6 38.8 4 1.00 30.5 30.9 38.6
32 1.02 29.9 30.0 39.1 2 1.00 30.5 31.0 38.6
16 1.01 30.2 30.5 38.6 1 1.00 30.6 30.9 38.7
The precise of transmitter mainly includes fixed offset, load-regulation characteristic, and the frequency characteristic. Because the calibration proposed in this paper is easy to achieve, it is reasonable that the precise of transmitter constant-current control is evaluated by load-regulation characteristic and the frequency characteristic after calibration. Because the error caused by load-regulation characteristic ranges from −0.5% to +2%, and the error caused by frequency characteristic ranges from −3.5% to −2.5%, the control error of constant-current control can be considered to be below 4%. E. Results on earth load
The output of the transmitter is connected to the earth load. The output waveforms with a conventional constant-voltage control method are shown in Fig. 16(a). The output waveforms with a constant-current control method, as proposed in this paper, are shown in Fig. 16(b). Figs. 16(c) and 16(d) show the details of Figs. 16(a) and 16(b), respectively. In Fig. 16, i L (in upper) and vL (in under) represent the current flowing through the earth load and the voltage of the earth load, respectively. Figs. 16(a) and 16(c) show that there are severe fluctuations ∆Ip and ∆In in the output current. ∆Ip and ∆In are approximately 15% of the average peak current. The output voltage is relatively flat with the voltage fluctuation less than 3%. However, the current waveforms in Figs. 16(b) and 16(d)
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are relatively flat with the current fluctuation less than 3%. The output voltage varies with the changes of the earth load. The voltage waveform has fluctuations ∆Vp and ∆Vn, which are approximately 15% of the average peak voltage. The experimental results show that the constant-current control method proposed in this paper can effectively suppress the load impedance change caused by the capacitance of earth load. It can also keep the output current stable. It is superior to the conventional constant-voltage control method.
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speed of constant-current control can effectively suppress load change caused by the capacitance of earth load. ACKNOWLEDGMENTS
This work is partially supported by the National Major Scientific Instruments Development Projects of China (No. 2011YQ030133), the National Natural Science Foundation of China (No. 41004027), and the Cooperation Innovation Projects of Education Ministry of China (Nos. OSR-02-02, OSP-02-03).
V. CONCLUSION 1Z. G. An, Q. Y. Di, C. M. Fu, C. Xu, and B. Cheng, J. Environ. Eng. Geophys.
A cascaded topology with constant-voltage control for one stage and constant-current control for another stage can achieve 40 kW constant-current transmitter. The capacitance of earth load is the main factor for the current waveform distortion when the transmitter frequency is below several Hz. The current waveform distortion caused by the capacitance of earth load can be suppressed when the constant-current circuit has a favorable responsive speed. The control error caused by the wire inductance should be considered when transmitter frequency is above 30 Hz, and the control error caused by the wire inductance can be calibrated by measured wire inductance and load resistance. The structure of the PWM constant-current controller and its parameter values are determined according to the hardware of the transmitting system. The ripple current of the inductance can lead to control error, and the control error should be suppressed by PWM constant-current controller. The loop gain of the open-loop transfer function is larger than 28 dB below 160 Hz, which shows a favorable responsive speed of the control system. The phase margin of the open-loop transfer function is 45◦, which shows that the control system is stable. The experimental results show that the control error is below 4% when the load ZL ranges from 11.1 Ω to 40 Ω, and the transmitting frequency is below 100 Hz; the responsive
18, 43 (2013). 2G. J. Wu, X. Y. Hu, G. P. Huo, and X. C. Zhou, J. Earth Sci. 23, 757 (2012). 3S.
B. Singh, G. A. Babu, B. Veeraiah, and O. P. Pandey, J. Geol. Soc. India 74, 697 (2009). 4Q. Y. Jiang, Ph.D. thesis, Central South University, Changsha, 2010. 5Q. H. Zhen, Q. Y. Di, and H. B. Liu, Chin. J. Geophys. 56, 3751 (2013). 6F. S. Shi, Prog. Geophys. 24, 1109 (2009). 7P. R. Lin, P. Guo, F. S. Shi, C. J. Zheng, Y. Li, J. H. Li, and B. L. Xu, Acta Geosci. Sin. 31, 149 (2010). 8Y. H. Cheng, K. Chen, L. B. Bai, and J. Yang, Rev. Sci. Instrum. 85, 025110 (2014). 9S. Gautam and R. Gupta, IEEE Trans. Power Electron. 29, 1480 (2014). 10B. H. Hwang, Y. C. Jhang, J. J. Chen, and Y. S. Hwang, Microelectron. J. 42, 291 (2011). 11Y. Qiu, X. Y. Chen, and H. L. Liu, IEEE Trans. Power Electron. 25, 1537 (2010). 12Y. Qiu, H. L. Liu, and X. Y. Chen, IEEE Trans. Ind. Electron. 57, 1670 (2010). 13J. Selvaraj and N. A. Rahim, J. Renewable Sustainable Energy 1, 053102 (2009). 14B. Zeng, H. B. Xu, K. Chen, and J. G. Chen, Rev. Sci. Instrum. 84, 104705 (2013). 15S. A. Alexander and T. Manigandan, J. Renewable Sustainable Energy 6, 013128 (2014). 16A. Patel, K. V. Nagesh, T. Kolge, and D. P. Chakravarthy, Rev. Sci. Instrum. 82, 045111 (2011). 17O. D. Cortázar, D. Ganuza, J. M. Fuente, M. Zulaika, A. Pérez, and D. E. Anderson, Rev. Sci. Instrum. 84, 054706 (2013). 18C. A. Dias, J. Geophys. Res. 77, 4945 (1972). 19J. S. He, J. Cent. South Univ. 41, 1065 (2010).
