1. Theory

Definition: Ebers-Moll and Gummel-Poon are the classic large-signal compact models of the bipolar transistor; Gummel-Poon, the default BJT model of SPICE, extends Ebers-Moll with base-charge effects (Gummel and Poon; TU Delft).
Ebers-Moll: Two coupled diodes and two controlled current sources, linked by the reciprocity relation $α_{F} I_{E S} = α_{R} I_{C S}$ , describe the transistor in forward active, reverse active, saturation and cut-off (Fabian and Zutic; Gummel and Poon).
Gummel-Poon: The transfer current is divided by a normalized base charge $q_{b}$ , which captures the Early effect through $q_{1}$ and high injection through $q_{2}$ ; separate leakage terms in the base current reproduce the fall of gain at low current (Qucs; MathWorks).
Dynamic model: Depletion and diffusion charges, a bias-dependent transit time and excess phase give the frequency and transient response (Qucs; MathWorks).
Limits: The model has no quasi-saturation, avalanche or self-heating, which motivated VBIC, HICUM and Mextram (background knowledge).

2. Working Principle

Charge control: The collector current equals the injected minority charge divided by the base transit time; Gummel and Poon expressed it through the integrated majority charge in the base, so any change in that charge changes the current (Gummel and Poon; background knowledge for the integral).
Early effect: Reverse base-collector bias narrows the neutral base and lowers $q_{1}$ , raising the collector current with $V_{C E}$ (Qucs; TU Delft).
High injection: When injected carriers exceed the base doping, $q_{2}$ dominates and the collector current grows only as the square root of $I_{F}$ , halving the Gummel-plot slope (TU Delft; computed).
Base current: The ideal part $I_{F} / B F$ adds to recombination leakage with emission coefficient NE, so the gain is low at small currents and peaks at moderate currents (Qucs; computed).

3. Applications

Circuit simulation: Default BJT model in SPICE-class simulators for discrete and integrated transistors (Qucs; MathWorks).
Analog design: Hand analysis with $g_{m}$ , $r_{π}$ , $r_{o}$ and $f_{T}$ derived from the model parameters (TU Delft).
Switching circuits: Ebers-Moll saturation and charge storage in logic and power switching (Gummel and Poon; computed).
Vendor models: Datasheet-level Gummel-Poon cards for discrete transistors, alongside the diode cards on the Diode Compact Models page (background knowledge).

