Forward-Active Terminal Currents Collector current: (electron diffusion current density) x (emitter area) diff J n AE qd n n po A E V E V th ------------------------------ e W (why minus sign? is by def. positive-in, opposite to x direction) ase current: (reverse-injected hole diffusion current density) x(emitter area) (minus sign again reflects positive-in convention for vs. x direction) diff J pe AE qd p p neo A E V E V th ------------------------------ ( e 1) W E
Forward-Active Current Gains Emitter current: Kirchoff s current law --> -( ) diff diff ( J n J pe )A E qd p p neo A E ------------------------------ qd n A n po E ------------------------------ e V E V th W E W The ratio of collector current to the magnitude of the emitter current is defined as alpha-f -------- qd n n po A E ------------------------------ W -------------------------------------------------------------------------------- qd p p neo A E ------------------------------ qd n A α F ------------------------------ n po E W E W α F --> 1... typically, α F 0.99.
Current Gains (cont.) The ratio of collector current to base current can be found in terms of α F : 1 α F ------ I α C --------------- F α F Solving for as a function of, we find that α F --------------- I β 1 α F F. A typical value is β F 100... with an uncertainty of /- 50% since it is a sensitive function of the parameters and internal dimensions of the JT.
The Saturation Region V CE(sat) 0.1 V (approx) from the characteristics --> both the emitter-base and the base-collector junctions are forward-biased Law of the Junction --> find minority carrier concentrations in the emitter, base, and the collector p ne (x) n p (x) p nc (x) emitter base collector edge of n buried layer W E x E x E 0 W W x C x electrons from -E junction electrons from -C junction oth junctions are injecting and both are also collecting... since the electric field in the depletion region remains in the same direction under forward bias. Separate the electron diffusion current in the base into two components: one due to the emitter-base junction (with zero bias on the base-collector junction) and the other due to the base-collector junction: diff J n n po ( e V E V th 1) n po ( e V C V th 1) qd n ----------------------------------------------- qd W n ------------------------------------------------ W
Ebers-Moll Model Electron diffusion current in the base: multiply by the emitter area I diff I S ( e V E V th 1) I S ( e V C V th 1) I 1 I 2 Emitter current : three components 1. - I 1 due to injection of electrons from the emitter-base junction, 2. - I 1 / β F due to reverse injection of holes into the emitter, and 3. I 2 due to collection of electrons from the base-collector junction. 1 I 1 ( I 1 β F ) I 2 1 ----- I1 I β 2 F 1 ------ I1 I α 2 F Collector current : three components (by symmetry) 1. - I 2 due to injection of electrons from the base-collector junction, 2. - I 2 / β R due to reverse injection of holes into the collector, and 3. I 1 due to collection of electrons from the emitter-base junction I 2 I 1 I 2 ------ I 1 β 1 1 ------ 1 I2 I R β 1 ------ I2 R α R β R α R / ( 1 - α R ) is the reverse current gain
Ebers-Moll Model (cont.) Standard form for Ebers-Moll equations: define two new constants Emitter current: S I S / α F and S I S / α R, S ( e V E V th 1) α R S ( e V C V th 1) Collector current: α F S ( e V E V th 1) S ( e V C V th 1) The collector current and the emitter current represent two diodes with currentcontrolled current sources coupling the emitter and the collector branches
Carrier Fluxes in Saturation oth junctions injecting and collecting; holes injected into collector recombine with electrons upon reaching the n buried layer For bias condition shown, > 0... injection from emitter-base junction still dominates. (could have 0 or even < 0) hole diffusion flux n polysilicon n emitter majority hole flux from base contact n buried layer n-type collector minority hole diffusion flux majority electron flux from collector contact to recombine with hole diffusion flux,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, majority electrons electron diffusion p-type base majority electron flux from collector contact supplying injection into base majority electron flux to collector contact
Ebers-Moll Equivalent Circuit C α F I F I R Diode Currents: I F S (e V E V th 1) I R S (e V C V th 1) I F α R I R E This model for the JT applies to general device structures, with the four parameters S, S, α F, and α R being linked by reciprocity α F S α R S I S Ebers-Moll must be simplified for hand calculation
Forward Active Model I R is negligible --> can neglect current through -C diode C I F α F C α F I F I R E Emitter current control I F α R I R C β F E E ase current control Eliminate forward-biased diode by replacing with a 0.7 V battery: β F - _ 0.7 V C _ 0.7 V C β F E E
Saturation Include both diodes in the circuit... both as batteries C _ 0.7 V _ 0.1 V E note that the batteries make the controlled current sources irrelevent to the circuit.
Small-Signal Model of the Forward-Active npn JT Transconductance (same concept as for MOSFET): g m i C v E Ebers-Moll (forward-active): i C I S e v E V th i C i C i c slope g m V E V E v be v E 0.2 0.4 0.6 V E v E Evaluating the derivative, we find that g m I S V E V th ------- e ------- V th V th
Input Resistance The collector current is a function of the base current in the forward-active region (recall β F ). At the operating point, we define β o i C i and so i c β o i b. (Note that the DC beta β F and the small-signal β o are both highly variable from device to device) Since the base current is therefore a function of the base-emitter voltage, we define the input resistance r π as: 1 r π i v E i i C i C v E 1 ----- gm β o Solving for the input resistance r π β o β o V th ------ -------------- g m ktβ o ------------ q For a high input resistance (often desirable), we need a high current gain or a low DC bias current.
Output Resistance The Ebers-Moll model has perfect current source behavior in the forward-active region -- actual characteristics show some increase: V An V CE Why? ase width shrinks due to encroachment by base-collector depletion region Approximate model: introduce Early voltage V An to model increase in i C Model: i C I S e v E V th v CE 1 --------- V An Output resistance: 1 r o i C v CE --------- V An
Numerical Values of Small-Signal Elements i b i c base collector v be v π r π g m v π r o v ce emitter Transconductance: 100 µa, V th 25 mv --> g m 4 ms 4 x 10-3 S Note: g m varies linearly with collector current and is independent of device geometry, in contrast to the MOSFET Input resistance: β o 100, 100 µa, V th 25 mv --> r π 25 kω Output resistance: 100 µa, V An 35 V --> r o 350 kω V An Early voltage increases with increasing base width and decreases with decreasing base doping.