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Constant-current control method of multi-function electromagnetic transmitter Kaichang Xue,1 Fengdao Zhou,1,2 Shuang Wang,1 and Jun Lin1,2,a)
1
College of Instrumentation and Electrical Engineering, Jilin University, Changchun 130021, China Key Laboratory for Geo-Exploration and Instrumentation of Ministry of Education, Jilin University, Changchun 130021, China 2
(Received 24 June 2014; accepted 4 February 2015; published online 24 February 2015) Based on the requirements of controlled source audio-frequency magnetotelluric, DC resistivity, and induced polarization, a constant-current control method is proposed. Using the required current waveforms in prospecting as a standard, the causes of current waveform distortion and current waveform distortion’s effects on prospecting are analyzed. A cascaded topology is adopted to achieve 40 kW constant-current transmitter. The responsive speed and precision are analyzed. According to the power circuit of the transmitting system, the circuit structure of the pulse width modulation (PWM) constant-current controller is designed. After establishing the power circuit model of the transmitting system and the PWM constant-current controller model, analyzing the influence of ripple current, and designing an open-loop transfer function according to the amplitude-frequency characteristic curves, the parameters of the PWM constant-current controller are determined. The open-loop transfer function indicates that the loop gain is no less than 28 dB below 160 Hz, which assures the responsive speed of the transmitting system; the phase margin is 45◦, which assures the stabilization of the transmitting system. Experimental results verify that the proposed constant-current control method can keep the control error below 4% and can effectively suppress load change caused by the capacitance of earth load. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4908191]
I. INTRODUCTION
Electromagnetic prospecting methods are the methods that obtain underground information by using electromagnetic theory with an excitation source. An excitation source includes a natural source and an artificial source. Since power is controllable and the induction signal is stronger, the artificial source provided by the electromagnetic transmitter is widely used in electromagnetic prospecting. To improve mineral exploration’s depth and precision, it is necessary to improve the transmitter’s power and precision of constant-current control, and it is beneficial for practical application.1–3 The higher the transmitter’s power, the deeper the effective prospecting depth will be. Meanwhile, the higher the precision of the constant-current control, the higher the prospecting accuracy will be. To improve the power of the transmitter, it is necessary to improve the rated power of the power components. However, with the improvement of the rated power of the power components, the operating frequency decreases, the responsive speed decreases, and then the precision of the transmitter’s constant-current control decreases. Therefore, there are certain constraints between the transmitter’s power and its control precision. The transmitter, developed by Central South University of China and used for widearea electromagnetic sounding, achieves the output power of 200 kW by using a constant-voltage generator and rectifying diodes to acquire the fixed bus voltage.4 However, it cannot achieve the function of constant-current control for open-loop a)Author to whom correspondence should be addressed. Electronic mail:
lin_jun@ jlu.edu.cn. 0034-6748/2015/86(2)/024501/10/$30.00
control. The Institute of Geology and Geophysics under the Chinese Academy of Sciences has developed the transmitter for the Surface Electromagnetic Prospecting (SEP) system. The transmitter achieves constant-voltage control by adjusting the generator excitation.5 However, it has not been shown to have the function of constant-current control. The Institute of Geophysical and Geochemical Exploration under the Chinese Academy of Geological Sciences has developed a DEM-T70 transmitter, which is part of the array and high-power electromagnetic prospecting system.6,7 The DEM-T70 transmitter achieves the output power of 70 kW by using a variablevoltage generator and rectifying diodes to acquire the variable bus voltage. It achieves constant-current control by controlling the generator excitation current. Since the responsive speed of the generator is generally slow, the response time is longer than 0.1 s. The American company, Zonge International, has developed a multi-function electromagnetic instrument, GDP32II, and the GGT-30 transmitter is part of GDP32II. The GGT-30 transmitter achieves the output power of 30 kW by using a constant-voltage generator at 400 Hz, multiple voltagerange transformers, and rectifying thyristors to adjust the bus voltage. It achieves constant-current control by controlling the rectifying thyristors’ conduction angle. The maximum outof-control time is 0.42 ms. However, the GGT-30’s multiple voltage-range transformers operate in 400 Hz, which is a low frequency and leads to heavy weight. As to Canadian Phoenix’s multi-function electromagnetic instrument V8, the TXU-30 transmitter achieves the output power of 20 kW by using the full-controlled insulated-gate bipolar transistor (IGBT) as the main power components to obtain the variable bus voltage. The TXU-30 achieves constant-current control
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with the pulse width modulation (PWM) technique, and the maximum out-of-control time is about 50 µs. PWM constantcurrent control can achieve higher precision and responsive speed than generator excitation control and controlled rectifier control. However, the output of TXU-30 is slightly lower. In addition, the TXU-30 has another two main limitations. First, it is commercial equipment; its intellectual property rights belong to Canadian Phoenix. Phoenix Company has just published that TXU-30 adopts the PWM technique for constant-current control. However, few introductions show the precision and responsive speed of constant-current control, and the process of constant-current control, which limits the technical exchanges in academia. Second, China is the most widely used area for electromagnetic transmitters, and its mainstream generators are three-phase 380 V generators, while the TXU-30 uses a three-phase 220 V generator. After three-phase 220 V rectifier, the bus voltage is 310 V, and TXU30 can adopt 600 V IGBTs. The upper limiting frequency of 600 V IGBTs can reach 40 kHz, and the corresponding maximum out-of-control time is about 25 µs. However, when the input is three-phase 380 V, the bus voltage is 540 V after rectifier, and the IGBTs’ voltage should be 1200 V. The upper limiting frequency of 1200 V IGBTs is about 20 kHz, and the corresponding maximum out-of-control time is about 50 µs. Hence, the constant-current control is more difficult when 380 V input is adopted rather than 220 V input. The distributed electromagnetic prospecting system was developed in the authors’ institution. Its multi-function transmitter adopts the new technology in the field of power electronics.8–17 The transmitter uses full-controlled components, IGBT, and relative control methods as key technology. Output power is 40 kW and the maximum out-of-control time is 56 µs. It uses three-phase 380 V generator as a power supply. The constant-current control method design will be illustrated in the following sections.