Ebers-Moll Model

Name	Formula	Related Formulae	Variables/Constants
Ebers-Moll injection model	$I_{E} = I_{E S} (e^{V_{B E} / V_{T}} - 1) - α_{R} I_{C S} (e^{V_{B C} / V_{T}} - 1), I_{C} = α_{F} I_{E S} (e^{V_{B E} / V_{T}} - 1) - I_{C S} (e^{V_{B C} / V_{T}} - 1)$	$α_{F} I_{E S} = α_{R} I_{C S} = I_{S} (reciprocity), I_{B} = I_{E} - I_{C}$	$I_{E}, I_{C}, I_{B} : e mi tt er, co l l ec t or an d ba se c u r r e n t s (n p n, f l o w in g o u t o f t h e e mi tt er an d in t o t h e co l l ec t or)$ $I_{E S}, I_{C S} : e mi tt er an d co l l ec t or j u n c t i o n s a t u r a t i o n c u r r e n t s$ $α_{F}, α_{R} : f or w a r d an d r e v er se co mm o n - ba se c u r r e n t g ain s$ $V_{B E}, V_{B C} : j u n c t i o n v o l t a g es$ $V_{T} : t h er ma l v o l t a g e k_{B} T / q$ $I_{S} : t r an s p or t s a t u r a t i o n c u r r e n t; F abian an d Z u t i c : f or w a r d an d r e v er se c u r r e n t s j_{f} = j_{se} (e x p (q V_{b e} / k_{B} T) - 1) an d j_{r} = j_{sc} (e x p (q V_{b c} / k_{B} T) - 1), t er mina l c u r r e n t s j_{e} = j_{f} - α_{r} j_{r} an d j_{c} = α_{f} j_{f} - j_{r}, j_{b} = j_{e} - j_{c}, an d r ec i p r oc i t y α_{f} j_{se} = α_{r} j_{sc}; G u mm e l an d P oo n : t h e E b er s - M o l l m o d e l (1954) a s t h e main l a r g e - s i g na l bi p o l a r m o d e l, p h y s i cs - ba se d an d v a l i d in a c t i v e, s a t u r a t e d an d c u t - o f f o p er a t i o n, b u t l a c k in g e f f ec t s s u c h a s t h e E a r l y e f f ec t$ $(co m p u t e d : I_{S} = 1 \times 1 0^{- 16} A w i t h β_{F} = 100 (α_{F} = 0.990) an d α_{R} = 0.5 g i v es I_{E S} = 1.01 \times 1 0^{- 16} A an d I_{C S} = 2.0 \times 1 0^{- 16} A)$
Saturation currents and base transport factor	$j_{se} = j_{g b}^{n} + j_{g e}^{p}, j_{sc} = j_{g b}^{n} + j_{g c}^{p}, j_{g b}^{n} = \frac{q D _{nb}}{L _{nb}} n_{0 b} coth (\frac{w _{b}}{L _{nb}})$	$α_{T} = \frac{1}{c o s h ( w _{b} / L _{nb} )}, β = \frac{α}{1 - α}$	$j_{se}, j_{sc} : e mi tt er an d co l l ec t or s a t u r a t i o n c u r r e n t d e n s i t i es$ $j_{g b}^{n} : e l ec t r o n g e n er a t i o n c u r r e n t in t h e ba se$ $j_{g e}^{p}, j_{g c}^{p} : h o l e g e n er a t i o n c u r r e n t s in e mi tt er an d co l l ec t or$ $D_{nb} : e l ec t r o n d i f f u s i o n coe f f i c i e n t in t h e ba se$ $L_{nb} : e l ec t r o n d i f f u s i o n l e n g t h$ $n_{0 b} : e q u i l ib r i u m e l ec t r o n d e n s i t y in t h e ba se$ $w_{b} : e f f ec t i v e ba se w i d t h$ $α_{T} : ba se t r an s p or t f a c t or$ $β : co mm o n - e mi tt er g ain; F abian an d Z u t i c : t h ese e x p r ess i o n s an d, in t h e i d e a l c a se, α_{f} j_{se} = α_{r} j_{sc} = α_{t} j_{g b}^{n}; t h e β - α r e l a t i o n i s ba c k g r o u n d k n o w l e d g e$ $(co m p u t e d : w_{b} / L_{nb} = 0.1 g i v es α_{T} = 0.9950, w hi c h a l o n e w o u l d a l l o w β o f ab o u t 200)$
Collector-emitter saturation voltage	$V_{C E, s a t} = V_{T} ln [\frac{1 + ( β _{f or ce d} + 1 ) / β _{R}}{1 - β _{f or ce d} / β _{F}}]$	$β_{f or ce d} = \frac{I _{C}}{I _{B}} < β_{F}$	$V_{C E, s a t} : co l l ec t or - e mi tt er v o l t a g e in s a t u r a t i o n$ $β_{f or ce d} : r a t i o o f co l l ec t or t o ba se c u r r e n t se t b y t h e c i r c u i t$ $β_{F}, β_{R} : f or w a r d an d r e v er se co mm o n - e mi tt er g ain s (E b er s - M o l l r es u l t, ba c k g r o u n d k n o w l e d g e); G u mm e l an d P oo n : E b er s - M o l l co v er s s a t u r a t i o n$ $(co m p u t e d : β_{F} = 100, β_{R} = 1 an d β_{f or ce d} = 10 g i v e 67 mV a t 27^{\circ} C; t h e o hmi c d r o p a cr oss R C a dd s t o t hi s in a r e a l d e v i ce)$

