238 BELL SYSTEM TECHNICAL JOURNAL 



For purely electrical measurement, we can get rid of the two ideal 

 transformers by taking half the impedance of Cm and Lm through each 

 transformer as shown in Fig. 14E. Since we have left two opposing 

 transformers of equal ratio they can be eliminated and the network of 

 Fig. 14F results. This is shown in balanced form. Figure 14G shows 

 the same network expressed in lattice form which is easily done by 

 using the equivalences of Fig. 8. This represents a crystal of twice 

 the impedance of the fully plated crystal in each series arm of the 

 lattice with the static capacitances Co in the other arm. If we connect 

 terminal 1 to 3 and 4 to 2, or in other words we use a completely 

 plated crystal, the equivalent circuit reduces to that for a fully plated 

 crystal as shown in Fig. 14H. 



The networks of Fig. 14, F and G, represent the two plate crystals 

 for transmission through the crystal, but do not give a four-terminal 

 equivalence. For example, if we measure the crystal between termi- 

 nals 1 and 3 we should not expect any impedance due to the vibration 

 of the crystal since there is no field applied perpendicular to the 

 thickness. The representation of Fig. 14, F or G, would not indicate 

 this. The same sort of problem arises when it is desirable to obtain 

 a four-terminal representation of a transformer and can be solved by 

 using a lattice network representation with positive and negative 

 inductance elements. The same procedure can be employed for a 

 crystal and the steps are shown in Fig. 15. 



We start with the lattice representation of Fig. 14G but employ 

 the series form of the impedance of a crystal shown in Fig. 1. The 

 series capacitance is divided into two parts, Co/2 and a negative 

 capacitance necessary to make the total series capacitance equal to 

 Co plus Ci. This negative capacitance and the antiresonant circuit are 

 lumped as one impedance 2Z in Fig. 15B. Now by the network 

 equivalence of Fig. 8, we can take the series capacitances Co/2 outside 

 the network. We can also add an impedance Z/2 on the ends of the 

 network provided we add a negative Z in series with all arms of the 

 network as shown in Fig. 15C. The network of Fig. 15C is equivalent 

 to that of Fig. 15A as far as transmission through it is concerned, but 

 is different if we measure impedances between any of the four termi- 

 nals. For example, if we measure the impedance between the termi- 

 nals 1 and 4, the impedance of the network reduces to that shown in 

 Fig. 15D. The impedance of the parallel circuit reduces to a plus Z 

 shunted by a minus Z which introduces an infinite impedance. Simi- 

 larly between 1 and 4, 2 and 3, and 2 and 4 the impedance becomes 

 infinite as it should be if we neglect the small static capacitances 

 existing in the crystal. If we take account of these the complete 



