The Coupled Current-Voltage Equations

∆E_gfrom E_g,GaAs(T ). Combining (3.12) and (3.19), we have

E_g,GaAs,bgn(T ) = E_g0^∗ − α^∗T

= 1.483 − 4.88 × 10⁻⁴T V.

(3.20)

This linear expression of E_g will simplify the whole problem largely but with small error. It is the reason why we use (3.20) in the following thermal stable problems. By substituting (3.20) into (3.13), the thermal-electrical feedback coefficient φ is obtained as

With (3.21), we can simplify (3.10) as

V_BE = kT We will call (3.24) the accurate I_C−V_BE equation. In this work, both the accurate and simplified I_C−V_BE equations will be solved and used to derive the thermal stable design of ballasting resistors.

3.2 The Coupled Current-Voltage Equations

Although the simplified IC−VBEequation (3.15) and the accurate IC−VBE equation (3.24) are both only functions of finger temperature T_i and current I_Ci apparently, from the discussion of Section 2.2 and (2.17), it can be found that the finger temperature is a

3.2 The Coupled Current-Voltage Equations

function of all finger powers P_j. The finger temperature T_iwill be affected by the power of other fingers and then change the behavior of the finger current ICi. The amount of the influence of the power of other fingers on the finger temperature T_iis determined by the values of coupling thermal resistance. Therefore, the coupling effect is originated from the finger temperature coupling dependence on the power of other fingers through the coupling thermal resistances.

For a multi-finger transistor, the fingers usually mean emitter fingers. Although the emitter fingers are separated in device structure, they are electrically connected by metal line to the contact pads. The voltage bias of all fingers is the same. We will assume V_C and V_BE are the same for all fingers in the following analysis. The dissipate power P_i for each finger is

P_i = I_CiV_C + I_BiV_BE

= I_CiV_C +I_Ci β V_BE

∼= ICiV_C.

(3.25)

Here, we neglect the power contribution of I_Bi and V_BE. It is because that β is quit large and base-emitter voltage is several times smaller than collector-emitter voltage under normal operation. Therefore, the finger temperature can be simplified as just a function of collector currents of all fingers, i.e. T_i = T_i(I_C1, I_C2, I_C3, · · · I_CN). Then, we can combine (2.15) and (2.17) to obtain the coupled temperature equation of the finger temperature T_ias a function of the finger current I_Cjand the derivative of T_iwith respect to the finger current I_Cj. For the constant thermal conductivity case, they are

T_i = T_A+ V_C XN

j=1

R_thijI_Cj (3.26)

∂T_i

∂I_Cj = V_CR_thij. (3.27)

3.2 The Coupled Current-Voltage Equations For the temperature dependent thermal conductivity case, they are

T_i = T_A

"

1 − b − 1 T_A V_C

XN j=1

R_thijI_Cj

#⁻¹

b−1

(3.28)

∂T_i

∂I_Cj = µT_i

T_A

¶_b

V_CR_thij (3.29)

where

R_thij = R_th0ij µT_A

300

¶_b .

The calculation of the derivative of T_i with respect to the finger current I_Cj is necessary when solving the coupled I_C−V_BE equations by using the Newton-Raphson method which will be discussed in Section 3.3.

From the simplified I_C−V_BE equation (3.15) and the accurate I_C−V_BE equation (3.24), we found that both of them can be expressed as

V_BE = V_BE(I_Ci, T_i) i = 1, 2, 3 · · · N (3.30)

for a N-finger device. Because that the finger temperature is a function of currents of all fingers, the expression of (3.30) is a set of coupled functions of currents of all fingers. We subtract the equation of each finger in (3.30) by the equation of se-lected finger r, whose current is assumed constant, and define the goal function gir = g_ir(I_C1, I_C2, I_C3, · · · I_Cr−1, I_Cr+1, · · · I_CN) as the voltage difference V_bei− V_ber, where i = 1, 2, 3 · · · N but i 6= r.

g_ir = V_BE(I_Ci, T_i) − V_BE(I_Cr, T_r) i 6= r (3.31)

We call finger r as the reference finger. There are N sets of goal functions corresponding to the N fingers to be the reference finger in turn. The rank of the problem at hand is reduced by one, from N coupled equations to N − 1 goal functions. When solving the

3.2 The Coupled Current-Voltage Equations

coupled equations, we will set I_Cr as a known constant and only need to solve the other N − 1 unknowns ICi,i6=r actually. It has some advantages when solving large problems.

In order to achieve our goal to solve the coupled I_C−V_BE equation, we theoretically just need to find out a solution of I_Ci,i6=r to make the goal functions zero. In Section 3.3, the method to find out the solution of gir(ICi,i6=r) = 0 will be discussed.

In this work, we will use three models to calculate the I_C−V_BE curves. The models include the simple model, the accurate model with constant thermal conductivity, and the accurate model with temperature dependent thermal conductivity. For the simple model, using the simplified I_C−V_BE equation (3.15), we can write down the coupled equations as Because that (3.15) is already a first order approximation, it is unnecessary to use high order equation (3.28) for Ti. Therefore, substituting the constant thermal conductivity case equation (3.26) into T_i. we can write down the coupled equations and the goal function for a N-finger device as

V_BE = kT_A

From (3.34) and using (3.27), the derivative of g_ir with respect to the finger current I_Cj is where δ_ij is the Kronecker delta which is defined by

δ_ij ≡

3.2 The Coupled Current-Voltage Equations

The derivative of the goal function will be used in Section 3.3 to solve the coupled IC−VBE equations.

For the accurate model, using the accurate I_C−V_BE equation (3.24), we can write down the coupled equations and the goal function for a N-finger device as

V_BE = E_g0^∗ − φ^∗_iT_i+ R_EiI_Ci i = 1, 2, 3 · · · N (3.36)

From (3.38), the derivative of φ^∗_i with respect to the finger current I_Cj is

∂φ^∗_i

From (3.37) and using (3.39), the derivative of g_ir with respect to the finger current I_Cj is

With constant thermal conductivity, substituting (3.27) into (3.40), we obtain

∂g_ir

With temperature dependent thermal conductivity, substituting (3.29) into (3.40), we obtain