Gummel-Poon DC Model

Name	Formula	Related Formulae	Variables/Constants
Gummel-Poon transfer current and normalized base charge	$I_{T} = \frac{I _{F} - I _{R}}{q _{b}}, q_{b} = \frac{q _{1}}{2} (1 + 1 + 4 q_{2})$	$I_{F} = I S (e^{V_{B E} / (N F V_{T})} - 1), I_{R} = I S (e^{V_{B C} / (N R V_{T})} - 1), q_{2} = \frac{I _{F}}{I K F} + \frac{I _{R}}{I K R}$	$I_{T} : t r an s f er c u r r e n t$ $I_{F}, I_{R} : f or w a r d an d r e v er se d i f f u s i o n c u r r e n t s$ $q_{b} : n or ma l i z e d maj or i t y ba se c ha r g e (1 a t z er o bia s)$ $q_{1} : E a r l y - e f f ec t f a c t or$ $q_{2} : hi g h - inj ec t i o n f a c t or$ $I S : t r an s p or t s a t u r a t i o n c u r r e n t (d e f a u l t 1 \times 1 0^{- 16} A)$ $N F, N R : e mi ss i o n coe f f i c i e n t s (d e f a u l t 1), I K F$ $I K R : f or w a r d an d r e v er se k n ee c u r r e n t s (d e f a u l t in f ini t e); Q u cs, M a t hW or k s an d T U D e l f t : t h e s am e s t r u c t u r e; M a t hW or k s : a s m oo t h e d s q u a r e r oo t w i t h ϵ = 1 \times 1 0^{- 4} an d an e x p o n e n t ia l l in e a r i z e d ab o v e 80 t h er ma l v o l t a g es; T U D e l f t : hi g h inj ec t i o n ha l v es t h e s l o p e o f I_{C} (V_{B E}) o n t h e G u mm e l pl o t, w i t h I K F a s t h e k n ee; G u mm e l an d P oo n : an in t e g r a l c ha r g e - co n t r o l m o d e l b u i l t o n t h e ba se c ha r g e$ $(co m p u t e d : q_{b} = 1.618 a t I_{F} = I K F, so I_{C} = 0.618 I K F t h er e; w e l l ab o v e I K F, q_{b} g r o w s a s I_{F} / I K F an d I_{C} a s I_{F} I K F, ha l v in g t h e G u mm e l - pl o t s l o p e)$
Base currents and low-current gain	$I_{B} = \frac{I _{F}}{B F} + I S E (e^{V_{B E} / (N E V_{T})} - 1) + \frac{I _{R}}{B R} + I S C (e^{V_{B C} / (N C V_{T})} - 1)$	$I_{C} = I_{T} - \frac{I _{R}}{B R} - I S C (e^{V_{B C} / (N C V_{T})} - 1)$	$B F, B R : i d e a l ma x im u m f or w a r d an d r e v er se g ain s (d e f a u l t s 100 an d 1), I S E$ $N E : ba se - e mi tt er l e ak a g e s a t u r a t i o n c u r r e n t an d e mi ss i o n coe f f i c i e n t (d e f a u l t s 0 an d 1.5), I S C$ $N C : ba se - co l l ec t or l e ak a g e p a r am e t er s (d e f a u l t s 0 an d 2); Q u cs : i d e a l an d l e ak a g e ba se c u r r e n t s I_{B E I} = I_{F} / B_{F}, I_{B E N}, I_{B C I}, I_{B C N}; M a t hW or k s : t h e t er mina l c u r r e n t s I_{B} an d I_{C} in t hi s f or m$ $(co m p u t e d : I S = 1 \times 1 0^{- 16} A, B F = 100, I S E = 1 \times 1 0^{- 15} A an d N E = 1.5 (a ss u m e d) g i v e β o f 38.6, 69.5 an d 89.2 a t V_{B E} = 0.5, 0.6 an d 0.7 V, a t c u r r e n t s o f 25 nA, 1.2 μ A an d 57 μ A)$
Early effect	$q_{1} = (1 - \frac{V _{B C}}{V A F} - \frac{V _{B E}}{V A R})^{- 1}$	$r_{o} \approx \frac{V A F + V _{C E}}{I _{C}}, I_{C} \approx I_{S} e^{V_{B E} / V_{T}} (1 + \frac{V _{C E}}{V A F})$	$V A F, V A R : f or w a r d an d r e v er se E a r l y v o l t a g es (d e f a u l t in f ini t e)$ $r_{o} : o u tp u t r es i s t an ce$ $V_{C E} : co l l ec t or - e mi tt er v o l t a g e; Q u cs an d M a t hW or k s : q_{1} in t hi s f or m; T U D e l f t : t h e E a r l y e f f ec t a s ba se - w i d t h m o d u l a t i o n b y t h e co l l ec t or - ba se v o l t a g e, V A F f r o m e x t r a p o l a t in g I_{C} (V_{C E}) t o t h e v o l t a g e a x i s, an d t h e s im pl i f i e d f or m s f or I_{C} an d r_{o}$ $(co m p u t e d : V A F = 20 V an d V_{B C} = - 2.3 V g i v e q_{1} = 0.897, r ai s in g I_{C} b y 11.5 p er ce n t; a t 1 mA an d V_{C E} = 3 V, r_{o} = 23 kΩ)$
Current-dependent base resistance	$r_{b b^{'}} = R B M + \frac{R B - R B M}{q _{b}} (I R B infinite)$	$r_{b b^{'}} = R B M + 3 (R B - R B M) \frac{t a n z - z}{z t a n ^{2} z}, z = \frac{1 + 144 I _{B} / ( π ^{2} I R B ) - 1}{( 24/ π ^{2} ) I _{B} / I R B}$	$r_{b b^{'}} : ba se r es i s t an ce$ $R B : z er o - bia s ba se r es i s t an ce$ $R B M : minim u m ba se r es i s t an ce a t hi g h c u r r e n t$ $I R B : c u r r e n t a t w hi c h r_{b b^{'}} i s ha l f w a y t o R B M$ $z : a ux i l ia r y v a r iab l e; Q u cs : r_{b b^{'}} f a l l s f r o m R B t o w a r d R B M, r e a c hin g t h e mi d p o in t a t I_{B} = I R B, w i t h t h e z e x p r ess i o n; M a t hW or k s : b o t h f or m s$ $(co m p u t e d : a t I_{B} = I R B, z = 1.213 an d t h e f r a c t i o n 3 (tan z - z) / (z tan^{2} z) = 0.506, co n f i r min g t h e mi d p o in t)$

