The ratio of a change in collector current to the change in the parameter value that caused it is called a stability factor, A stability factor is thus a measure of how sensitive collector bias current is to changes ill a parameter value. Stability factors can be defined for each of the three parameter,s we have discussed: S = due (6-3) t:.st; S(:) = (6-4) t:.V’,f: S({3) = t:.(6-5) A{3 To illustrate, if the value of S() for a certain bias circuit is 20, then a change in from 0.01 /LA to 0.02 p.A. (= 0.01 /LA) will cause a change in equal 20(0.01 p.A) = 0.2 /LA. An ideal stability factor would have value zero, implying I/O change in I( , for (/II)’ change in parameter value. The actual value depends on the components used in the bias circuit and. can never be O. We therefore seek bias designs that make the stability factors :is small as practical. Using calculus, it can be shown that approximate values of till’: stability factors for the circuit in figure 6-1 arc found Iron <‘(I ) – ({3 + t)(t ,. NI/IR,.) _ R, -I- Rjl J (“I() – – (f:3 + I) + RI/II?, . R, + R”I/j \'(V)= -{3 • /II. Ru + R,(f:3 + I) S({3) = PIUS(lC/lO) {31{3″ RI/ = Rill R” lc, = initial value of Ie {31 = initial value of {3 {3~= larger value of (3 (I(,l/rJ) = value of S(I(fI() for {3= f:3~ Note that Sl//l,J is negative, so when an increase in temperature causes Vii’. to decrease, AV/I” is itself a negative quantity and 1e = S( V/l1)(t V’II) , being the product of two negative quantities, is a positive quantity. Equations 6-6 through 6-8 show that each stability factor is reduced (improved) when HI: is made large. This confirms our intuitive analysis of the beneficial contribution of emitter resistance in providing current feedback. S(/(‘I//I) and S({3) arc improved h making R” small. As we shall see presently, there arc other considerations that prevent us from making R” arbitrarily large and/or N/I arbitrarily small. As a rule of thumb, satisfactory stability can he realized by making the ratio less than 10. The stability equations can be applied to the “f zed-bias” circuit we studied earlier by letting R” equal 0 and R” equal the value of the series base resistor. Each of the stability equations gives the stability factor related to variation in one parameter only, so the total change in collector current over a certain temperature range can he approximated by where . A /II}S + A VI J/,S(V/I,J + A{3 S({3) (6-9) \ where , A : , and A{3 arc the total changes in the respective parameter values over the temperature range. The expression is an approximation because. all three parameters are changing simultaneously with temperature. To compute an individual stability factor whose value depends on another parameter, use a mid range or average value of the parameter. The next example illustrates the computation
Find the total change in the die collector current over the temperature range from locator 100°C for each of the circuits shown in Figure 6-3. Each transistor is biased at IE = 1 mA at loo C. Solution. Applying equations 6-6 through 6-8 to the unsuitability circuit of Figure 6-3(a), with Rf: = 0 and R/I = 1.4 MH, we find R£ + R8 RB S(JClm)R, + Rn1f3 = Rn1f3 = f3 The average f3 over the given temperature range is (60 + 140)/2 = 100, so we use S(lc8(;)100. -(3 -f3 -100 S R« + R f :(f3 + 1) = R,; = 1.4MO = -7.14 x 1O-~ S(f3) = (I x 10-3)(140) = 167 x (60)(140)Now, MC = (1.2 p.A) – (0.01 p.A) = 1.t9 p.A,: =.(0.54 V) – (0.74 V) = -0.2 V. and = 140 – 60 = 80. So, from equation 6-9 (1.19p.A)(IOO) + (-0.2)(-7.14 x 1O+ (80)(1.67 x 10-5). = (0.119 mA) + (0.014 mA) + (1.336 mA) = 2~47mA We see that the collector current increases by about 1.47 mA in the stabilized circuit (to 2.47 mA, more than double its original value). The greatest part of the change in is due to the change in f3. For the emitter-stabilized circuit of Figure 6-3(b), we have R8 = (130 kO) II (10 kO) = 9.29 kO. Applying the stability equations, we find S(lcno)+ (9.29 kO) = 9.42 (1 kO) + 9.29 kO 100 S(Vn,J (9.29 = -9.07 x 10-4 S(f3) (1 x 10 .1) 7 X 10-4) + (80)(1.15 X 10-6) = (0.0112 mA) + (0.181 mA) + (~.092 mA)0.284 mA The increase in collector current in the stabilized circuit is about 28%, much less
than that of the stabilized circuit over the same temperature range, Note particularly how stabilization has reduced the change in y the change -1.336 mA in the stabilized circuit versus 0.092 mA in the stabilized circuit. We concluded that the stabilized circuit could be used with transistors having a range of {3 from 60 to 140 were substituted into COIL.
