A cascade amplifier is any amplifier constructed from a series of amplifiers, where each amplifier sends its output to the input of the next amplifier in a daisy chain. 
A cascade is basically a differential amplifier with one input grounded and the side with the real input has no load. It can also be seen as a common collector (emitter follower) followed by a common base. Since the input side has no load there is no gain on that side and the Miller effect does not come into play. In addition, Vds or Vce stays fairly constant which reduces distortion. Its advantage over the cascode is that it does not require as much voltage headroom. Its disadvantage is since it has two legs it requires twice as much current as a cascode for similar performance.
The complication in calculating the gain of cascaded stages is the non-ideal coupling between stages due to loading. Two cascaded common emitter stages are shown below. Because the input resistance of the second stage forms a voltage divider with the output resistance of the first stage, the total gain is not the product of the individual (separated) stages.
The total voltage gain can be calculated in either of two ways. First way: the gain of the first stage is calculated including the loading of Ri2. Then the second-stage gain is calculated from the output of the first stage. Because the loading (output divider) was accounted for in the first-stage gain, the second-stage gain input quantity is the Q2 base voltage, VB2 = Vo1.
Second way: the first-stage gain is found by disconnecting the input of the second stage, thereby eliminating output loading. Then the Thevenin-equivalent output of the first stage is connected to the input of the second stage and its gain is calculated, including the input divider formed by the first-stage output resistance and second-stage input resistance. In this case, the first-stage gain output quantity is the Thevenin-equivalent voltage, not the actual collector voltage of the stage-connected amplifier. The second way includes interstage loading as an input divider in the gain of the second stage while the first way includes it as an output divider in the gain of the first stage.
By cascading a CE stage followed by an emitter-follower (CC) stage, a good voltage amplifier results. The CE input resistance is high and CC output resistance is low. The CC contributes no increase in voltage gain but provides a near voltage-source (low resistance) output so that the gain is nearly independent of load resistance. The high input resistance of the CE stage makes the input voltage nearly independent of input-source resistance. Multiple CE stages can be cascaded and CC stages inserted between them to reduce attenuation due to inter-stage loading.
Because of the Friis formulas for noise, most designers put the amplifier with the biggest gain first in the series.
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