Models of Transistors

Fig. 4.9 shows the nonlinear Ebers-Moll model for the n-р-n transistor.

Fig. 4.9. Ebers-Moll model

The currents of injection emitter І₁ and collector І₂ р-n junctions are the managers:

, (4.10)

. (4.11)

Sources of a current α _NI₁ and α_II₂ simulate the phenomena of extraction. Parameters α_N and α_I – are factors of transfer of currents of the transistor in normal and inverse modes.

The Ebers-Moll model more often is used for the analysis of the circuits with CB.

For the analysis of the circuits with CE use nonlinear Gummel-Poon model (Fig. 4.10).

Fig. 4.10. Gummel-Poon model

The source of a current I_CE simulates the phenomena of carry of the minority carriers through base:

, (4.12)

where I_S = α_NI_E0 = α_ІI_C0 – transfer reverse current (saturation) of the transistor.

The sources of a current I_BE and I_BC simulate the phenomena recombination in base in normal and inverse modes:

, (4.13)

, (4.14)

, . (4.15)

The nonlinear capacities С_E and С_C are the sums of the barrier and diffusion components of emitter and collector р-n junctions.

In addition to the already defined alpha current gain, α_N, the primary parameters include resistances in accord with an a. c. equi­valent circuit of a transistor (Fig. 4.11). This is what is known as the T-equi­valent circuit.

Fig. 4.11. T- equivalent linear circuit of a transistor

The emitter resistance r_E is the differential resistance pre­sented by the emitter junction. Similarly, r_C is the sum of the resistances presented by the collector junction and the collector region, but the latter is negligibly small in comparison with the former. The resistance r_B is the series re­sistance of the base.

At high fre­quencies we must also consider the capacitances of the emitter C_de and C_jc collector junctions, and this results in a more elaborate equivalent circuit.

The equivalent circuit derived by applying for a transistor with the common-emitter connection is shown in Fig. 4.12. Here the generator pro­duces a current βI_B, and the resistance of the collector junction is substantially smaller in comparison with what it is in the previous equivalent circuit, being r_C (1 - α) or r_C /(β+1).

Fig. 4.12. T - equivalent linear circuit of a transistor connected in the CE configuration

The h -parameters are usually given by ma­nufacturers in specifications and data sheets for transistors. The h -parameters are convenient to measure, and this is an important advantage because reference sources usually quote average values derived by measuring the parameters of a large number of transistors of a given type.

The relations between alternating currents and voltages in a transistor may be expressed in terms of the h -parameters as follows:

U_BE=h_11EI_B+h_12EU_CE = h_ieI_B + h_reU_CE, (4.16)

I_C = h_21EI_B + h_22EU_CE = h_feI_B + h_oeU_CE .

Equations (4.16) apply to the equivalent circuit shown in Fig. 4.13.

Fig. 4.13. Hybrid-parameter equivalent circuit of a transistor (a) and simple circuit (b)

For a CE circuit the h -parameters can be defined as follows:

Input impedance

h_l1E = ∆U_BE /∆I_B with U_CE = const.

It ranges in value from several hundred ohms to several kilohms.

Reverse voltage feedback ratio

h_12E = ∆U_BE /∆U_CE with I_B = const.

It usually is 10^-3-10^-4 which means that the voltage fed back from output to input is a few thousandths or ten-thousandths of the output voltage.

Forward current gain ratio

h_21E = β = ∆I_C /∆I_B with U_CE = const.

It is anywhere from a few tens to several hundreds.

Output admittance

h_22E = ∆I_C /∆U_CE with I_B = const.

It is equal to a few tenths or hundredths of a microsiemens so that the output resistance, 1/ h_22E, is a few tens of kilohms.

Sometimes a transistor may be represented by an equivalent pi-circuit in which the admittances (Fig. 4.14) are connected to the h -parameters in the following manner:

; g_N = βg_BE; ; g_CE = h_22E.

Fig. 4.14. Equivalent pi-circuit of a transistor

The constant-current generator g_NU_BE in this equivalent circuit accounts for the amplified current produced in the output circuit.

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