Staggered tuning is a technique used in the design of multi-stage tuned amplifiers whereby each stage is tuned to a slightly different frequency. In comparison to synchronous tuning (where each stage is tuned identically) it produces a wider bandwidth at the expense of reduced gain. It also produces a sharper transition from the passband to the stopband. Both staggered tuning and synchronous tuning circuits are easier to tune and manufacture than many other filter types. The function of stagger-tuned circuits can be expressed as a rational function and hence they can be designed to any of the major filter responses such as Butterworth and Chebyshev. The poles of the circuit are easy to manipulate to achieve the desired response because of the amplifier buffering between stages. Applications include television IF amplifiers (mostly 20th century receivers) and wireless LAN.
Rationale
Staggered tuning improves the bandwidth of a multi-stage tuned amplifier at the expense of the overall gain. Staggered tuning also increases the steepness of passband skirts and hence improves selectivity.
The value of staggered tuning is best explained by first looking at the shortcomings of tuning every stage identically. This method is called synchronous tuning. Each stage of the amplifier will reduce the bandwidth. In an amplifier with multiple identical stages, the 3 dB points of the response after the first stage will become the 6 dB points of the second stage. Each successive stage will add a further 3 dB to what was the band edge of the first stage. Thus the 3 dB bandwidth becomes progressively narrower with each additional stage. As an example, a four-stage amplifier will have its 3 dB points at the 0.75 dB points of an individual stage. The fractional bandwidth of an LC circuit is given by,
B = m − 1 Q {\displaystyle B={{\sqrt {m-1}} \over Q}}
where m is the power ratio of the power at resonance to that at the band edge frequency (equal to 2 for the 3 dB point and 1.19 for the 0.75 dB point) and Q is the quality factor.
The bandwidth is thus reduced by a factor of m − 1 {\displaystyle {\sqrt {m-1}}} . In terms of the number of stages m = 2 1 / n {\displaystyle m=2^{1/n}} . Thus, the four stage synchronously tuned amplifier will have a bandwidth of only 19% of a single stage. Even in a two-stage amplifier the bandwidth is reduced to 41% of the original. Staggered tuning allows the bandwidth to be widened at the expense of overall gain. The overall gain is reduced because when any one stage is at resonance (and thus maximum gain) the others are not, unlike synchronous tuning where all stages are at maximum gain at the same frequency. A two-stage stagger-tuned amplifier will have a gain 3 dB less than a synchronously tuned amplifier. Even in a design that is intended to be synchronously tuned, some staggered tuning effect is inevitable because of the practical impossibility of keeping all tuned circuits perfectly in step and because of feedback effects. This can be a problem in very narrow band applications where essentially only one spot frequency is of interest, such as a local oscillator feed or a wave trap. The overall gain of a synchronously tuned amplifier will always be less than the theoretical maximum because of this. Both synchronously tuned and stagger-tuned schemes have a number of advantages over schemes that place all the tuning components in a single aggregated filter circuit separate from the amplifier such as ladder networks or coupled resonators. One advantage is that they are easy to tune. Each resonator is buffered from the others by the amplifier stages so have little effect on each other. The resonators in aggregated circuits, on the other hand, will all interact with each other, particularly their nearest neighbours. Another advantage is that the components need not be close to ideal. Every LC resonator is directly working into a resistor which lowers the Q anyway so any losses in the L and C components can be absorbed into this resistor in the design. Aggregated designs usually require high Q resonators. Also, stagger-tuned circuits have resonator components with values that are quite close to each other and in synchronously tuned circuits they can be identical. The spread of component values is thus less in stagger-tuned circuits than in aggregated circuits.
Design Tuned amplifiers such as the one illustrated at the beginning of this article can be more generically depicted as a chain of transconductance amplifiers each loaded with a tuned circuit.
where for each stage (omitting the suffixes) gm is the amplifier transconductance C is the tuned circuit capacitance L is the tuned circuit inductance G is the sum of the amplifier output conductance and the input conductance of the next amplifier.
Stage gain The gain A(s), of one stage of this amplifier is given by;
A ( s ) = g m s L s 2 L C + s L G + 1 {\displaystyle A(s)={\frac {g_{\mathrm {m} }sL}{s^{2}LC+sLG+1}}}
where s is the complex frequency operator. This can be written in a more generic form, that is, not assuming that the resonators are the LC type, with the following substitutions,
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