Parametric amplifiers Parametric oscillator

1 parametric amplifiers

1.1 introduction
1.2 mathematical equation
1.3 advantages
1.4 other relevant mathematical results

parametric amplifiers
introduction

a parametric amplifier implemented mixer. mixer s gain shows in output amplifier gain. input weak signal mixed strong local oscillator signal, , resultant strong output used in ensuing receiver stages.

parametric amplifiers operate changing parameter of amplifier. intuitively, can understood follows, variable capacitor based amplifier.

q [charge in capacitor] = c x v

therefore

v [voltage across capacitor] = q/c

knowing above, if capacitor charged until voltage equals sampled voltage of incoming weak signal, , if capacitor s capacitance reduced (say, manually moving plates further apart), voltage across capacitor increase. in way, voltage of weak signal amplified.

if capacitor varicap diode, moving plates can done applying time-varying dc voltage varicap diode. driving voltage comes oscillator — called pump .

the resulting output signal contains frequencies sum , difference of input signal (f1) , pump signal (f2): (f1 + f2) , (f1 - f2).

a practical parametric oscillator needs following connections: 1 common or ground , 1 feed pump, 1 retrieve output, , maybe fourth 1 biasing. parametric amplifier needs fifth port input signal being amplified. since varactor diode has 2 connections, can part of lc network 4 eigenvectors nodes @ connections. can implemented transimpedance amplifier, traveling wave amplifier or means of circulator.

mathematical equation

the parametric oscillator equation can extended adding external driving force



e
(
t
)


{\displaystyle e(t)}

:

d

2


x


d

t

2





+
β
(
t
)



d
x


d
t



+

ω

2


(
t
)
x
=
e
(
t
)
.


{\displaystyle {\frac {d^{2}x}{dt^{2}}}+\beta (t){\frac {dx}{dt}}+\omega ^{2}(t)x=e(t).}

we assume damping



d


{\displaystyle d}

sufficiently strong that, in absence of driving force



e


{\displaystyle e}

, amplitude of parametric oscillations not diverge, i.e.,



α
t
<
d


{\displaystyle \alpha t<d}

. in situation, parametric pumping acts lower effective damping in system. illustration, let damping constant



β
(
t
)
=

ω

0


b


{\displaystyle \beta (t)=\omega _{0}b}

, assume external driving force @ mean resonance frequency




ω

0




{\displaystyle \omega _{0}}

, i.e.,



e
(
t
)
=

e

0


sin
⁡

ω

0


t


{\displaystyle e(t)=e_{0}\sin \omega _{0}t}

. equation becomes

d

2


x


d

t

2





+
b

ω

0





d
x


d
t



+

ω

0


2



[
1
+

h

0


sin
⁡
2

ω

0


t
]

x
=

e

0


sin
⁡

ω

0


t


{\displaystyle {\frac {d^{2}x}{dt^{2}}}+b\omega _{0}{\frac {dx}{dt}}+\omega _{0}^{2}\left[1+h_{0}\sin 2\omega _{0}t\right]x=e_{0}\sin \omega _{0}t}

whose solution roughly

x
(
t
)
=



2

e

0





ω

0


2



(
2
b
−

h

0


)




cos
⁡

ω

0


t
.


{\displaystyle x(t)={\frac {2e_{0}}{\omega _{0}^{2}\left(2b-h_{0}\right)}}\cos \omega _{0}t.}

as




h

0




{\displaystyle h_{0}}

approaches threshold



2
b


{\displaystyle 2b}

, amplitude diverges. when



h
≥
2
b


{\displaystyle h\geq 2b}

, system enters parametric resonance , amplitude begins grow exponentially, in absence of driving force



e
(
t
)


{\displaystyle e(t)}

.

advantages

1:it highly sensitive

2:low noise level amplifier ultra high frequency , microwave radio signal

3:the unique capability operate wireless powered amplifier doesn t require internal power source

other relevant mathematical results

if parameters of second-order linear differential equation varied periodically, floquet analysis shows solutions must vary either sinusoidally or exponentially.

the



q


{\displaystyle q}

equation above periodically varying



f
(
t
)


{\displaystyle f(t)}

example of hill equation. if



f
(
t
)


{\displaystyle f(t)}

simple sinusoid, equation called mathieu equation.