I C = I S ( 1 + V CE V A ) exp ( V BE V t ) I C = I S ( 1 + V A V CE ) exp ( V t V BE )
《模拟电子技术基础》董诗白，第四版，p30
（1）VBE正向电压，扩散运动形成IE：
Emitter的杂质（多子电子）浓度高，电子通过扩散（外加电场+浓度扩散）从Emitter到Base，形成了I E N I E N I E P I E P
其中Emitter的电子电流I E N I E N I B N I B N
{ I E = I EN + I EP I EN = I BN + I CN ⇒ I E = I BN + I CN + I EP { I E = I EN + I EP I EN = I BN + I CN ⇒ I E = I BN + I CN + I EP
（2）漂移运动形成I C I C V C B V C B V C ( N ) < V B ( P ) V C ( N ) < V B ( P )
从Emitter扩散到Base的电子（又叫非平衡少子），通过漂移（外加电场，Base为负电压），到达Collector区，形成I C N I C N I C B O I C B O
I C = I C N + I C B O I C = I C N + I C B O
（3）复合电流I B I B
Emitter扩散到Base的电子，由于Base的体积小，且杂质（多子空穴）浓度低，所以只有一小部分与之复合，形成了流出的电流IBN；同时还有VBE正向电压扩散电流IEP，以及流入的VCB反向电压少子漂移电流ICBO。
I B = I B N + I E P − I C B O I B = I B N + I E P − I C B O
（4）联立(1)(2)(3)，可得，
I E = I B + I C I E = I B + I C
在N型区（参杂磷元素，Ⅴ族，多出一个自由电子），p n p n N D N D n n n n n n
p n + N D = n n p n + N D = n n
在P型区（参杂硼元素，Ⅲ族，多出一个空穴，n p n p N A N A p p p p p p
n p + N A = p p n p + N A = p p
本征激发浓度n i n i n p 0 n p 0 N A N A p p p p
n i 2 = n p 0 ( N A + p p ) ≈ n p 0 N A (0) n i 2 = n p 0 ( N A + p p ) ≈ n p 0 N A ( 0 )
这是半导体的基本公式，有待整理半导体物理的基本知识
本征载流子浓度与材料，温度和禁带宽度有关，而与导带、价带、费米能级无关
热平衡状态下，电子浓度n 0 n 0 p 0 p 0
对于一个工作在放大区的NPN：
基区（P）的少子电子n p n p
n p ( 0 ) = n p 0 exp ( V B E V t ) (1) n p ( 0 ) = n p 0 exp ( V t V B E ) ( 1 )
n p ( W B ) = n po exp ( V B C V t ) ≈ 0 (2) n p ( W B ) = n po exp ( V t V B C ) ≈ 0 ( 2 )
这里n p ( W B ) n p ( W B ) V B C V B C n p ( W B ) n p ( W B ) n p ( x ) n p ( x )
n p + N A = p p (3) n p + N A = p p ( 3 )
由于以上关系，N A N A p p p p
基区（P）中的少子（电子）随浓度梯度扩散，并在反偏的集电结电场（CB=NP=正负）作用下形成集电极电流，其电流密度：
J n = q D n d n p ( x ) d x J n = q D n n p ( 0 ) W B (4) J n = q D n d x d n p ( x ) J n = q D n W B n p ( 0 ) ( 4 )
J n J n n n
D n D n
(4)结合(1)，可得集电极电流，
I C N = J n A = q D n n p ( 0 ) W B A = q A D n W B n p o exp ( V B E V t ) = I S exp ( V B E V t ) (5) I C N = J n A = q D n W B n p ( 0 ) A = W B q A D n n p o exp ( V t V B E ) = I S exp ( V t V B E ) ( 5 )
(5)结合(0)，可得集电极电流
I S = q A D n n p 0 W B = q A D n n i 2 W B = q A D n ‾ n i 2 Q B = k T A μ n ‾ n i 2 Q B I S = W B q A D n n p 0 = W B q A D n n i 2 = Q B q A D n n i 2 = Q B k T A μ n n i 2
