Incremental linear models

In our discussion on piece-wise linear models on the preceding section, the small voltage drop between base and emitter is small in comparison with the bias voltage and thus can be neglected.

However, the base-to-emitter voltage cannot be neglected when only the increments of voltage and currents is considered. Also, when calculating increments of current and voltage, it is often necessary to account for the small effects of variations in collector voltage on both the input and output circuits. For these reasons the incremental model for the transistor provides a better approximation than the piece-wise linear approximation.

The base-to-emitter voltage and the collector current are functions of the base current and collector-to-emitter voltage . In other words,

(3.62) and

(3.63) If and are changed in small increments, the resulting increment in can be expressed as (3.64) and the increment in can be written as

(3.65) The partial derivative in the first term of (3.64) has the dimensions of resistance and it is denoted as , and that in the second term is a dimensionless voltage ratio denoted as . It is also conve-nient to denote these derivatives in lower case letters with lower case subscripts. Then, (3.64) is expressed as

(3.66) Likewise, the partial derivative in the first term of (3.65) is a dimensionless current ratio denoted as , and that in the second term has the dimensions of conductance and it is denoted as . Then, (3.65) is expressed as

v_CE = Supply Voltage Load Voltage– = 35 24.24– = 10.76 V

Chapter 3 Bipolar Junction Transistors

(3.67) The relations of (3.66) and (3.67) along with suggest the circuit shown in Figure 3.58 known as the hybrid incremental network model for the transistor. It is referred to as hybrid model because of the mixed set of voltages and currents as indicated by the expressions of (3.66) and (3.67).

The input resistance is the slope of the input voltage and current characteristics and it accounts for the voltage drop across the base-emitter junction. Likewise, the output conductance

is the slope of the output current and voltage characteristics.

Figure 3.58. The hybrid incremental model for a transistor in the common-emitter configuration

The voltage amplification factor is related to the input characteristics caused by a change in , and the current amplification factor is related to the output characteristics caused by a change in .

Typical values for the parameters of relations (3.66) and (3.67) are , , , and , and since the value of is a very small number, the voltage source in Figure 3.58 can be replaced by a short circuit, and thus the model reduces to that shown in Figure 3.59.

Figure 3.59. The hybrid incremental model for the transistor with i_c = βi_b+g_ov_ce

i_e = i_b+i_c

r_n g_o

µv_ce r_n B i_b

v_be

g_o βi_b

E

i_c v_ce^C

i_e

µ

v_CE β

i_B

r_n = 2 KΩ µ = 5 10× ^–4

β = 100 g_o = 2 10× ^–5 Ω^–1 µ

µv_ce

r_n B i_b

v_be

g_o βi_b

E

i_c v_ce^C

i_e

µv_ce = 0

The transistor hybrid parameters^* provide us with a means to evaluate voltages, currents, and power in devices that are connected externally to the transistor. Let us, for example, consider the circuit of Figure 3.60 which is an incremental model for the transistor amplifier in Figure 3.53.

Figure 3.60. Transistor incremental model with external devices Now, we let represent the parallel combination and . Then,

(3.68)

and (3.69)

Hence, the voltage of the is proportional to the current flowing through the source, and from this fact we can replace the voltage source with a resistance , and thus the model of Figure 3.60 can be redrawn as shown in Figure 3.61.

Figure 3.61. The circuit of Figure 3.60 with the voltage source replaced by the resistance

The negative resistance is always much smaller than and thus the net input resistance to the transistor is always positive. Therefore, the negative resistance can be replaced by a short circuit, and assuming that the base current is unaffected by this assumption, the voltages and currents in the collector side of the circuit are not affected.

The current amplification is defined as

* We will introduce the h-equivalent transistor circuits in Section 3.15.

µv_ce

Chapter 3 Bipolar Junction Transistors

(3.70) where

and thus the current amplification is

(3.71) The parameters , , , and are normally denoted by the (hybrid) parameters^* as

, , , and . These designations along

with the additional notations , , , and , provide a symmetrical form for the relations of (3.66)and (3.67) as follows:

(3.72) or

(3.73) In (3.73) the subscript denotes the input impedance with the output short-circuited, the sub-script denotes the reverse transfer voltage ratio with the input terminals open-circuited, the subscript denotes the forward transfer current ratio with the output short circuited, and the

* For a detailed discussion of the , , , and parameters refer to Circuit Analysis II with MATLAB Applications, ISBN 0-9709511-5-9, Orchard Publications.

A_c i_c

subscript denotes the output admittance with the input terminals open-circuited. The second subscript indicates that the parameters apply for the transistor operating in the common-emitter mode. A similar set of symbols with the subscript replacing the subscript denotes the hybrid parameters for a transistor operating in the common-base mode, and a set with the subscript replacing the letter denotes the hybrid parameters for a transistor operating in the common-col-lector mode.

Values for the hybrid parameters at a typical quiescent operating point for the common-emitter mode are provided by the transistor manufacturers. Please refer to the last section of this chapter.

Table 3.3 lists the h-parameter equations for the three bipolar transistor configurations.

Example 3.15

For the amplifier circuit of Figure 3.62 it is known that , ,

, and . Find the small signal current amplification .

Figure 3.62. Transistor amplifier for Example 3.15 Solution:

The incremental model of this transistor amplifier is shown in Figure 3.63 where TABLE 3.3 h-parameter equations for transistors

Parameter Common-Base Common-Emitter Common-Collector o

Chapter 3 Bipolar Junction Transistors

Figure 3.63. The incremental model for the transistor circuit of Figure 3.62

and thus the magnitude of the equivalent resistance reflected into the input part of the circuit is

and since this is much smaller than , it can be neglected.

The current gain can be found from the relation (3.71). Then,

3.12 Transconductance

Another useful parameter used in small signal analysis at high frequencies is the transconductance, denoted as , and defined as

(3.74)

and as a reminder, we denote time-varying quantities with lower case letters and lower case sub-scripts. Thus, the transconductance is the slope at point on the versus characteris-tics at as shown in Figure 3.64.

An approximate value for the transconductance at room temperature is . This relation is derived as follows:

From (3.2)

Figure 3.64. The transconductance defined By substitution of (3.75) into (3.76) we get

(3.77) and with

(3.78) and thus we see that the transconductance is proportional to the collector current . Therefore, a transistor can be viewed as an amplifier with a transconductance of millimhos for each milli-ampere of collector current.