II. CHARACTERISTICS OF CURRENT WAVEFORM A. Requirements of current waveform
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FIG. 2. Equivalent load of transmitter.
100% duty square wave shown in Fig. 1(b), with a frequency ranging from 1 Hz to 10 kHz. B. Causes of current waveform distortion
Equivalent load of transmitter is shown in Fig. 2. ZL represents the equivalent load. The earth load is two holes on grand, and the distance between the two holes is 1-3 km. L L represents the inductance of the wire connecting two holes, and L L is 1-5 mH. Because the earth load includes resistance and capacitance, Fig. 2 adopts Debye model18 to represent the earth load. m = R1/(R1 + R2), and τ = (R1 + R2)CL. Generally, m < 0.3, τ > 0.1, and 10 Ω < R1 < 80 Ω. Because τ is about 0.110, the branch of CL and R2 should be considered to be below several Hz. When the transmitting frequency f is above 10 Hz, the earth load can be equivalent to R1 R2/(R1 + R2). When the transmitting frequency f is below 30 Hz, ωL L, ω = 2π f , is much less than R1, and the wire inductance can be ignored. When transmitting frequency is below several Hz, the output voltage vL and current i L are shown as Fig. 3(a), and ZL is shown as Eq. (1). When transmitting frequency is several Hz to 30 Hz, ZL can be represented by R1 R2/(R1 + R2). When transmitting frequency is above 30 Hz, the output voltage vL and current i L are shown as Fig. 3(b), and ZL is shown as Eq. (2), ZL = R1 1 − me−t /τ , (1) where t = 0 represents the moment of pulse switch and ZL = R1 R2/(R1 + R2) + jωL L,
(2)
The distributed electromagnetic prospecting system was developed by the authors’ institution and adopts DC resistivity method, induced polarization (IP) method, and controlled source audio-frequency magnetotelluric (CSAMT) method to obtain underground information. When DC resistivity and IP methods are adopted, the transmitting current waveform is 1/8 Hz, 50% duty square wave shown in Fig. 1(a). When CSAMT method is adopted, the transmitting waveform is
where ω = 2π f , f is the transmitting frequency. Therefore, when transmitting frequency is below several Hz, the output current waveform distortion is mainly caused by the resistance and capacitance of earth load, and ZL changes within a transmitting frequency, which is related to responsive speed of constant-current control. When transmitting frequency is above 30 Hz, the output current waveform distortion is mainly caused by wire inductance, ZL does not change
FIG. 1. Current waveforms. (a) 50% duty square wave. (b) 100% duty square wave.
FIG. 3. Output voltage and current waveforms. (a) Below several Hz. (b) Above 30 Hz.
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within a transmitting frequency, and ZL is just related to the precision of constant-current control. C. Impact of current waveform distortion
The current waveform distortion mainly impacts the following three aspects. (1) Due to the use of artificial sources in CSAMT method and the limitation of the distance between transmitter and receivers, the plane-wave approximation for the electromagnetic field is no longer reasonable with frequency decreasing, and the electromagnetic wave generally begins to enter the transition area and near area below n × 10 Hz. In this case, the full-area expression of the electric field is shown as Eq. (3)19 and adopted to obtain the apparent resistivity ρs. Equation (3) contains current I so that the current waveform distortion will affect the results of the near-field measurement, IdL ρs 3cos2ϕ − 2 + eik r (1 − ikr) , (3) 3 2πr where dL is the length of the electric dipole in x direction, r is the distance from the receiving point to the center of the electric dipole, E x is the value of the electric field, ϕ is the angle between r and x direction, k is the wave number of the electromagnetic wave, and I is the transmitting current. (2) The expression of apparent resistivity ρs in DC resistivity method is shown as Eq. (4). Equation (4) also involves current I. Therefore, the current waveform distortion affects the measuring accuracy of apparent resistivity, Ex =
ρs = K · ∆U/I,
(4)
where K is a coefficient for the device, ∆U is the voltage between two electrodes, and I is the transmitting current. (3) In an IP method, the expression of polarizability η s is deduced with the excitation waveform shown in Fig. 1(a). Therefore, the current waveform distortion does not accord with the prerequisite of IP measurement. From the above, the constant-current control method has practical value in the near-field measurement of CSAMT, in the measurement of apparent resistivity in DC resistivity method, and in keeping the prerequisite of IP measurement. III. DESIGN OF CONSTANT-CURRENT CONTROL
FIG. 4. Cascaded topology for constant-current control.
