68 INSTRUMENTATION IN SCIENTIFIC RESEARCH [Chap. 1 



Two different applications of this type of capacitive transducer can 

 be distinguished: 



1. The stage following the transducer responds to the difference of 

 the partial capacitances C x — C 2 . Usually an a-c voltage E is applied 

 between plates P 1 and P 2 , and the difference of the partial voltages 

 E x — E 2 is measured. The partial voltages are 



_ EC i) „ EC-, 



l 7-» 7T ^° 



or 



e£ + a; <Z — x 



Ei = E ; — E 9 = E — 



1 2d 2 2d 



Therefore, the difference of the partial voltages is 



AE = E, -E 2 =E X - 

 1 2 d 



The relationship between the output signal AE and the displacement 

 of the middle electrode x is linear and independent of the capacitor 

 plate area or the dielectric constant. Also the magnitude of each 

 partial voltage, E x or E 2 , varies linearly with the displacement x of 

 the middle electrode. The sensitivity of the system is 



Ax d 



If stray capacitance cannot be neglected (e.g., the capacitance in the 

 input of the subsequent stage), the sensitivity will be reduced, and 

 nonlinearity will arise between the displacement x and the output. 

 According to Reisch, 1 the output voltage is 



9 r 1 



AE = E' 



d \ + B 



1 -B\aJ \l - B) 



where B( = CJCj) is the ratio of the stray capacitanceC s to the capac- 

 itance C x or C 2 if the middle plate M is in the center position. For 

 B = I and xjd = \, the sensitivity is reduced by | and the deviation 

 from linearity is 1 per cent. 



2. The subsequent stage responds to the ratio C x fC 2 (e.g., balanced 

 bridges, ratio meters). For such systems, the relationship between 

 the output signal and the displacement of the middle electrode is 



C x d — x 



1 S. Reisch, Z. Hochfrequenztech., 38, 104 (1931). 



