Analysis of CE, CB and CC Amplifiers using re Model

Analysis of CE, CB and CC Amplifiers using r_e Model

AU : Dec.-18

• The r-parameters (resistance -parameters) are perhaps easier to work with than h-parameters. Let us see the r-parameter model for transistor. It is shown in the Fig. 6.10.1. It consists of five r-parameters as listed in Table 1.4. The rg is the ac emitter resistance, rc is the a.c. collector resistance and rb is the a.c. base resistance. The current source is represented by

 

1. Relationships of r-Parameters and h-Parameters

• Data sheets often specify the common emitter h-parameters of the transistor. In such situations, we have to convert these parameters into r-parameters to analyze transistor amplifier using r-parameters. The conversion formulae are as follows :

 

2. Simplified r-Parameter Model for BJT

• In a forward biased condition of base-emitter junction rb is very small and its effect is also small enough to neglect it. Thus it can be replaced by a short circuit. The collector-base junction is always reverse biased in the active region. In this biasing state, a.c. collector resistance rc is usually several hundred kilo-ohms, and hence it can be replaced by an open circuit. This gives the simplified r-parameter model for BJT as shown in the Fig. 6.10.2.

 

Ex. 6.10.1 Determine the r-parameters of a transistor if the h-parameters are hie = 1.1 k Ω hfe = 50, hoe = 25 µA/V and hre = 2.5 × 10^-4

Sol. : Referring Table 6.10.1 and equation (6.10.1) through equation (6.10.5) we have,

 

3. r-Parameter Model for CB Configuration

• The simplified r-parameter shown in the Fig. 6.10.3 is nothing but the r-parameter model for common base configuration. However, in this section we are going see the determination of r-parameter model for common base configuration in a different way.

• The re model employs a diode and controlled current source to duplicate the behaviour of a transistor in the active region. The Fig. 6.10.3 shows the common base transistor configuration and its rg model. The forward biased base emitter junction in the active region is replaced by the PN junction diode. The forward biased base emitter junction causes collector current to flow which intum depends upon the base current. This input, output current relationship is represented by a controlled current source whose value is

• We know that, a.c. resistance of diode can be determined by the equation

r_ac = 26mV / I_D  … (6.10.6)

where I_D is the d.c. current through the diode at the Q point. This same equation can be used to find the a.c. resistance of the diode in Fig. 6.10.3 (b). If we substitute diode current equal to I_E we get

• The subscript e or rg is chosen to emphasize that it is the d.c. level of emitter current that determines the a.c. resistance of the diode in the Fig. 6.10.3 (b). Replacing the diode by its equivalent resistance we get the r-parameter model for circuit, as shown in the Fig. 6.10.3.

 • Input Resistance : Due to the isolation that exists between input and output circuit of Fig. 6.10.3, it is fairly obvious that the input resistance for the common base configuration of a transistor is simply r_e . That is

R₁ = r_e         ... (6.10.8)

• The typical value of R;, for common base configuration range from a few ohms to a maximum of about 50 Ω .

Output Resistance : To determine the output resistance we have to make signal zero i.e. I_e = 0. When I_e = 0, ɑ I_e is also zero and output is open-circuited. Therefore,

R_o = ∞  ... (6.10.9)

Voltage Gain : The voltage gain is given as

 A_V  = V_o / V_i

• It can be determined considering the load connected at the output terminals as shown in the Fig. 6.10.4.

• Looking at Fig. 6.10.4,

Current Gain : The current gain is given as

 

Ex. 6.10.2 Determine the value of r_e when d.c. emitter current at operating point is 2.5 mA.

Sol. : The re is given as

Ex. 6.10.3 Determine the If, Ri, A^ A? R° and power gain for the common base amplifier shown in Fig. 6.10.5.

Sol. :

The r-parameter equivalent circuit for a given amplifier is shown in Fig. 6.10.6. It is drawn by short circuiting d.c. source and all capacitors, and replacing transistor by its r-parameter model.

To determine rg we have to first find IE. It is calculated as follows :

 

4. r-Parameter Model for CE Configuration

• We know, for common emitter configuration, the input terminals are the base and the emitter and output terminals are collector and emitter. The Fig. 6.10.7 shows the common emitter configuration and its rg model. Here, also the rg model employs a diode and controlled current source to duplicate the behaviour of a transistor in the active region. The forward biased base emitter junction in the active region is represented by the diode and controlled current source is still connected between the collector and the base terminals.

• Looking at the model we have current through diode equal to

• The input resistance is given by

• The voltage Vbe is across the diode resistance as shown in Fig. 6.10.8. The level of rg is still determined by the d.c. current I£. Using Ohm's law gives

• The typical value of Riz for common emitter configuration range from a few hundred ohms to maximum about 6-7 kilo ohms.

• Output resistance : The rg model in Fig. 6.10.9 does not include an output resistance. However, if it is available from a graphical analysis or from data sheets it can be included as shown in Fig. 6.10.9.

• When input is short circuited, Ib = 0 and I = 0 and output resistance is given as

R_o = r_o         ... (6.10.14)

• If r_o is ignored from r_e model of CE, the output resistance is ∞

Voltage Gain : The voltage gain is given as

A_v = V_o / V_i

• It can be determined considering the load connected at the output terminals and output resistance as shown in Fig. 6.10.10.

• The negative sign in the resulting equation for indicates that a 180 ° phase shift occurs between the input and output signals.