Charge, Frequency and Temperature

Name	Formula	Related Formulae	Variables/Constants
Forward transit time and diffusion charge	$τ_{F} = T F [1 + X T F (\frac{I _{F}}{I _{F} + I T F})^{2} exp (\frac{V _{B C}}{1.44 V T F})], Q_{B E} = \frac{τ _{F} I _{F}}{q _{b}}$	$Q_{B C} = T R \cdot I_{R}, ω_{0} = \frac{1}{P T F \cdot \frac{π}{180} \cdot T F} (excess phase)$	$τ_{F} : bia s - d e p e n d e n t f or w a r d t r an s i t t im e$ $T F : i d e a l f or w a r d t r an s i t t im e$ $X T F : bia s - d e p e n d e n ce coe f f i c i e n t$ $I T F : hi g h - c u r r e n t p a r am e t er$ $V T F : ba se - co l l ec t or v o l t a g e d e p e n d e n ce (d e f a u l t in f ini t e, r e m o v in g t h e e x p o n e n t ia l)$ $Q_{B E}, Q_{B C} : d i f f u s i o n c ha r g es$ $T R : r e v er se t r an s i t t im e$ $P T F : e x cess p ha se in d e g r ees a t 1/ (2 π T F)$ $ω_{0} : cor n er o f t h e e x cess - p ha se f i l t er (ω_{0} f or m, ba c k g r o u n d k n o w l e d g e); Q u cs : τ_{F} in t hi s f or m, C_{B E, d i f f} f r o m Q_{B E} = I_{F} T_{F F} / Q_{B}, C_{B C, d i f f} = T_{R} g_{I R}, an d a f i l t er 3 ω_{0}^{2} / (s^{2} + 3 ω_{0} s + 3 ω_{0}^{2}) o n t h e f or w a r d c u r r e n t; M a t hW or k s : t h e s am e τ_{F} d i v i d e d b y q_{b}; T U D e l f t : a d i f f er e n t g r o u p in g (see t h e Q / A)$ $(co m p u t e d : w i t h X T F = 2 (a ss u m e d) an d V T F in f ini t e, τ_{F} = 1.5 T F a t I_{F} = I T F in t h e S P I C E f or m an d 2 T F in t h e T U D e l f t g r o u p in g)$
Small-signal parameters and cut-off frequencies	$g_{m} = \frac{I _{C}}{V _{T}}, r_{π} = \frac{B F}{g _{m}}, c_{π} = C J E + T F \cdot g_{m}, f_{T} = \frac{g _{m}}{2 π ( c _{π} + c _{μ} )}$	$f_{ma x} = \frac{f _{T}}{8 π r _{b} c _{μ}}, f_{T, ma x} \approx \frac{1}{2 π T F}$	$g_{m} : t r an sco n d u c t an ce, r_{π} : in p u t r es i s t an ce, c_{π} : ba se - e mi tt er c a p a c i t an ce$ $C J E : ba se - e mi tt er d e pl e t i o n c a p a c i t an ce, c_{μ} : ba se - co l l ec t or c a p a c i t an ce (ab o u t C J C)$ $f_{T} : t r an s i t i o n f r e q u e n cy$ $f_{ma x} : ma x im u m osc i l l a t i o n f r e q u e n cy$ $r_{b} : ba se r es i s t an ce; T U D e l f t : t h ese s im pl i f i e d h y b r i d - p i f or m s, U_{T} = 25.26 mV a t 293 K, f_{T} l imi t e d b y d e pl e t i o n c a p a c i t an ce a t l o w c u r r e n t an d b y T F a t hi g h c u r r e n t, an d t h e m y N P N p a r am e t er s I S = 0.5 fA, B F = 100, T F = 1 ns, C J E = 5 pF, C J C = 1 pF, V A F = 20 V, I T F = 20 mA$ $(co m p u t e d : m y N P N a t 1 mA an d 293 K g i v es g_{m} = 39.6 mS, r_{π} = 2.52 kΩ, c_{π} = 44.6 pF an d f_{T} = 138 MHz, a g ain s t f_{T, ma x} = 159 MHz; w i t h an a ss u m e d r_{b} = 100 Ω, f_{ma x} = 235 MHz)$
Temperature scaling	$I S (T) = I S (\frac{T}{T _{0}})^{X T I} exp [(\frac{T}{T _{0}} - 1) \frac{q E G}{k _{B} T}]$	$B F (T) = B F (\frac{T}{T _{0}})^{X T B}, I S E (T) = I S E (\frac{T}{T _{0}})^{- X T B} (\frac{I S ( T )}{I S})^{1/ N E}$	$T_{0} : m e a s u r e m e n t t e m p er a t u r e (T N O M, 27^{\circ} C)$ $X T I : I S t e m p er a t u r e e x p o n e n t (d e f a u l t 3)$ $E G : a c t i v a t i o n e n er g y (d e f a u l t 1.11 eV)$ $X T B : g ain t e m p er a t u r e e x p o n e n t (d e f a u l t 0); M a t hW or k s : t h ese e q u a t i o n s, t h e s am e E_{g} (T), V_{J} (T) an d C_{J} (T) f or m s a s t h e d i o d e m o d e l, an d B R sc a l e d l ik e B F; T U D e l f t : β_{F} (T) = B F (T / T_{0})^{X T B}; Q u cs : X T I, X T B an d E G d e f a u l t s; d i o d e t e m p er a t u r e f or m s o n t h e D i o d e C o m p a c t M o d e l s p a g e$ $(co m p u t e d : I S r i ses b y a f a c t or 728 f r o m 27 t o 77^{\circ} C; X T B = 1.5 (a ss u m e d) r ai ses B F b y 54 p er ce n t f r o m 300 t o 400 K)$

