Double-tuned amplifier

A double-tuned amplifier is a

There is a critical value of transformer

The circuit shown consists of two stages of

A further detail that may be seen in this kind of amplifier is the presence of

The capacitors connected between the bottom of the transformer secondary windings and ground do not form part of the tuning. Rather, their purpose is to decouple the transistor bias resistors from the AC circuit.

Properties

Double tuning, as compared to single tuning, has the effect of widening the bandwidth of the amplifier and steepening the

There is a critical value of coupling at which the gain of the amplifier is a maximum at resonance. Below this critical value, there is a single peak in the frequency response with the amplitude peaking at resonance and the peak decreasing as k decreases. Such a response is said to be undercoupled, At values of k above critical coupling the response starts to split into two peaks. These peaks become narrower and further apart as k increases and the gap between them (centred on the resonant frequency) becomes progressively deeper. Such a response is said to be overcoupled.[3]

A critically coupled amplifier has a response that is

This expression applies only to small fractional bandwidths.[6]

Analysis

The circuit can be represented in a more generic way by replacing the amplifiers with a generalised transconductance amplifier as shown.

where (omitting the stage number suffixes),

g_m is the transconductance of the amplifiers

G_o is the output conductance of the amplifiers

G_i is the input conductance of the amplifiers.

Typically, a design will make the resonant frequencies and Qs on the primary and secondary sides identical, such that,

${\displaystyle \omega _{0}=\omega _{0\mathrm {p} }={1 \over {\sqrt {L_{\mathrm {p} }C_{\mathrm {p} }}}}=\omega _{0\mathrm {s} }={1 \over {\sqrt {L_{\mathrm {s} }C_{\mathrm {s} }}}}}$

and,

${\displaystyle Q=Q_{\mathrm {p} }={1 \over L_{\mathrm {p} }G_{\mathrm {o} }}=Q_{\mathrm {s} }={1 \over L_{\mathrm {s} }G_{\mathrm {i} }}}$

where ω₀ is the resonant frequency expressed in units of angular frequency and the subscripts p and s refer respectively to components on the primary and secondary side of the transformer.

Stage gain

With the above assumptions, the voltage gain, A of one stage of the amplifier can be expressed as

${\displaystyle A=A_{0}\ {\frac {2kQ}{\ 4Q\delta -i\left[\ 1+k^{2}Q^{2}-4Q^{2}\delta ^{2}\ \right]\ }}\ ,}$

where

${\displaystyle \ i\ }$ is the imaginary unit ${\displaystyle \left(\ i^{2}\equiv -1\ \right),}$

${\displaystyle \ A_{0}\equiv {\frac {g_{\mathrm {m} }}{2{\sqrt {G_{\mathrm {o} }G_{\mathrm {i} }\ }}}}\ }$

is the maximum gain the stage can possibly deliver, and

${\displaystyle \ \delta \equiv {\frac {\ \omega -\omega _{0}\ }{\omega _{0}}}\ }$

is the frequency expressed as the fractional frequency deviation from the resonant frequency.

Peak frequency

With less than critical coupling, there is one peak in the response occurring at resonance. Above critical coupling, there are two peaks at frequencies given by

${\displaystyle \delta _{\mathrm {H} },\delta _{\mathrm {L} }=\pm {1 \over 2Q}{\sqrt {k^{2}Q^{2}-1}}}$

where δ_L and δ_H are respectively the low and high frequencies of the peaks expressed as fractional deviation.

With critical coupling or above, the peaks reach the maximum gain available from the amplifier.

Critical coupling

Critical coupling occurs when the two peaks just coincide. That is, when

${\displaystyle k^{2}Q^{2}-1=0}$

or

${\displaystyle k={1 \over Q}}$^[7]

References

Bibliography

• Bakshi, Uday A.; Godse, Atul P., Electronic Circuit Analysis, Technical Publications, 2009
  ISBN 8184310471
  .
• Bhargava, N. N.; Gupta, S. C.; Kulshreshtha D. C., Basic Electronics and Linear Circuits, Tata McGraw-Hill, 1984
  ISBN 0074519654
  .
• Chattopadhyay, D., Electronics: Fundamentals and Applications, New Age International, 2006
  ISBN 8122417809
  .
• Gulati, R. R., Monochrome and Colour Television, New Age International, 2007
  ISBN 8122416071
  .