Resistor sheet resistance

The minimum length l and width id of a resistor are calculated from the given resistance R, the sheet resistance R in ohms per square, dissipated power P, and permissible power dissipation per square inch P by use of the formulas u> = /(P R)/P R and l = u>R/R. The capacitance of film capacitors is given by C = 0.225D(W — 1 )A/t, where C is the capacitance in picofarads. D the dielectric constant. N the number of plates, A die area in square inches, and / the dielectric thickness in inches. 

The functional phase in thick-film resistors is a mixture of electrically conducting (or semiconducting) ceramic powders such as ruthenium dioxide (RUO2), bismuth ruthenate (Bi2Ru207), lead ruthenate (Pb2Ru206), and Ag-Pd-PdO mixtures for use in air-fired pastes and tantalum nitride (TaN) for nitrogen-fired pastes. The resistance of thick-film resistors is specified in terms of sheet resistance, which has units of ohms/square (Q/D). 

The concept of sheet resistance is illustrated in Figure 27.17a, which shows a sheet resistor of length I, width w, and thickness d, having a resistance, R given by... 

Thick-film resistors are available with sheet resistance values in the range from 0.1 to 10 M Q/D. By blending different quantities of conductive material and an electrically insulating glass the resistivity is controlled. For a high sheet resistance formulation the ratio of conductor to glass would be about 70/30. 

Traditionally cracked carbon film resistors have been used for general purpose applications. The films are made by cracking a hydrocarbon in the absence of oxygen on the rods. The sheet resistance is only partly determined by the thickness of the film. The resistivity of the material can vary strongly, depending on the proportion of amorphous carbon and the amount of contaminants in the film. The TCR usually becomes more negative when the resistivity increases. 

For resistive films, often the sheet resistivity is used. This is the resistance of a square area on the resistor surface. 

If the length is equal to the width (the sample is a square), the electrical resistance is the same as the sheet resistivity independent of the actual dimensions of the sample. This is the basis of the units of sheet resistivity, ohms/square/unit thickness. For thick-film resistors, the standard adopted for unit thickness is 0.001 in. or 25 mm of dried thickness. The specific units for thick-film resistors are S2/D/0.001 (read ohms per square per mil) of dried thickness. For convenience, the units are generally referred to as, simply,... 

The most common types of resistor material are nichrome (NiCr) and tantalum nitride (TaN). Although NiCr has excellent stability and TCR characteristics, it is susceptible to corrosion by moisture if not passivated by sputtered quartz or by evaporated silicon monoxide (SiO). TaN, on the other hand, may be passivated by simply baking in air for a few minutes. This feature has resulted in the increased use of TaN at the expense of NiCr, especially in military programs. The stabihty of passivated TaN is comparable to that of passivated NiCr, but the TCR is not as good unless annealed for several hours in a vacuum to minimize the effect of the grain boundaries. Both NiCr and TaN have a relatively low maximum sheet resistivity on alumina, about 400 S2/ for NiCr and 200 / for TaN. This requires lengthy and complex patterns to... 

The range of resistor values is wider. Thin-hlm designers are usually limited to a single value of sheet resistivity from which to design all of the resistors in the circuit. 

The first resistor ink system with a wide range of sheet resistivities was developed in 1958 by J. D Andrea. This palladium and silver system (PdO/ Ag) had a high firing temperature influence on the sheet resistivity, caused by the complicated chemical-dynamical process. The wide range of resistances that this composition could achieve was one main reason for the rapid growth of thick-film technology since then. 

Resistor inks are available in a broad range of sheet resistivities (1 Q/a to more than 10 m 2/n). The sheet resistivity is derived from the paste bulk resistance normalized to a certain tiiickness (e.g., 25 pm dried thickness). Therefore, TFRs are determined by their width-to-length ratio and the characteristics of the paste applied (Equation 9.1). Figure 9.11 shows a typical rectangular TFR design. 

Resistors with low sheet resistivities increase their value during HVP trimming (Figure 9.43a). 

The resistance value of TFRs is defined by Equation 9.1 in Subsection 9.3.1. Flowever, the termination that is overlapped by the resistor layer has an impact on the resistance (Figure 9.11). If silver-containing inks are used, the sheet resistivity changes owing to Ag-diffusion into the resistor body. The shorter the resistor, the more the diffusion influences the value of the resistor. This must be taken into consideration by introducing a correction factor in the design. 

The noise index NI in dB is measured by a band-pass filter (bandwidth of 1 kHz between 618 Hz and 1618 kHz) [51]. A noise index NI = 0 dB corresponds to 1 pV noise voltage per 1 V DC voltage. For ruthenate-based TFRs, the noise index exhibits an increase proportional to the sheet resistivity and a decrease proportional to the resistor area [47,52]. 

Over the used frequency range, a TFR shows a complex impedance Z. It is influenced by geometrical dimensions (g) and by the conduction mechanism into the resistor layer (r) (Equation 9.25). For resistors with low sheet resistivities (<100 Q/n), the imaginary part is inductive (Figure 9.51a) whereas with higher sheet resistivities (>100 Q/o), it becomes capacitive (Figure 9.51b). For low and high sheet resistivities, the different RF behavior can be explained by the different conduction mechanisms [47,52,53]. 

The equivalent circuit of printed resistors is shown in Figure 9.52 [53]. Depending on the sheet resistivity of fhe paste, the model will be simplified by omitting some elements. The capacitors C represent the influence of the ground plane. They are equal to the capacitance of a transmission line of identical dimensions (width, length, height above groxmd, etc.). Therefore, C reflects the equivalent line type (microstrip-like and stripline-like). 

Vest, R.W., A Model for Sheet Resistivity of Ru02 Thick Film Resistors, IEEE Transactions on CHMT, Vol. 14, No.2, June 1991, pp. 396-404. 

Resistor Type Sheet Resistivity (per square) Temperature Coefficient (ppm/°C)... 

Low-value resistors, typically less than 10 OJcsr (sheet resistivity) for applications such as surge arrestor chcuitry... 

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