Important Questions

Ques: How does the Gummel-Poon model extend the Ebers-Moll model?

By relating the transfer current to the majority charge in the base. Gummel and Poon note that Ebers-Moll, introduced in 1954, is physics-based and covers active, saturated and cut-off operation but misses effects such as the Early effect, and that charge control suits transient analysis. Their 1970 model divides the transfer current by $q_{b} = (q_{1} /2) (1 + 1 + 4 q_{2})$ , where $q_{1}$ carries the forward and reverse Early voltages and $q_{2}$ the knee currents IKF and IKR (Qucs). It also adds leakage base currents with emission coefficients NE and NC, a bias-dependent transit time and a current-dependent base resistance. With $q_{b} = 1$ , no leakage and constant transit times, Gummel-Poon reduces to the transport form of Ebers-Moll (background knowledge), which is why both use the same IS, BF and BR. Reference: H. K. Gummel and H. C. Poon, ‘An Integral Charge Control Model of Bipolar Transistors’, Bell Labs publication page (May 1970; abstract only; Bell System Technical Journal 49, 827-852, background knowledge) (fetch-tool summary); Qucs project, ‘Qucs Technical Papers’, BJT (SPICE Gummel-Poon) model section, qucs.sourceforge.net/tech/node70 (fetch-tool summary).

Ques: Why is the saturation voltage of a BJT only tens of millivolts?

