The overall gain of a multistage amplifier is the product of gain of individual stages.

\[A={{A}_{1}}\times {{A}_{2}}\times …..{{A}_{n}}\]

\[A=\frac{{{V}_{o1}}}{{{V}_{in1}}}\times \frac{{{V}_{o2}}}{{{V}_{in2}}}\times ……\times \frac{{{V}_{on}}}{{{V}_{inn}}}\]

Need of Cascade Amplifier

Amplification capacity of a single stage amplifier is limited and cannot meet the required specifications. A single stage amplifier uses limited transistor parameters because of which it cannot provide very high voltage and current gains and also it does not match its input impedance with the source and output impedance with the load. In order to overcome these limitations, two or more single stage amplifiers are connected in cascade. The cascade connection of amplifiers (i.e., multistage amplifiers) provides desired amplification.

N-stage Cascade Amplifier

Figure 2: n-stage cascaded CE amplifier.

The transistor circuit which involves more than one stage (or) multiple stages of amplification is called a multistage amplifier or cascaded amplifier. The block schematic of a multistage amplifier is shown in figure (1). The overall gain of a multistage amplifier is the product of gain of individual stages.

\[A={{A}_{1}}\times {{A}_{2}}\times …..{{A}_{n}}\] \[A=\frac{{{V}_{o1}}}{{{V}_{in1}}}\times \frac{{{V}_{o2}}}{{{V}_{in2}}}\times ……\times \frac{{{V}_{on}}}{{{V}_{inn}}}\]

Figure (2) illustrates the block schematic of an ‘n’ stage cascaded CE amplifier. The n-stages of the CE amplifiers are connected in such a way that, the output voltage of one amplifier is input to neighboring amplifier. The overall gain of an n-stage cascaded amplifier can be obtained as follows,

Voltage Gain (A_V): The general expression for the voltage gain of an amplifier circuit is given by,

\[{{A}_{V}}=\frac{\text{Output Voltage }}{\text{Input Voltage }}\]

Stage 1: Voltage gain of the first stage is expressed as,

\[{{A}_{v1}}=\frac{\text{Output Voltage of the first stage  }}{\text{Input Voltage of the first stage }}\]

\[{{A}_{V1}}=\frac{{{V}_{2}}}{{{V}_{1}}}\]

Stage 2: Voltage gain of second stage is expressed as,

\[{{A}_{V2}}=\frac{\text{Output Voltage of the second stage  }}{\text{Input Voltage of the second stage }}=\frac{{{V}_{3}}}{{{V}_{2}}}\]

Thus, the overall voltage gain of n-stage amplifier is the product of all the individual stages.

\[{{A}_{V}}=\frac{{{V}_{2}}}{{{V}_{1}}}\times \frac{{{V}_{3}}}{{{V}_{2}}}\times \frac{{{V}_{4}}}{{{V}_{3}}}……\times \frac{{{V}_{n}}}{{{V}_{n-1}}}\]

\[={{A}_{V1}}\times {{A}_{V2}}\times {{A}_{V3}}\times ……{{A}_{Vn}}\] \[{{A}_{V}}={{A}_{V1}}\times {{A}_{V2}}……\]

It can also be determined by using the relation,

\[{{A}_{V}}=\frac{{{A}_{1}}{{R}_{cn}}}{{{R}_{i1}}}\]

Where,

C₁ – Current gain of the ‘n’ stage amplifier

R_cn — Effective load impedance at the collector of n^th stage

R_il – Input impedance of first stage.

Current Gain (A_l): The general expression for current gain of an amplifier is given by,

\[{{A}_{I}}=\frac{\text{Output current }}{\text{Input current}}\]

Stage 1: The expression for current gain of first stage is given by,

\[{{A}_{I1}}=\frac{\text{Output current of the first stage  }}{\text{Input current of the first stage }}\]

\[=\frac{{{I}_{o}}}{{{I}_{b1}}}=\frac{-{{I}_{cn}}}{{{I}_{b1}}}\]

\[{{A}_{I1}}=\frac{-{{I}_{cn}}}{{{I}_{b1}}}\]

Where,

\[\frac{-{{I}_{cn}}}{{{I}_{b1}}}\text{ }-\text{ Base to collector gain of the first stage}\]

Stage 2: The expression for current gain of second stage is given by,

Therefore the overall current gain of an ‘n’ stage amplifier is expressed as,

\[{{A}_{I2}}=\frac{{{I}_{b1}}}{{{I}_{c1}}}=\frac{{{I}_{c2}}}{{{I}_{c1}}}…..\frac{{{I}_{cn}}}{{{I}_{{{c}_{n-1}}}}}\]

\[{{A}_{l}}={{A}_{I1}}\times {{A}_{I2}}\times ……\]

Power Gain (A_P): The power gain of an n-stage cascade amplifier is given as,

\[{{\overline{A}}_{p}}=\frac{\text{Output power of last or }{{\text{n}}^{\text{th}}}\text{stage  }}{\text{Input power of the first stage }}\]

\[=\frac{{{\overline{V}}_{o}}{{\overline{I}}_{o}}}{{{\overline{V}}_{1}}{{\overline{I}}_{b1}}}\]

\[{{\overline{A}}_{p}}={{\overline{A}}_{V}}.{{\overline{A}}_{l}}\]

Thus, the overall gain of an n-stage cascade amplifier is  \({{\overline{A}}_{p}}={{\overline{A}}_{V}}.{{\overline{A}}_{l}}\).

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