load are in series, the currents flowing through E1 and E2 are the same as the values of output positive current pulse and negative current pulse. When keeping the bus voltage of 1# output full-bridge a fixed voltage E1 and keeping the bus current flowing through 2# output full-bridge a fixed current I2, the values of output positive current pulse and negative current pulse can be kept to I2. It is the overall design of constantcurrent control that 1# output full-bridge adopts a constantvoltage source, and 2# output full-bridge adopts a constantcurrent source. The current I2 can be kept to the desired value by adjusting the voltage E2. This overall design not only can achieve the constant-current control but also can keep the voltages of E1 and E2 to be half of output voltage. It is easier to achieve 1000 V by two 500 V sources than to achieve a 1000 V source directly. The related hardware of voltage source E1 and current source I2 is shown in Fig. 5. Voltage source E1 is achieved by 1# AC-DC power supply and 1# constant-voltage circuit, and current source I2 is achieved by 2# AC-DC power supply and 2# constant-current circuit. 1# and 2# AC-DC power supply are the same. The AC-DC power supply converts three-phase 380 V AC to 500 V DC. The AC-DC power supply adopts a highfrequency switching power supply technology. U1 is 50 Hz rectifier. U2 is a full-bridge, which works at 18 kHz, and the output of AC-DC power supply can be adjusted by adjusting the duty cycle of U2. U3 is a high-frequency transformer, and it makes the input and output isolated. Because E1 and I2 are in a floating ground in the cascaded topology shown in Fig. 4, the isolation of input and output is necessary. U4 is high-frequency rectifier, and U5 is LC filter. 1# constant-voltage circuit and 2# constant-current circuit adopt the same topology, Buck step-down chopper, and adopt different control strategies. 1#
A. Overall design
The constant-current control adopts a cascaded topology shown in Fig. 4. ZL is the load of transmitter. VT11, VT12, VT13, and VT14 make up 1# output full-bridge, and VT21, VT22, VT23, and VT24 make up 2# output full-bridge. The bus voltages of 1# and 2# output full-bridge are E1 and E2, respectively, and the maximum values of E1 and E2 are 500 V. When VT11, VT14 and VT21, VT24 turn on, the load voltage is vL = E1 + E2, and the output is a positive current pulse. When VT12, VT13 and VT22, VT23 turn on, the load voltage is vL = −(E1 + E2), and the output is a negative current pulse. Because 1# output full-bridge, 2# output full-bridge, and the
FIG. 5. Related hardware of voltage source E 1 and current source I2.
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FIG. 7. Simplified analysis model. (a) Before t = 0. (b) t > 0.
FIG. 6. Related equivalent hardware of constant-current design.
constant-current circuit adopts constant-voltage control, and the feedback voltage is output E1. 2# constant-current circuit adopts constant-current control, and the feedback current is the current flowing through the inductance in Buck topology. When the average current ILo in Buck inductance is I2, the average current of output is also I2. Because E1 is just used for increasing the output power and the constant-current control in 2# constant-current circuit is the key, it is reasonable that constant-current design adopts the equivalent hardware in Fig. 6, in which only 2# AC-DC power, 2# constant-current circuit, and 2# output full-bridge are considered. Before analysis, several definitions are made as follows. Vi represents the equivalent output voltage of AC-DC power supply, and vd represents the voltage of switching node. i Lo and ILo represent instantaneous value and average value of the current flowing through inductance L. i o and Io represent the instantaneous value and average value of the current flowing through the bus of output full-bridge, respectively. E2 represents the bus voltage of Output full-bridge. vL represents the output voltage on ZL. i L represents instantaneous value of the output current flowing through ZL. PWM control receives the reference current value Iref from transmitter controller and keeps ILo = ILref by adjusting the duty cycle of V25; where ILref is corresponding to Iref and is the desired current flowing through L. Because i L is the aim of constant-current control, it is widely believed that PWM controller should adopt i L as feedback signal. However, i L does not flow through IGBT V25, D1, and L directly, and it is more reliable that i Lo is used as feedback signal rather than i L. The error between i Lo and i L will be analyzed in the following. B. Responsive speed and precision
The responsive speed should be analyzed when the equivalent load ZL changes within a transmitting frequency. When the corresponding transmitting frequency is below several Hz, ZL is shown as Eq. (1). A simplified analysis will be done in the following. Because the analysis result is worse than practical situation, the simplified analysis is reasonable. Figure 7 is the simplified model. ILo represents the current flowing through L, and PWM controller always keep ILo as a constant. e2 represents the bus voltage, and it is not a constant. When V21 and V24 turn on for a long time as Fig. 7(a), it is reasonable that ZL = R1, ILo = i o = −i L, and e2 = ILo R1. When V22 and V23 turn on at t = 0 as Fig. 7(b), i o = i L, and ZL is always higher than (1 − m)R1. When ZL takes (1 − m)R1, the analysis is more stringent, and Eq. (5) can be listed. i L(t) is shown as Eq. (6),
de2(t) e2(t) , e2(0) = ILo R1, (5) + i L(t),i L(t) = (1 − m) R1 dt m −t /[(1−m)R 1C] i L(t) = ILo + ILo e . (6) 1−m The second item on the right of Eq. (6) is the error. The maximum amplitude of error is decided by m, and m is decided by earth load. However, the decay time constant is (1 − m)R1C, and C can be changed. If C takes 200 µF, (1 − m)R1C = 11 ms on the worst conditions that R1 = 80 Ω and m = 0.3. The decay time constant is much less than the period corresponding to several Hz’s transmitting frequency. Therefore, the responsive speed can achieve the requirement. However, this analysis is related to a premise that PWM controller always keeps ILo as a constant. This premise needs that the PWM controller has the corresponding responsive speed, which will be considered in Secs. III C and III F. The precision should be analyzed when ωL L is not much less than earth load R. On this condition, transmitting frequency is above 30 Hz, earth load R = R1 R2/(R1 + R2), and the bus voltage of output full-bridge is a constant E2 within a transmitting frequency. Within a switching cycle, the inflow charge of capacitance C is equal to the outflow charge, namely, ILo = Io. The main waveforms are shown as Fig. 8. When vL changes from −E2 to +E2 at t 0, and t 0 = 0, Eq. (7) can be listed before t = t 4. i L(t) is shown as Eq. (8), ILo = C
T di L(t) + i L(t)R, i L( ) = −i L(0), dt 2 where T is the period, and T = 1/ f , ( ) E2 E2 − LR t i L(t) = + i L(0) − e L , R R E2 = L L
i L(0) =
E2 1 − e−RT /(2L L) · . R −1 − e−RT /(2L L)
FIG. 8. Main waveforms of transmitting system.