Because both junctions are forward biased and their voltages nearly cancel. In the Ebers-Moll picture, Fabian and Zutic’s terminal currents $j_{c} = α_{f} j_{f} - j_{r}$ and $j_{e} = j_{f} - α_{r} j_{r}$ let the collector current fall below $β_{F} I_{B}$ when the collector junction also injects. Solving for the voltage gives $V_{C E, s a t} = V_{T} ln {[1 + (β_{f or ce d} + 1) / β_{R}] / (1 - β_{f or ce d} / β_{F})}$ (background knowledge). For $β_{F} = 100$ , $β_{R} = 1$ and a forced gain of 10 this is 67 $mV$ at 27 $^{\circ} C$ (computed). A low reverse gain, typical of planar transistors with a lightly doped collector, raises it. Measured values of 0.1 to 0.3 $V$ include the collector series resistance, which the Gummel-Poon model adds as RC. The stored charge in saturation, set by TR, also slows turn-off. Reference: J. Fabian and I. Zutic, ‘The Ebers-Moll model for magnetic bipolar transistors’, arXiv:cond-mat/0409196 (conventional model recovered by setting the spin terms to zero) (fetch-tool summary).

Ques: How does the normalized base charge model high injection?

Through $q_{2} = I_{F} / I K F + I_{R} / I K R$ inside $q_{b} = (q_{1} /2) (1 + 1 + 4 q_{2})$ (Qucs). At low current $q_{2}$ is negligible and $q_{b}$ equals the Early factor $q_{1}$ . At $I_{F} = I K F$ , $q_{b} = 1.618$ , so the collector current is 0.618 times IKF (computed). Far above the knee $q_{b} \approx q_{2}$ , so $I_{C} \approx I_{F} I K F$ , which grows as $exp (V_{B E} /2 V_{T})$ (computed). TU Delft describes the same effect: when the excess minority density in the base exceeds the doping, the slope of the Gummel plot falls by a factor of two, with IKF setting the knee. Because the base current keeps its ideal slope, the gain falls at high current. MathWorks smooths the square root with $ϵ = 1 0^{- 4}$ to keep $q_{b}$ well behaved during iteration. Reference: Qucs project, ‘Qucs Technical Papers’, BJT (SPICE Gummel-Poon) model section, qucs.sourceforge.net/tech/node70 (fetch-tool summary); A. Montagne, ‘Bipolar transistors’, Structured Electronics Design web book, TU Delft (2023) (fetch-tool summary); MathWorks, ‘SPICE NPN’, Simscape Electrical block reference page (fetch-tool summary).

Ques: Which form of the forward transit-time equation is correct?

The SPICE form is $τ_{F} = T F [1 + X T F (I_{F} / (I_{F} + I T F))^{2} exp (V_{B C} / (1.44 V T F))]$ , as written by Qucs and by MathWorks, which also divides the resulting charge by $q_{b}$ . TU Delft writes $T F [1 + (X T F I_{t f} / (I_{t f} + I T F))^{2}] exp (V_{b^{'} c^{'}} / (1.44 V T F))$ , squaring XTF and applying the exponential to the whole bracket. For an assumed XTF = 2 at $I_{F} = I T F$ with VTF infinite, the SPICE form gives 1.5 TF and the TU Delft grouping 2 TF (computed); with finite VTF the TU Delft form would also scale the constant TF with $V_{B C}$ . The two independent simulator descriptions agree, and in SPICE source code XTF multiplies the squared current ratio (background knowledge), so the SPICE form is far more likely to be correct and the TU Delft version a transcription slip (my reading). Reference: Qucs project, ‘Qucs Technical Papers’, BJT (SPICE Gummel-Poon) model section, qucs.sourceforge.net/tech/node70 (fetch-tool summary); MathWorks, ‘SPICE NPN’, Simscape Electrical block reference page (fetch-tool summary); A. Montagne, ‘Bipolar transistors’, Structured Electronics Design web book, TU Delft (2023) (fetch-tool summary).

Ques: Why does the current gain fall at low collector current?

Because the leakage base current has a larger emission coefficient. The Gummel-Poon base current adds $I S E (exp (V_{B E} / (N E V_{T})) - 1)$ to the ideal $I_{F} / B F$ (Qucs; MathWorks). With NE = 1.5 against NF = 1, this term falls more slowly than the collector current as $V_{B E}$ drops. For IS = $1 \times 1 0^{- 16}$ $A$ , BF = 100 and an assumed ISE = $1 \times 1 0^{- 15}$ $A$ , the gain is 89.2 at 57 $μ A$ , 69.5 at 1.2 $μ A$ and 38.6 at 25 $nA$ (computed). On a Gummel plot the base current bends upward at low bias, and NE between 1.5 and 2 reflects recombination in the emitter-base depletion region (background knowledge). Combined with the high-injection fall above IKF, this gives the typical bell-shaped gain against collector current. Reference: Qucs project, ‘Qucs Technical Papers’, BJT (SPICE Gummel-Poon) model section, qucs.sourceforge.net/tech/node70 (fetch-tool summary); MathWorks, ‘SPICE NPN’, Simscape Electrical block reference page (fetch-tool summary).

