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For a CE amplifier derive the expressions for Av, AI, Zi and Zo.
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TRANSISTOR AMPLIFICATION IN AC DOMAIN:

• The transistor can be employed as an amplifying device, that is, the output ac power is greater than the input ac power.
• The factor that permits an ac power output greater than the input ac power is the applied DC power.
• The amplifier is initially biased for the required DC voltages and currents. Then the ac to be amplified is given as input to the amplifier.
• If the applied ac exceeds the limit set by dc level, clipping of the peak region will result in the output.
• Thus, proper (faithful) amplification design requires that the dc and ac components be sensitive to each other’s requirements and limitations.
• The superposition theorem is applicable for the analysis and design of the dc and ac components of a BJT network, permitting the separation of the analysis of the dc and ac responses of the system.

BJT TRANSISTOR MODELING:

• The key to transistor small-signal analysis is the use of the equivalent circuits (models).
• A Model is a combination of ircuit elements like voltage or current sources,resistors,capacitors etc that best approximates the behavior of a device under specific operating conditions.
• Once the model (ac equivalent circuit) is determined, the schematic symbol for the device can be replaced by the equivalent circuit and the basic methods of circuit analysis applied to determine the desired quantities of the network. Hybrid equivalent network – employed initially.
• The only drawbak is that It is defined for a set of operating conditions that might not match the actual operating conditions.

COMMON EMITTER  BIAS CONFIGURATION:

• Applying KVL to the input side:

$V_{i} = I_{b}\beta_{re} + I_{e}R_{E}$ ..

$V_{i} = I_{b} \beta_{ re} +( \beta+1) I_{b}R_{E}$

• Input impedance looking into the network to the right of RB is

$Z_{i} = V_{i} \dfrac I \beta_{ re}+ ( \beta +1)R_{E}$

• Since b>>1,$( \beta +1) = \beta$

Thus

$Z_{i} = V_{i} I_{b} (\beta_{re}+R_{E})$

• Since RE is often much greater than re,

$Z_{i} = \beta R_{E}, Z_{i} = R_{B}||Z_{b}$

Zo the Output impedance is determined by setting Vi to zero, Ib = 0 and b Ib can be replaced by open circuit equivalent.

• The result is that the output impedance is given by

$Z_{o} = R_{C}$

• AV : We know that,

$V_{o} = - I_{o}R_{C} = - \beta I_{b}R_{C} = - \beta \dfrac{(V_{i})}{(Z_{b})}R_{C}$

$A_{V} =\dfrac{V_{o}}{ V_{i}} = - \beta \dfrac{R_{C}}{Z_{b}}$

• Substituting,

$Z_{i} = \beta(re + R_{E}) A_{V} =\dfrac {V_{o}} { V_{i}} = - \beta[ \dfrac{R_{C}} {(re + R_{E}})]$

• RE >>re,

$A_{V} =\dfrac{ V_{o}} { V_{i}} = \beta\dfrac{R_{C}} {R_{E}}$

Phase relation: The negative sign in the gain equation reveals a 180 degree phase shift between input and output.

• Since in a BJT transitor model we have the transistor such that Common emitter amplifiers give the amplifier an inverted output and can have a very high gain that may vary widely from one transistor to the next. The gain is a strong function of both temperature and bias current, and so the actual gain is somewhat unpredictable.
• The current gain is given as

$A_i=\dfrac{i_{out}}{i_{in}}$