D n ‾ D n D n D n
Q B Q B
D n ‾ = k T q μ n ‾ D n = q k T μ n
μ n ‾ μ n m 2 / m 2 /
可是I C N I C N I C I C I C B O I C B O
但是Base区x = W B x = W B n p o n p o
J n = q D n n p ( 0 ) − n p o W B = n p o exp ( V B E V t ) − n p o W B = n p o ( exp ( V B E V t ) − 1 ) W B J n = q D n W B n p ( 0 ) − n p o = W B n p o exp ( V t V B E ) − n p o = W B n p o ( exp ( V t V B E ) − 1 )
I C = J n A = I S [ exp ( V BE V t ) − 1 ] (6) I C = J n A = I S [ exp ( V t V BE ) − 1 ] ( 6 )
这个表达式于二极管的表达式完全相同
I C = I S [ exp ( V B E V t ) − 1 ] I C = I S [ exp ( V t V B E ) − 1 ]
I D = I S [ exp ( V D V t ) − 1 ] I D = I S [ exp ( V t V D ) − 1 ]
由于V C E V C E W B W B I C I C
I C = I S [ exp ( V B E V t ) − 1 ] ( 1 + V C E V A ) I C = I S [ exp ( V t V B E ) − 1 ] ( 1 + V A V C E )
其中V A V A
（1）基区复合电流
I B 1 I B 1 Q e Q e
Q e = 1 2 n p ( 0 ) W B q A Q e = 2 1 n p ( 0 ) W B q A
I B 1 = Q e τ b = 1 2 n p ( 0 ) W B q A τ b (6) I B 1 = τ b Q e = 2 1 τ b n p ( 0 ) W B q A ( 6 )
τ b τ b
(6)结合(1)，可得
I B 1 = Q e τ b = 1 2 n p o W B q A τ b exp ( V B E V t ) (7) I B 1 = τ b Q e = 2 1 τ b n p o W B q A exp ( V t V B E ) ( 7 )
结/表面/通道复合电流的非线性
I B 1 ∝ exp ( V B E n E V t ) I B 1 ∝ exp ( n E V t V B E )
（2）发射区少子扩散电流
I B 2 I B 2
I B 2 = q A D p L P p n E ( 0 ) I B 2 = L P q A D p p n E ( 0 )
D p D p
L P L P
p n E ( 0 ) p n E ( 0 )
{ I B 2 = q A D p L P p n E ( 0 ) p n E ( 0 ) = p n E o exp ( V B E V t ) p n E o = n i 2 N D ⟹ I B 2 = q A D p L P n i 2 N D exp ( V B E V t ) (8) ⎩ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎧ I B 2 = L P q A D p p n E ( 0 ) p n E ( 0 ) = p n E o exp ( V t V B E ) p n E o = N D n i 2 ⟹ I B 2 = L P q A D p N D n i 2 exp ( V t V B E ) ( 8 )
(7)结合(8)，得到I B I B
I B = I B 1 + I B 2 = ( 1 2 n p o W B q A τ b + q A D p L P n i 2 N D ) exp ( V B E V t ) I B = I B 1 + I B 2 = ( 2 1 τ b n p o W B q A + L P q A D p N D n i 2 ) exp ( V t V B E )
n E n E n E n E
（1）在华成英，董诗白《模拟电子技术基础》中
直流系数β ‾ β ：
β ‾ = I C N I B ′ = I C − I C B O I B + I C B O ⇒ I C = β ‾ I B + ( 1 + β ‾ ) I C B O β = I B ′ I C N = I B + I C B O I C − I C B O ⇒ I C = β I B + ( 1 + β ) I C B O
从Base到Emitter的电流（流出总电流），是I B + I C B O I B + I C B O