Ques: Why do the MathWorks resistance defaults differ from SPICE?

Because the MathWorks block scales resistances by area and sets small non-zero values. Qucs, following SPICE, gives RB, RC and RE defaults of zero, and the base resistance only appears when RB is set. MathWorks lists RB = 1, RE = $1 \times 1 0^{- 4}$ and RC = 0.01 $m^{2} Ω$ , divided by AREA times SCALE, so a device of unit area has 1 $Ω$ of base, 0.1 $mΩ$ of emitter and 10 $mΩ$ of collector resistance (computed). These are too small to matter in most circuits, but they change results at very high currents and remove internal nodes that a zero value would collapse. They are probably chosen for numerical robustness rather than physics (my reading). MathWorks also uses VTF both for $N F k_{B} T / q$ and for the transit-time parameter, which should not be confused. Model cards should set RB, RE and RC explicitly. Reference: Qucs project, ‘Qucs Technical Papers’, BJT (SPICE Gummel-Poon) model section, qucs.sourceforge.net/tech/node70 (fetch-tool summary); MathWorks, ‘SPICE NPN’, Simscape Electrical block reference page (fetch-tool summary).

Ques: How are fT and fmax estimated from Gummel-Poon parameters?

With the hybrid-pi forms TU Delft gives: $g_{m} = I_{C} / V_{T}$ , $c_{π} = C J E + T F g_{m}$ , $c_{μ} = C J C$ , $f_{T} = g_{m} / [2 π (c_{π} + c_{μ})]$ and $f_{ma x} = f_{T} / (8 π r_{b} c_{μ})$ . For TU Delft’s myNPN at 1 $mA$ and 293 $K$ , $g_{m} = 39.6$ $mS$ and $c_{π} = 44.6$ $pF$ , so $f_{T} = 138$ $MHz$ , close to the 159 $MHz$ limit $1/ (2 π T F)$ for TF = 1 $ns$ (computed). With an assumed 100 $Ω$ base resistance, $f_{ma x}$ is 235 $MHz$ (computed). At low current CJE dominates and $f_{T}$ rises with current; at high current XTF, ITF and VTF raise the transit time and $f_{T}$ falls, the Kirk-effect roll-off (TU Delft; Qucs). Real extraction fits TF, XTF, ITF and VTF to measured $f_{T}$ against current at several $V_{C E}$ (background knowledge). Reference: A. Montagne, ‘Bipolar transistors’, Structured Electronics Design web book, TU Delft (2023) (fetch-tool summary); Qucs project, ‘Qucs Technical Papers’, BJT (SPICE Gummel-Poon) model section, qucs.sourceforge.net/tech/node70 (fetch-tool summary).

References

H. K. Gummel and H. C. Poon, “An Integral Charge Control Model of Bipolar Transistors,” Bell System Technical Journal, vol. 49, no. 5, pp. 827-852, May 1970 (abstract on the Bell Labs publications page). Available: https://nokia.com/bell-labs/publications-and-media/publications/an-integral-charge-control-model-of-bipolar-transistors
Qucs project, “Qucs Technical Papers,” bipolar junction transistor (SPICE Gummel-Poon) section. Available: https://qucs.sourceforge.net/tech/node70.html
J. Fabian and I. Zutic, “The Ebers-Moll model for magnetic bipolar transistors,” arXiv:cond-mat/0409196. Available: https://arxiv.org/abs/cond-mat/0409196
A. Montagne, “Bipolar transistors,” Structured Electronics Design, TU Delft web book, 2023. Available: https://analog-electronics.tudelft.nl/webbook/SED/_build/html/ModelingActiveDevices/ActiveDevices_Bipolar_transistors.html
MathWorks, “SPICE NPN,” Simscape Electrical documentation, block reference page. Available: https://www.mathworks.com/help/sps/ref/spicenpn.html

Quartz 5

Explorer

Ebers-Moll and Gummel-Poon Models