**n Stage Cascaded Amplifier:**

As already discussed, several amplifier stages are usually cascaded to increase the overall voltage gain of the amplifier. However, sometimes cascading is done to obtain the desired output and input impedance for specific applications. Block diagram of an n stage cascaded amplifier is given in Fig. 16.1. The first stage is driven by a voltage source V_{s} (or a current source I_{s}). The output of first stage is fed to the input of the second stage, the output of second stage is supplied to the input of the third stage and so on. The output of the nth or last stage is fed to the load R_{L}. Actual voltage available at the input of the first stage is V_{in} (V_{in} equals signal voltage V_{s} if source resistance R_{s} is negligible) and the voltage available at the output terminals of the last stage is V_{out}. Then the ratio V_{out}/V_{in }provides the voltage gain of the n-stage cascaded amplifier. Thus by designing properly a cascaded amplifier, a weak input signal V_{in} of just a few microvolts can be amplified giving output voltage V_{out} of several volts.

Here we take up R-C coupled amplifier, being the most popular amplifier employed for audio-frequency amplification. A typical two-stage R-C coupled amplifier using N-P-N transistors in common emitter configuration (the most suitable transistor configuration for cascading) is given in Fig. 16.2. Typical values of circuit components are given in the brackets.

The block diagram of a two stage R-C coupled transistor amplifier shown in Fig. 16.2 indicating the various voltages, currents and resistances involved is given in Fig. 16.3. The biasing arrangements and coupling capacitors have been omitted for sake of simplicity and clarity.

**1. Voltage Gain:** The overall voltage gain of the amplifier is given by the product of the voltage gains of the individual stages. This is proved as below :

Voltage gain of first stage,

where A_{v1} is the magnitude of the voltage gain of the first stage and θ_{1} is the phase angle between output and input voltages of this stage.

Similarly the voltage gain of nth stage,

Thus the overall voltage gain of the complete n-stage cascaded amplifier is given as

where A_{v} is the magnitude of the voltage gain and θ is the phase angle between the output and input voltages of the amplifier.

Since

Hence

Thus it is concluded that the magnitude of the overall voltage gain equals the product of the magnitudes of the voltage gains of the individual stages and the resultant phase shift of the amplifier equals the sum of the phase shifts introduced by individual stages.

The voltage gain of kth stage (an intermediate stage) of an n-stage CE cascaded amplifier shown in Fig. 16.4 is given as

where R_{LK} is the effective load at the collector of the kth stage.

The quantities in above equation (i.e. A_{ik}, R_{Lk} and R_{in k}) are evaluated by starting with the last stage and proceeding backward. Thus for the nth stage, current gain A_{in} and input resistance R_{in n} are given as

where R_{Ln} is the effective load impedance of the final (i.e. nth) stage and equals R_{cn}.

For the (n – 1)th stage, effective load impedance R_{in n-1 } equals R_{cn-1} || R_{in n}. So

Now the current gain A_{i (n – )}, of the last but one stage [i.e. (n – 1)th stage] is obtained using Eq. (16.11) by replacing R_{L n} by R_{L n – 1}. Similarly the input impedance R_{in n – 1} of the (n – 1)th stage is determined from Eq. (16.12) by replacing A_{in} by A_{i n – 1} and R_{L n } by R_{L n – 1.} Thus proceeding backward, base-to-collector current gains and input impedances of every stage including the first one can be determined. Then the voltage gain of each stage can be determined by using Eq. (16.10).

Overall voltage gain of the n-stage cascaded amplifier may then be calculated from Eq. (16.6).

**2. Current Gain:** Overall voltage gain of an n-stage cascaded amplifier can be obtained, without determining the voltage gain of individual stage, from the following equation

where A_{i} is the overall current gain of the n-stage cascaded amplifier and

where I_{cn} ≡ I_{n}, the collector current of the nth stage.

Now let us obtain equations so as to calculate current gain A_{i} in terms of circuit parameters.

Since

where A_{i1} is the base-to-collector current gain of the first stage and is equal to -I_{ci}/I_{bi} while A′_{i2}, A′_{i3} etc. are the collector-to-collector current gains of second, third etc. stages.

For kth stage, collector-to-collector current gain A′_{ik} is given as

Similarly base-to-collector current gain of kth stage A_{ik} is given as

Now from Fig. 16.4,

where R_{in k} is the input impedance of the kth stage.

Hence

Base-to-collector current gain A_{ik} is determined by starting with the output stage and proceeding backward to the kth stage, as indicated in connection with Eqs. (16.11), (16.12) and (16.13). The collector-to-collector current gains are then determined by using Eq. (16.20) and the overall current gain of the n-stage amplifier is determined by using Eq. (16.16).

**3. Input Impedance:** The input impedance of the complete cascaded amplifier is determined, as discussed above, by starting with the last stage and proceeding backward as follows.

Determine

R_{L n – 1 }is the effective load impedance of the (n – 1)th stage.

Calculate A_{(n – 1),} R_{in (n – 1) } and R_{L n – 2} from the above equations. Proceed in this manner to find the effective input impedance (R_{in}) of the first stage.

**4. Output Impedance:** The output impedance of each transistor stage and of the overall amplifier is determined starting with the first stage and using equation

Then 1/Y_{out 1} provides the corresponding output impedance R_{out 1}.

The output impedance R′_{out k} of the kth stage is the parallel combination of the output impedance R_{out k} of kth stage and R_{c k}. The effective source impedance of the (k + 1)th stage is also R′_{out k}.

**5. Power Gain:** The overall power gain of the n-stage cascaded amplifier is given as