Current Gain : The current gain is given as

• Using the derived values of input resistance (P rg), the collector current (P Ib), and the output resistance (rQ), the equivalent model for common emitter configuration can be given as shown in the Fig. 6.10.11.

a. Analysis of Common Emitter Fixed Bias Configuration

• Let us consider a common emitter amplifier with a bias, as shown in Fig. 6.10.12.

• By short-circuiting C₁, C₂ and dc power supply, and replacing transistor with its equivalent r-parameter model (Refer Fig. 6.10.11) we get the r-parameter equivalent circuit for given transistor amplifier as shown in Fig. 6.10.11.

• To find the rg we have to first determine the IE. It can be calculated as follows :

D.C. Analysis :

b. Analysis of Common Emitter Voltage Divider Bias Configuration

• Let us consider a common emitter amplifier with a voltage divider bias, as shown in Fig. 6.10.15.

• By short circuiting C₁, C₂, C_E and d.c. power supply, and replacing transistor with its equivalent r-parameter model we get the r-parameter equivalent circuit for given transistor amplifier, as shown in Fig. 6.10.16.

• To find the re we have to first determine the IE. It can be calculated as follows :

• D.C. Analysis : If β R_E > 10 R₂ we can use approximate approach.

100 × 11 K > 10 × 10 K

110 K > 100 K

Using the approximate approach we have,

The equivalent circuit shown in Fig. 6.10.16 and Fig. 6.10.13 are same. Thus by using same analysis we have,

Voltage Gain :

 

Ex. 6.10.4 For the circuit shown below, find i) d.c. bias levels ii) d.c. voltages across the capacitors iii) a.c. emitter resistance iv) voltage gain and v) state of the transistor.

Sol. : i) DC bias levels

DC voltage across R₂ (V₂)

v) State of the transistor

As calculated, V_C = 10.446 V and V_E = 2.3 V

Since V_C >V_E, the transistor is in active state.

c. Analysis of Common Emitter with Unbypassed R_E

• Let us consider a common emitter amplifier with unbypassed RE, as shown in Fig. 6.10.18.

• By short-circuiting C₁, C₁ and d.c. power supply, and replacing transistor with its equivalent r-parameter model we get the r-parameter equivalent circuit for given transistor amplifier, as shown in Fig. 6.10.19.

D.C. Analysis :

Output Resistance : With V_i set to zero, I_b = 0, β I_b = 0 and (1 + β) I_b = 0, and can be replaced by open circuit.

R_o = R_C

= 3.3 kΩ  ... (6.10.21)

Voltage Gain : A_V = V_o / V_i

where

Substituting value of I_c from equation (6.10.19) we get

Approximate A.C. Analysis :

• The Fig. 6.10.20 shows the approximate r-parameter equivalent circuit for common emitter amplifier with unbypassed RE, In the approximate A.C. analysis we have assumed r_o = ∞

Input Resistance :

Note : The analysis approach (Exact or approximate) can be selected by testing following two conditions :

1. r_o ≥ 10 (R_C + R_E) and 2. r_o ≥ 10 R_C

If both the conditions are satisfied we can use approximate approach.

d. Analysis of Common Emitter with Collector Feedback Configuration

• Let us consider a common emitter amplifier with a collector feedback configuration, as shown in the Fig. 6.10.21.

• Applying Miller's theorem (Refer section 6.7.1) we have,

• Therefore, the r-parameter equivalent circuit for given amplifier circuit is as shown in the Fig. 6.10.22.

D.C. Analysis :

Output Resistance

When V_i = 0, I_b = 0 and β I_b = 0 and they can be treated as open circuit.

5. r-Parameter Model for CC Configuration

• For the common collector configuration the model defined for the common emitter configuration with unbypassed RE is applied rather than defining a model for the common collector configuration. In common collector configuration the output is taken from the emitter terminal. Thus resistance Pre appears between input and output, and the direction of current source gets exactly reverse to maintain current flow from collector to emitter. The Fig. 6.10.23 shows the modified model of common-emitter configuration which can be used to analyse common collector amplifier.

• Let us consider a common collector amplifier as shown in the Fig. 6.10.24.

• By short circuiting Clz C2 and d.c. power supply, and replacing transistor with its equivalent r-parameter model we get the r-parameter equivalent circuit for given transistor amplifier, as shown in the Fig. 6.10.25.

D.C. Analysis :

Output Resistance :

• When V_i is set to zero, Ife becomes theoretically zero which eventually makes Ie zero. Students may get confused while calculating R _o due to this theoretical aspect.

• Practically, in this case it is necessary to consider the effect of Ife at the output side, while calculating R _o. This can be mathematically expressed as,

We know that

I _b = V_i / R _i

The I_b can be expressed interms of I_e as,

I_e = (β + 1) I_b = (β + 1) V_i / R_i

Substituting R_i in above equation we get,

The relationship of Ie and Vi given by above equation can be represented by network shown in the Fig. 6.10.26

Approximate A.C. Analysis

• The Fig. 6.10.27 shows the approximate r-parameter equivalent circuit for common collector amplifier. In the approximate A.C. analysis we have assumed r_o = ∞

The relationship of Ie and Vi given by above equation can be represented by network shown in the Fig. 6.10.28

Review Questions

1. Explain the rg transistor model.

2. Give the relationship between r-parameter and h-parameters.