Multistage amplifiers
Three identical 1 MHz stages do not make a 1 MHz amplifier: the cascade cuts off at 510 kHz. Each stage adds gain and eats up bandwidth.
01Why more than one stage
A single stage rarely meets everything that is asked of an amplifier: high gain, high input resistance so as not to load the source, and low output resistance to drive the load. Each configuration is good at one thing, so they are combined in cascade, each doing its own job:
| Stage | Function | Typical configuration |
|---|---|---|
| Input | Not loading the source, rejecting the common mode, low noise | Differential pair, follower, FET |
| Gain | Providing almost all the voltage amplification | Common emitter or common source with an active load |
| Output | Delivering current to a low-resistance load | Emitter follower, complementary pair |
This is exactly the structure of an integrated operational amplifier: a differential pair, a gain stage and an output stage.
02The gain of a cascade
The total gain is the product of the gains of each stage, but measured with the real load that the next one places on it. Between two stages a divider is formed between the output resistance of one and the input resistance of the other:
The typical mistake is to calculate each stage unloaded and multiply: a stage with a gain of 100 and a 10 kΩ output, loaded by a 10 kΩ input, delivers only 50. A follower inserted between the two solves the problem, at the cost of one more stage.
03How the stages are connected
| Coupling | Advantage | Problem |
|---|---|---|
| Capacitive | Each stage is biased separately | DC does not pass; each capacitor adds a low-frequency cutoff |
| Direct | Amplifies down to DC; no large capacitors: it is the only option in an integrated circuit | The DC levels of one stage set those of the next, and drift is amplified |
| Transformer | Matches impedances and isolates | Heavy, expensive and band-limited: today only in radio frequency and isolation |
04Classic combinations
- Darlington: two transistors where the emitter of the first feeds the base of the second. They behave like a single one with a β of around \( \beta_1 \beta_2 \), about 10,000, and twice the base-emitter voltage. It is used in output stages and in power switches.
- Cascode: a common emitter followed by a common base. The common base keeps the collector voltage of the first fixed, so the first has almost no voltage gain and therefore barely suffers from the Miller effect: the collector-base capacitance, which in a common emitter with a gain of 100 is seen multiplied by 101 from the input, is multiplied by barely 2 in the cascode. Result: the same \( g_m R_C \) with much more bandwidth.
- Common emitter followed by common collector: the gain of the first and the low output resistance of the second.
05The bandwidth shrinks
Each stage has its own high-frequency cutoff. When they are cascaded, the cutoffs multiply and the overall response falls off sooner than that of each one. For \( n \) identical stages with a pole at \( f_H \):
That is why it is not a good idea to build an enormous gain with many identical stages: each added stage gives gain but eats up bandwidth. The design balances the number of stages, the gain of each one and its own bandwidth.
\( n \) identical stages, each with an open-loop gain, a high-frequency pole and a loading loss at each junction (given by the ratio between the input of the next stage and the output of the previous one). The graph compares the response of a single stage with that of the cascade.
06In the lab
Build two common emitters coupled by a capacitor and measure the total gain. Insert an emitter follower between them and compare with the product of the unloaded gains.
Measure the high-frequency cutoff of a common emitter with a 10 kΩ source resistance, and then that of a cascode with the same input transistor and the same load.
With two identical stages, measure the bandwidth of each one and that of the cascade, and compare it with the 64 % predicted by the formula.
07Common mistakes
- Multiplying unloaded gains. The loading of each stage on the previous one must be included.
- Assuming the total bandwidth is that of one stage. With several stages it is smaller.
- Direct coupling without checking the DC levels. One stage can end up saturated by the voltage of the previous one.
- Forgetting the Miller effect. A high-gain stage multiplies its collector-base capacitance as seen from the input.
08Self-assessment
A stage with a gain of 50 and a 5 kΩ output drives a 20 kΩ input. What does it deliver?
\( 50 \times 20/25 = 40 \).
What bandwidth do three identical stages of 1 MHz each have?
\( 1 \text{ MHz} \times \sqrt{2^{1/3} - 1} \approx 510 \) kHz.
Why does the cascode have more bandwidth than the common emitter?
Because the input transistor has almost no voltage gain and its collector-base capacitance is not multiplied by the Miller effect.
What is the approximate β of a Darlington made of two transistors with β = 80?
About 6400.
09Further reading
- Adel S. Sedra and Kenneth C. Smith. Microelectronic Circuits. 7th ed., Oxford University Press, 2014. Multistage amplifiers and frequency response, with the Miller effect and the cascode.
- Paul R. Gray, Paul J. Hurst, Stephen H. Lewis and Robert G. Meyer. Analysis and Design of Analog Integrated Circuits. 5th ed., Wiley, 2009. Output stages and the transistor combinations of an integrated amplifier.
- Jacob Millman and Arvin Grabel. Microelectronics. 2nd ed., McGraw-Hill, 1987. The classic text of the course, very complete on coupled multistage amplifiers and their frequency response.