I B + I C B O = I B N + I E P I B + I C B O = I B N + I E P
I C B O I C B O ( 1 + β ‾ ) I C B O ( 1 + β ) I C B O
交流系数β 定义：
β = Δ i C Δ i B β = Δ i B Δ i C
推导，
i C i B ∣ ( 1 + β ‾ ) I C B O → 0 = I C + Δ i C I B + Δ i B = β ‾ I B + β Δ i B + ( 1 + β ‾ ) I C B O I B + Δ i B ≈ β ‾ I B + β Δ i B I B + Δ i B ∣ Δ i B → 0 , β ‾ = β ≈ β i B i C ∣ ∣ ∣ ∣ ∣ ( 1 + β ) I C B O → 0 = I B + Δ i B I C + Δ i C = I B + Δ i B β I B + β Δ i B + ( 1 + β ) I C B O ≈ I B + Δ i B β I B + β Δ i B ∣ ∣ ∣ ∣ ∣ Δ i B → 0 , β = β ≈ β
（2）在Paul R. Gray的《模拟集成电路的分析与设计》中，正向电流放大系数βF
β F = I C I B = I C I B 1 + I B 2 I C N I B N + I E P β F = I B I C = I B 1 + I B 2 I C I B N + I E P I C N
g m = ∂ I C ∂ V B E = ∂ ( I S exp ( V B E V t ) ( 1 + V C E V A ) ) ∂ V B E = I C V t g m = ∂ V B E ∂ I C = ∂ V B E ∂ ( I S exp ( V t V B E ) ( 1 + V A V C E ) ) = V t I C
r π = ∂ V B E ∂ I E = β V t I C = β g m r π = ∂ I E ∂ V B E = β I C V t = g m β
r e = ∂ V B E ∂ I E = β β + 1 g m = α g m r e = ∂ I E ∂ V B E = g m β + 1 β = g m α
r o = ∂ V C E ∂ I C = V A I C r o = ∂ I C ∂ V C E = I C V A
C b e = C d + C j = τ b I C V t + 2 A E C j e 0 = g m τ b + 2 A E C j e 0 C b e = C d + C j = τ b V t I C + 2 A E C j e 0 = g m τ b + 2 A E C j e 0
C c b = C j = A c C j c 0 ( 1 + V C B φ c 0 ) 1 3 C c b = C j = ( 1 + φ c 0 V C B ) 3 1 A c C j c 0
C c s = A T C J s 0 1 + V C S φ S 0 C c s = 1 + φ S 0 V C S A T C J s 0
利用公式（6），对于电流比
{ I C = J n A = I S [ exp ( V B E V t ) − 1 ] I C 1 / I C 2 = p { I C = J n A = I S [ exp ( V t V B E ) − 1 ] I C 1 / I C 2 = p
Δ V B E = V B E 1 − V B E 2 = K T q ln ( p I C + I s I C + I s ) Δ V B E = V B E 1 − V B E 2 = q K T ln ( I C + I s p I C + I s )
当I s I s Δ V B E Δ V B E I S I S I C I C I C I C I C I C I C I C V B E V B E
I C ∝ q V B E 2 k T I C ∝ 2 k T q V B E
（Precision Temperature Sensors in CMOS Technology, Michiel A.P. Pertijs, page18）
二极管不能用来产生Δ V B E Δ V B E
ln ( I B ) ∝ q V B E 2 k T l n ( I B ) ∝ 2 k T q V B E
总之，当I C I C I S I S V B E − I C V B E − I C
已知
I S = k T A μ n ‾ n i 2 Q B I S = Q B k T A μ n n i 2
表示其中温度相关的量，I S I S
I S ( T ) = k T A μ n ‾ ( T ) n i 2 ( T ) Q B ( T ) I S ( T ) = Q B ( T ) k T A μ n ( T ) n i 2 ( T )
（1）本征激发浓度
