2292 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 60, NO. 7, JULY 2011 A Compensation Method to Measure the Mutual Inductance at Low Frequency Zhengkun Li, Zhonghua Zhang, Qing He, Yaqiong Fu, Jianting Zhao, Bing Han, Shisong Li, Jiang Lan, and Chen Li Abstract—Traditionally, the most precise way to get the mutual inductance value is by a Campbell calculable mutual inductor. Here, an approach of measuring the mutual inductance at low fre- quency with compensation method is described. A multi-channel direct digital frequency synthesis source is developed to provide stable and precise phase. The mutual inductance is traced to the value of the frequency and resistance directly. The measurement uncertainty is 0.43 ppm at present and could be improved further. Index Terms—Compensation, low frequency, multi-channel di- rect digital frequency synthesis (DDS), mutual inductance, phase accuracy. I. I NTRODUCTION N ATIONAL Institute of Metrology (NIM) has proposed a “Joule Balance” method to measure the Planck constant h [1], [2], and the mutual inductance measurement is a key point for this approach. Traditional method to get the mutual inductance value falls into two methods. One is based on Campbell calculable mutual inductor [3], [4], and the other is the bridge method, such as Campbell Bridge, Foster Bridge, etc. [3]. A. Campbell provided an idea to establish a standard of mutual inductance with calculable coils in 1907 as shown in Fig. 1. The primary of the Campbell mutual inductor consists of two equivalent coils in series with the secondary between them as shown in Fig. 2, all three being coaxial. The advantage of this arrangement is that with proper di- mension and distance between primary and secondary, the secondary coil can be placed in such a position that the mean circumference field of the secondary coil due to the primary coils is zero. Thus, the mutual inductance per turn will be practically constant over the whole section of a secondary coil whose axial and radial depths are both small; and the secondary Manuscript received June 11, 2010; revised November 4, 2010; accepted November 25, 2010. Date of publication January 6, 2011; date of current ver- sion June 8, 2011. This work was supported by the China Ministry of Science and Technology 2006BAF06B01 and National Natural Science Foundation of China 50677065. The Associate Editor coordinating the review process for this paper was Dr. Wan-Seop Kim. Z. Li is with the National Institute of Metrology, Beijing 100013, China (e-mail: lzk@nim.ac.cn). Z. Zhang, Q. He, and J. Zhao are with the National Institute of Metrology, Beijing 100013, China. Y. Fu is with the China Jiliang University, Hzangzhou 310018, China. B. Han is with the Hebei University, Baoding 710049, China. S. Li, J. Lan, and C. Li are with the Tsinghua University, Beijing 100084, China. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIM.2010.2099270 Fig. 1. Campbell mutual inductor. Fig. 2. Outline of the Campbell mutual inductor. may consist of a many-layered coil whose dimensions and position need not to be known with high accuracy [3]. P. W. Harrison and G. H. Rayner established a primary standard of mutual inductance based on Campbell’s idea with uncertainty of 2 × 10 −6 (k = 2) in 1966 [4]. NIM built a similar one in 1986 with uncertainty of 5 × 10 −6 (k = 2) [5]. Campbell inductor has been used to determine the resistance via the Campbell Bridge [6], [7]. The principle of Campbell Bridge is shown in Fig. 3. There are other bridges, such as A. Campbell’s mutual- inductance bridge [8], Lynch’s and Eastwood’s inductance bridges [9], etc., which are used to measure resistance or ca- pacitance in terms of mutual inductance for its high accuracy at that time. With the Quantum Hall resistance and Cross capacitor established with high accuracy, it is possible that the mutual inductance could be determined by QHR standard or capaci- tance with these bridges. However, the resistors and capacitors 0018-9456/$26.00 © 2010 IEEE