Resistors, thick-film 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). 

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. 

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 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. 

Thick-film resistors can be processed with a tolerance of about 25%. Laser trimming increases the resistance value. Therefore, a resistor is designed to a lower value than desired and will be trimmed to its target value later on. Besides the resistance value required, the power dissipation density is required to design a thick-film resistor. The power dissipation density (Pdensiiy in mW/mm ) is a paste property, which is specified in the data sheet. It is typically related to a 50% trim cut (maximum allowable trim length) and application on prefired alumina. For a stable resistor, the minimum area Ag is determined by the maximmn circuit power dissipation requirement, as in Equation 9.3 ... 

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. 

Materials are also available that can combine mixed values of resistance on one layer. Polymer thick films (PTE) are applied by successive screen print/cure cycles so that aU the desired resistance values can be made on one layer. Also, some sheet capacitance manufacturers laminate a layer of resistor foil as one electrode in their construction. Therefore, a fabricator can mix resistance and capacitance on one sheet type raw material. 

Resistor Noise. The random motion of current carriers causes spontaneous fluctuations in electric current, termed electrical noise. Current noise in thick-film resistors is called excess noise, as it is considered the excess noise over thermal noise. Thermal noise is predictable and is a function of resistance and temperature, whereas current noise has no definite relationship. Noise in thick-film resistors depends on the sheet resistance value, thickness of the resistor print, and geometry of the resistor. Figure 8.59 shows a typical plot of measured noise versus resistor area for a 200-kQ/sq resistor. Current noise decreases as resistor area increases. Resistor noise also depends on processing conditions. Generally, noise decreases with an increase in peak firing temperature. The typical noise value for commercial thick-film resistors is shown in Table 8.20. 

Statistical loading curve models have been proposed that take into account the segregated nature of thick-film resistors. A systematic microstmcture development study has been conducted by Vest for RuOj model thick-film resistors. Vest proposed a statistical loading curve to explain the sheet resistivity variation as a function of volume fraction of the conductive phase and particle sizes of both the conductive phase and glass. Several theoretical models have heen developed to explain the variation of resistivity as a function of volume fraction of the conductive phase. A comprehensive model to explain the sheet resistivity variation in thick-film resistors in terms of composition and physicochemical properties of powders over the entire resistance range has not heen developed to date. 

FIGU RE 8.68 Effect of glass particle size on sheet resistance of model thick-film resistors. 

R. W. Vest, AModel for Sheet Resistivity of RuOj Thick Film Resistors, IEEE Trans. CHMT, vol. 14, no. 2, pp. 396-406,1991. 

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