n i 2 ( T ) ∝ T 3 exp ( − q V g ( T ) k T ) n i 2 ( T ) ∝ T 3 exp ( k T − q V g ( T ) )
这里V g 0 V g 0
V g ( T ) = V g 0 − α T V g ( T ) = V g 0 − α T
（2）电子迁移率，这里的∗ n ∗ ∗ n ∗
μ n ‾ ( T ) ∝ T − n μ n ( T ) ∝ T − n
（3）Gummel number略微与温度有关，因为Base的宽度是与温度相关的。
Q B ( T ) = W B ( T ) N A Q B ( T ) = W B ( T ) N A
结合以上因素，
I S ( T ) = C T η exp ( − q V g 0 k T ) I S ( T ) = C T η exp ( − k T q V g 0 )
η = 4 − n η = 4 − n
所以
I c ( T ) = C T η exp ( − q V g 0 k T ) exp ( q V B E ( T ) k T ) = C T η exp ( q ( V B E ( T ) − V g 0 ) k T ) I c ( T ) = C T η exp ( − k T q V g 0 ) exp ( k T q V B E ( T ) ) = C T η exp ( k T q ( V B E ( T ) − V g 0 ) )
两边同时取对数，则有
ln [ I c ( T ) ] = ln ( C ) + η ln ( T ) + q ( V B E ( T ) − V g 0 ) k T ln [ I c ( T ) ] = ln ( C ) + η ln ( T ) + k T q ( V B E ( T ) − V g 0 )
k T q ln [ I c ( T ) ] = k T q ln ( C ) + η k T q ln ( T ) + V B E ( T ) − V g 0 q k T ln [ I c ( T ) ] = q k T ln ( C ) + η q k T ln ( T ) + V B E ( T ) − V g 0
V B E ( T ) = V g 0 + k T q ln [ I c ( T ) ] − k T q ln ( C ) + η k T q ln ( T ) V B E ( T ) = V g 0 + q k T ln [ I c ( T ) ] − q k T ln ( C ) + η q k T ln ( T )
不知为何，写成了经验公式：
V B E ( T ) = V g 0 ( 1 − T T r ) + T T r V B E ( T r ) − η K T q ln ( T T r ) + K T q ln [ I C ( T ) I C ( T r ) ] V B E ( T ) = V g 0 ( 1 − T r T ) + T r T V B E ( T r ) − η q K T ln ( T r T ) + q K T ln [ I C ( T r ) I C ( T ) ]
V B E ( T ) = V g 0 − T T r [ V g 0 − V B E ( T r ) ] − η k T q ln ( T T r ) + k T q ln [ I C ( T ) I C ( T r ) ] V B E ( T ) = V g 0 − T r T [ V g 0 − V B E ( T r ) ] − η q k T ln ( T r T ) + q k T ln [ I C ( T r ) I C ( T ) ]
在以下文献中，A Curvature-Corrected Low-Voltage Bandgap Reference, Made Gunawan, 1993. 也是bandgap中的情况，是用PTAT电流作为偏置电流源的。则有：
I C ( T ) I C ( T r ) = T T r I C ( T r ) I C ( T ) = T r T
V B E ( T ) = V g 0 − T T r [ V g 0 − V B E ( T r ) ] − ( η − 1 ) K T q ln ( T T r ) V B E ( T ) = V g 0 − T r T [ V g 0 − V B E ( T r ) ] − ( η − 1 ) q K T ln ( T r T )
这里T r T r V g 0 V g 0 η η
V g ( 0 ) = 1170 m V V B E ( T r ) = 0.65 V T R = 300 K η = 3.6 V g ( 0 ) V B E ( T r ) T R η = 1 1 7 0 m V = 0 . 6 5 V = 3 0 0 K = 3 . 6
β F β F
β F ( T ) = β F 0 ( T T r ) X T B β F ( T ) = β F 0 ( T r T ) X T B