(7)
(8)
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i L(0) = −0.9999E2/R on the worst conditions that L L = 5 mH, R = 10 Ω, and T = 1/100 s. Therefore, i L(t) can be approximately shown as Eq. (9). ILo is shown as Eq. (10), E2 (9) 1 − 2e−Rt /L L , R T /2 T /2 1 1 = Io = i o(t)dt = i L(t)dt T/2 T/2 0 0 ( ) ( ) RT E2 4L L − 2L E2 4L L 4L L = 1+ ≈ 1− . e L− R RT RT R RT (10) i L(t) ≈
ILo
Based on Fourier series, the fundamental frequency component vLb of vL(t) is shown as Eq. (11). Therefore, the fundamental frequency component i Lb of i L(t) can be shown as Eq. (12), vLb = i Lb =
4 · π
E2 R2 + (ωL L)
4 E2 sin (ωt) , π
(11) ) ωL L sin (ωt − θ) , θ = arctan . R 2 (
(12) The relationship between ILo and |i Lb| is shown as Eq. (13) by combining Eq. (10) and Eq. (12). k( f ) is the error about frequency, |i Lb| =
4 ILo k( f ), k( f ) = π
1 1 + (2π f L L/R)
· 2
1 . 1 − 4 f L L/R (13)
k( f ) = 1.19 on the worst conditions that L L = 5 mH, R = 10 Ω, and f = 100 Hz. k( f ) cannot be neglected. However, it is easy to be calibrated. First, the voltage E2 and ILo should be read from meters when transmitting frequencies are about 30 Hz and 100 Hz. Second, L L and R should be calculated by combining Eq. (10) and the two sets of E2 and ILo. This calculation can be achieved in transmitter controller. Finally, k( f ) can be calculated by Eq. (13) in transmitter controller, and the error caused by wire inductance L L can be calibrated. Because the expression for calculating the apparent resistivity ρs in CSAMT is not related to transmitting current when transmitting frequency is above n × 10 Hz, the upper limitation transmitting frequency 100 Hz is chosen in the precision analysis.
FIG. 9. PWM constant-current controller.
fluctuation in the duty cycle of the PWM signal, so special consideration needs to be taken for the ripple current. (2) The limitation between responsive speed and switching losses. The responsive speed of constant-current control is related to whether the circuit before C in Fig. 6 can be represented as a current source ILo or not. The responsive speed is fast when the switching frequency is high; however, high switching frequency leads to great switching losses. Therefore, the switching frequency should not be set too high. The PWM constant-current controller needs to meet the requirements of the responsive speed with a certain switching frequency. The structure of the designed PWM constant-current controller is shown in Fig. 9. Regulator 1 is an active low pass filter for suppressing the ripple current ∆ILo of inductance L. Regulator 2 is a Proportional-Integral (PI) regulator with two poles and a zero and is used to meet the requirements of responsive speed and the stability of the loop-circuit. The PWM generator generates the controlling signal d for the switching component V25. Vp-p is the peak-to-peak value of the sawtooth waveform Vtri. i Lo1 represents the output of Hall current sensor which is used for inducing the current i Lo, and Iref represents the preset current signal. D. Model of the system
There are two aspects that need special consideration, when the PWM constant-current controller is designed.
Based on the hardware of the transmitter and the PWM constant-current controller, the system block diagram shown in Fig. 10 is built. The current i Lo is converted to the voltage signal i Lo1 through the Hall current sensor, and Kio is the gain of the Hall current sensor. The error signal er can be acquired by processing i Lo1 with regulator 1 and regulator 2, as shown in Fig. 9. Fc1(s) and Fc(s) are the transfer functions of regulator 1 and regulator 2. The controlling signal d can be acquired by processing the error signal er with the PWM generator. d, the
(1) The impact of inductance L’s ripple current. The value of inductance L is limited by the weight and current ripple; the current ripple decreases when the value increases, but the weight increases when the value increases, both of which are mutually restricted. Generally, the trade-off inductance value adopts the value that makes ∆ILo/ILo = 0.4 in engineering. ∆ILo is the peak-to-peak value of the inductance’s ripple current. ∆ILo can cause significant
FIG. 10. System block diagram.
C. Design of the PWM controller
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FIG. 11. Equivalent circuit for acquiring Gid(s).
