Current-voltage characteristics

The following assumptions are involved when deriving ideal current voltage characteristics of BJT

• Low level injection
• Uniform doping in each region with the abrupt junctions
• One dimensional current
• Negligible recombination generation in the space charge regions
• Negligible electric fields outside the space charge regions.

It is important to characterize minority diffusion currents induced by the injection of carriers. With regard to the PN junction diode, a key relation is diffusion equation.

A solution of this equation is given below, and two boundary conditions are used to solve and find C₁ and C₂.

The following equations apply to emitter and collector region, respectively, and origins 0, 0', and 0'' apply to base, collector, and emitter.

A boundary condition of emitter is given below:

The values of constants A₁ and B₁ are zero because of the following conditions of emitter and collector regions as  and .

Because A₁ = B₁ = 0, values of Δn_E(0'') and Δn_c(0') are A₂ and B₂, respectively.

Expressions of I_En and I_Cn can be evaluated.

Because insignificant recombination takes place, the second derivative of Δp_B(x) is zero. There is therefore a linear relationship among excess hole density and x.

Following are the boundary conditions of Δp_B.

Substitute into above linear relation.

With this result, now derive value of I_Ep.

Use expressions of I_Ep, IEn, Δp_B(0), and Δp_B(W) to develop an expression of emitter current.

Likewise, an expression of collector current is derived.

An expression of base current is found with previous results.

Punchthrough

When base collector voltage reaches a certain value, base collector depletion region boundary meets the base-emitter depletion region boundary. When in this state transistor effectively has no base. The device loses all gain when in this state.

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