DC component of d, is proportional to er. Fm is the scaling factor of the PWM generator, and Fm = 1/Vp-p. The controlling signal of d operates on the switching component V25 in Fig. 6, causing the noise voltage vn, which produces a ripple current on inductance L. The switching node voltage vd represents the composite result of the AC and DC components. Gid(s) represents the relationship between the switching voltage vd and the current i Lo. Fig. 11 shows the equivalent circuit for calculating Gid(s). Rc, as shown in Fig. 11, is the equivalent series resistance of capacitance C. The meanings of the other letters in Fig. 11 are the same as those shown in Fig. 6. Gid(s) is shown as Eq. (14), Gid(s) =
i Lo [1 + s (R + Rc) C]/[LC (R + Rc)] . (14) = R R cC+L vd s2 + s LC(R+R + (R+RRc)LC c)
Fc1(s) and Fc(s) are shown as Eq. (15) and Eq. (16), respectively. The meanings of the symbols shown in Eqs. (15) and (16) are identified in Fig. 9, Fc1(s) = ωp1 =
Kc1(1 + s/ωz1) R22 + R23 , ωz1 = , (1 + s/ωp1) R22 R23C21 R12 (R22 + R23) 1 , Kc1 = , R23C21 R11 R21
(15)
Fig. 12 shows the main waveforms for analysis of the ripple current. First, the noise component vn produces the ripple current ∆i Ln, which is the AC components of the i Lo. Second, the ∆i Ln produces the AC components of er. Third, the AC components of er lead that practical d is not the desired value. The desired value d ref is determined by the reference Iref and is corresponding to the average value of er. The average value of er is represented by dotted line in Fig. 12(c). Therefore, the practical value d = d ref + d n, where d n represents the error caused by the AC components of er. Finally, d n leads that ILo has an error. Based on Fourier series, the noise component vn of vd is shown as Eq. (18). Because the voltage of C can be considered as a constant, the ripple current ∆i Ln can be shown as Eq. (19), ) ( ∞ 2Vi sin jdπ , (18) vn = vn j cos ( jωst) , vn j = jπ j=1 ∆i Ln =
∞ ( vn j π ) cos jωst − j . jωs L 2 j=1
(19)
Because the fundamental frequency is much higher than others, the ripple current ∆i Ln can be shown as Eq. (20), approximately, ( vn1 π) ∆i Ln ≈ cos ωst − . (20) ωs L 2 The error component d n can be shown as Eq. (21) from Fig. 10. The error of ILo can be shown as Eq. (22), d n = kdd n, d n = ∆i Ln KioFmFc1(s)Fc(s),
(21)
where ωz1, ωp1, and Kc1 are zero, pole, and gain of Fc1(s), respectively;
where kd is between −1 and +1 because the intersection of vtri and er is not corresponding to a fixed angle of er,
Kc (1 + s/ωz2) 1 , ωz2 = , R s 1 + s/ωp2 32C31 1 C31 + C32 , Kc = , ωp2 = R32C31C32 R31 (C31 + C32)
ILref − ILo Vid ref/R − Vid/R −d n = = , ILref Vid ref/R d − dn
Fc(s) =
(16)
where ωz2, ωp2, and Kc are zero, high-frequency pole, and gain of Fc(s), respectively. The expression of the open-loop transfer function G(s) is as follows: G(s) = KioFc1(s)Fc(s)FmViGid(s).
(22)
where ILref is the desired value of Iref . There are two conclusions can be acquired from Eqs. (20)–(22). First, the error of ILo can be decreased by decreasing
(17)
Other confirmed parameters of the hardware are displayed in Table I. f s is the switching frequency, and ωs = 2π f s. E. Analysis of the ripple current
The ripple current flowing through L has an effect on precise of constant-current control, and it should be analyzed. TABLE I. Known parameters. Parameters Value Parameters Value
Vi 500 V C 2 ×10−3 F
Fm 0.42 V−1 Rc 0.13 Ω
Kio 0.05 V/A f s(ω s) 18 kHz(113 krad/s)
L 0.5 × 10−3 H R 5 Ω-5 kΩ
FIG. 12. Main waveforms of ripple current analysis. (a) vd. (b) i Lo. (c) PWM generator.
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the gain of Fc1(s) · F(s) at switching frequency. Second, the error is high at low duty cycle, and is small at high duty cycle. Therefore, when the output voltage is between 500 V and 1000 V, it is a reasonable method to increase the duty cycle of 2# constant-current circuit by adjusting the output voltage E1 of 1# constant-voltage circuit. When the output voltage is below 500 V, it is a reasonable method to increase the duty cycle that ZL is only connected to the output of 2# output full-bridge, and adjusting the output voltage Vi of 2# AC-DC power supply. F. Design of open-loop transfer function
Figure 13 is the schematic diagram for designing the open-loop transfer function, which can be achieved using the following steps. (1) Draw the amplitude-frequency characteristic diagram of Go(s) = KioFmViGid(s). According to the parameters in Table I, Gid(s) can be approximately simplified as 1 Kid (1 + s/ωzo) ,ωzo ≈ , 2 RC + sωpo/Q + ωpo 1 L 1 1 , Kid ≈ ,Q ≈ , ωpo ≈ LC LC R Rc C
Gid(s) =
s2
(23)
where ωzo, ωpo, Kid, and Q are zero, second-order pole, gain, and quality factor of Gid(s), respectively. Go(s) is drawn with a different load R in MATLAB by using the simplified expression of Gid(s). After analyzing, it is proven that Go(s) has little to do with load R when the angular frequency is above ωpo. The gain can be considered to be approximately 28 dB at ωpo. The amplitude-frequency characteristic of Go(s) can be expressed as shown in Fig. 13(a). (2) Determine the positions of zero and pole of Fc1(s). The amplitude-frequency characteristic of Fc1(s) is shown in Fig. 13(b). The changing rate of Fc1(s) is −20 dB/dec between the pole ωp1 and the zero ωz1. The following five aspects should be considered when determining the zero and pole of Fc1(s). (a) Fc1(s) is used for suppressing the ripple current of inductance L. To have a favorable effect
FIG. 13. Design of open-loop transfer function. (a) Go(s). (b) Fc1(s). (c) Fc(s). (d) G(s).
of suppression, the attenuation should not be less than 5 times (i.e., 14 dB) at the switching angular frequency ωs. Therefore, ωz1 should be larger than 6 times of ωp1. (b) When the distance between zero and pole is more than 10 times, the phase-frequency characteristic will deteriorate rapidly between pole and zero. Therefore, ωz1 should not exceed ωp1 for 10 times. (c) The gain in low frequency will decrease when the zero and pole of Fc1(s) are set before ωpo, which results in a low responsive speed. Therefore, the zero and pole should be set after ωpo. (d) Go(s) has a second-order pole at ωpo. Adding pole within the angular frequency ranging below 10 times of ωpo will cause the rapid deterioration of G(s)’s phase-frequency characteristic. Therefore, ωp1 should be larger than 10 times of ωpo. (e) The current ripple of i Lo is mainly at the switching angular frequency, namely, ωs = 113 krad/s. ωz1 needs to be less than ωs so that Fc1(s) can achieve a favorable effect on the ripple suppression. From the above, ωp1 = 1.5 × 104 rad/s and ωz1 = 9 × 104 rad/s are taken. (3) Determine the positions of zero and poles of Fc(s). The amplitude-frequency characteristic of Fc(s) is shown in Fig. 13(c). Since the changing rate of Go(s) is +20 dB/dec between ωzo and ωpo, the gain decreases at low frequency. Therefore, the zero of Fc(s) is set at ωpo to increase the gain below ωpo, namely, taking ωz2 = 103 rad/s. The highfrequency pole ωp2 can be used to suppress the highfrequency interference of the system. When pole is close to ωz1, it may cause a premature negative phase shift. Meanwhile, when pole is far away from ωz1 and ωs, it may cause an insufficient suppressive range of highfrequency interference. Therefore, ωp2 = 3 × 105 rad/s is taken. (4) Determine the gain of Fc1(s) and Fc(s). When G(s) is drawn with 0 dB of Fc1(s) at low frequency (ω < ωp1) and 0 dB of Fc(s) at middle frequency (ωz2 < ω < ωp2), the amplitude-frequency characteristic of G(s) is shown as in Fig. 13(d). The gain of G(s) is no less than 28 dB below 103 rad/s (i.e., 160 Hz), which makes the responsive speed meet the requirements. The crossing angular frequency ωcross = 2 × 104 rad/s. ωcross is one fifth of the switching angular frequency ωs, which keeps the loop stable to a certain extent. Therefore, it is reasonable to
FIG. 14. Bode plot of open-loop transfer function.
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TABLE II. Parameters of PWM constant-current controller.
TABLE III. Fixed offset of constant-current control.
Parameters Value (kΩ)
R 11 10
R 12 10
R 21 20
R 22 4
R 23 20
Parameters Value
C 21 3.3 nF
R 31 33 kΩ
R 32 36 kΩ
C 31 33 nF
C 32 0.1 nF
determine Kc1 and Kc when the gain of Fc1(s) in low frequency is 0 dB and the gain of Fc(s) in middle frequency is 0 dB. (5) Check the stability of the loop. According to the analysis above, Bode plot of G(s) is drawn in MATLAB, as shown Fig. 14. It can be seen from Fig. 14 that when the amplitude-frequency characteristic is 0 dB, the corresponding phase margin is 45◦, which meets the requirement of loop stability. When R takes 5 Ω, 30 Ω and 5 kΩ, the gain is higher than 28 dB below 103 rad/s (i.e., 160 Hz). Therefore, it achieves a favorable response rate with R between 5 Ω and 5 kΩ. (6) Determine the parameters of the PWM constant-current controller. According to the zeros, poles, and gains of Fc1(s) and Fc(s), the parameter values shown in Table II can be acquired.
IV. EXPERIMENTAL RESULTS
The experiment was conducted on the transmitting system shown in Figs. 4 and 5. In Fig. 6, V25 and D1 are IGBT and freewheeling diode in FF200R12KT4 module, respectively; V21 − V24 use 2MBI200U4H-170 module; the Hall current sensor uses CS050B-4 V. The switching frequency f s is 18 kHz, and the constant-current control chip is SG3525. The measured waveforms and the average current are obtained using a TDS1012B oscilloscope and its own measuring function. A. Dynamic characteristics
Figure 15 shows the waveforms of the dynamic characteristics test. Iref is a step signal shown in the under part of Fig. 15. i Lo, shown in the upper, is the corresponding current waveform when R = 11.1 Ω. It can be seen from Fig. 15 that when Iref
ILref (A) ILo (A) Error (%) ILref (A) ILo (A) Error (%)
5 5.1 +2 25 26.1 +4.4
10 10.5 +5 30 30.8 +2.7
15 15.9 +6 35 35.3 +0.8
20 21.1 +5.5 40 39.2 −2
changes from low to high, the delay time t d of the current i Lo is 0.7 ms. The rise time t r is 0.3 ms, the transition time t s is 3 ms, and the overshoot δp is 10%. The current i Lo will drop to zero quickly when Iref changes from high to low. The dynamic experimental results show that the stepsignal’s response time of the constant-current circuit is 3 ms, namely, that when the disturbance is below 330 Hz, it can achieve a favorable effect of suppression. Therefore, it is reasonable that the average current ILo is viewed as a constantcurrent source. B. Fixed offset
ILo is shown in Table III when ILref ranges from 5 to 40 A. Table III is acquired on the condition that ZL = 25 Ω, and the transmitter frequency is 11 Hz. The error ranges from −2% to +6%. Because the error is fixed, the error can be calibrated by using the data in Table III. k(ILref ), shown as Eq. (24), is used for defining the fixed offset of transmitter, k(ILref ) = ILo/ILref .
(24)
C. Load-regulation characteristic
ILo is shown in Table IV when ZL ranges from 11.1 Ω to 40 Ω. Table IV is acquired under the condition that ILref = 20 A, and the transmitter frequency is 11 Hz. The error ranges from +4.5% to +7.5%. The error includes two parts, one is the fixed error +5.5% in Table III, another is the error caused by the load. The error caused by the load ranges from −0.5% to +2%, and is related to load-regulation characteristic. D. Frequency characteristic
The frequency characteristic is measured on the conditions that L L = 4.7 mH and R = 25.6 Ω. The accurate values of L L and R are obtained by Agilent U1733C. Two sets of E2 and ILo are obtained from transmitter meters to obtain the estimated values of L L and R. E2 = 440 V and ILo = 16.2 A when transmitting frequency is 32 Hz. E2 = 475 V and ILo = 16.0 A when transmitting frequency is 128 Hz. Transmitter TABLE IV. Results of the load-regulation characteristic. Z L (Ω) ILo (A) Z L (Ω) ILo (A) FIG. 15. Waveforms of the dynamic characteristics test.
11.1 21.3 20.0 21.5
12.5 21.2 25.0 20.9
14.3 21.3 33.3 21.2
16.7 21.4 40 21.2
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FIG. 16. Waveforms of the experiments on the earth load. (a) Waveforms of constant-voltage. (b) Waveforms of constant-current. (c) Detailed waveforms of constant-voltage. (d) Detailed waveforms of constant-current.
controller figures out that L L = 5.1 mH and R = 26.7 Ω by two sets of data and Eq. (10). Then, transmitter controller can figure out k( f ) with L L = 5.1 mH, R = 26.7 Ω, and Eq. (13). Finally, ILref can be obtained by Eqs. (13) and (24), and ILref = |i Lb|/[(4/π) · k( f ) · k(ILref )], where |i Lb| is the desired fundamental frequency amplitude. Table V shows the main data of the desired |i Lb| is 40 A. |i Lb| is obtained by Fast Fourier Transfer (FTT) of i L. The error of |i Lb| ranges from −3.5% to −2.5%. k( f ) ranges from 1.00 to 1.10 when f ranges from 1 Hz to 128 Hz, which means that the error caused by frequency change can reach 10%. However, the fluctuation of |i Lb| is 1% when the calibration method proposed in Sec. III B is adopted.
TABLE V. Results of the frequency characteristic. f (Hz) k( f ) ILref (A) ILo (A) |i Lb| (A) f (Hz) k( f ) ILref (A) ILo (A) |i Lb| (A)
128 1.10 27.9 28.3 39.1 8 1.01 30.5 30.8 38.6
64 1.05 29.2 29.6 38.8 4 1.00 30.5 30.9 38.6
32 1.02 29.9 30.0 39.1 2 1.00 30.5 31.0 38.6
16 1.01 30.2 30.5 38.6 1 1.00 30.6 30.9 38.7
The precise of transmitter mainly includes fixed offset, load-regulation characteristic, and the frequency characteristic. Because the calibration proposed in this paper is easy to achieve, it is reasonable that the precise of transmitter constant-current control is evaluated by load-regulation characteristic and the frequency characteristic after calibration. Because the error caused by load-regulation characteristic ranges from −0.5% to +2%, and the error caused by frequency characteristic ranges from −3.5% to −2.5%, the control error of constant-current control can be considered to be below 4%. E. Results on earth load
The output of the transmitter is connected to the earth load. The output waveforms with a conventional constant-voltage control method are shown in Fig. 16(a). The output waveforms with a constant-current control method, as proposed in this paper, are shown in Fig. 16(b). Figs. 16(c) and 16(d) show the details of Figs. 16(a) and 16(b), respectively. In Fig. 16, i L (in upper) and vL (in under) represent the current flowing through the earth load and the voltage of the earth load, respectively. Figs. 16(a) and 16(c) show that there are severe fluctuations ∆Ip and ∆In in the output current. ∆Ip and ∆In are approximately 15% of the average peak current. The output voltage is relatively flat with the voltage fluctuation less than 3%. However, the current waveforms in Figs. 16(b) and 16(d)
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are relatively flat with the current fluctuation less than 3%. The output voltage varies with the changes of the earth load. The voltage waveform has fluctuations ∆Vp and ∆Vn, which are approximately 15% of the average peak voltage. The experimental results show that the constant-current control method proposed in this paper can effectively suppress the load impedance change caused by the capacitance of earth load. It can also keep the output current stable. It is superior to the conventional constant-voltage control method.
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speed of constant-current control can effectively suppress load change caused by the capacitance of earth load. ACKNOWLEDGMENTS
This work is partially supported by the National Major Scientific Instruments Development Projects of China (No. 2011YQ030133), the National Natural Science Foundation of China (No. 41004027), and the Cooperation Innovation Projects of Education Ministry of China (Nos. OSR-02-02, OSP-02-03).
V. CONCLUSION 1Z. G. An, Q. Y. Di, C. M. Fu, C. Xu, and B. Cheng, J. Environ. Eng. Geophys.
A cascaded topology with constant-voltage control for one stage and constant-current control for another stage can achieve 40 kW constant-current transmitter. The capacitance of earth load is the main factor for the current waveform distortion when the transmitter frequency is below several Hz. The current waveform distortion caused by the capacitance of earth load can be suppressed when the constant-current circuit has a favorable responsive speed. The control error caused by the wire inductance should be considered when transmitter frequency is above 30 Hz, and the control error caused by the wire inductance can be calibrated by measured wire inductance and load resistance. The structure of the PWM constant-current controller and its parameter values are determined according to the hardware of the transmitting system. The ripple current of the inductance can lead to control error, and the control error should be suppressed by PWM constant-current controller. The loop gain of the open-loop transfer function is larger than 28 dB below 160 Hz, which shows a favorable responsive speed of the control system. The phase margin of the open-loop transfer function is 45◦, which shows that the control system is stable. The experimental results show that the control error is below 4% when the load ZL ranges from 11.1 Ω to 40 Ω, and the transmitting frequency is below 100 Hz; the responsive
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