Processing math: 3%

Friday, 15 December 2017

Hartley oscillator

Hartley oscillator

This is under the category of tuned oscillator or resonant circuit oscillator
Operation
When Vcc is applied ,the collector current begins to flow and the drop in collector voltage is coupled through capacitor C and L2 .Thus the capacitor charges to its minimum voltage ,this voltage acts as initial excitation for tank circuit ,causing a current to flow in the LC circuit  .This current induces damped harmonic oscillation across L1 and this circuit acts as input to base of transistor
This damped signal is amplified and appears at collector which is coupled as feedback to the tank circuit  C and L1. The feedback voltage across L2 is in phase with the input voltage across L1 results in sustained oscillation
The feedback voltage is in phase shift with input voltage since 180° phase shift produced by the transistor and another 180° being provided by the tank circuit
Capacitor block ,dc component of the collector circuit ,but coupled c signal
As a results of this dc is out of tank circuit this energy loss due to tank circuit  is reduced and hence the oscillation is more stable

WRITING NODE EQUATION
      \frac{V_π}{r_π}= \frac{V_π}{SL_1}+ (V_π-V_c)S_c=0 
      g_mV_π+\frac{V_c}{R}= \frac{V_c}{SL_2}+ (V_c-V_π)S_c=0 
\end{array}\right]

= \left[\begin{array}{cc}V_π\\V_c\\ \end{array}\right]=0

\left[\begin{array}{ccc}   (\frac{1}{r_π}+\frac{1}{SL_1}+S_c)&-S_c\\  (g_m-S_c)&\frac{1}{R}+\frac{1}{SL_2}+S_c\\ \end{array}\right]=0
will continue.......
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Tuesday, 12 December 2017

TUNED COLLECTOR OSCILLATOR

TUNED COLLECTOR OSCILLATOR

It is used for high frequency application .It is called tuned -tuned collector oscillation because  the tuned circuit is connected to the collector .Capacitor  C and primary coil L form the tuned circuits it forms the load impedance and determine frequency of oscillation
the output voltage developed across the tuned circuit is inductively coupled to base through secondary coil    L_{1}
the feedback appears across base emitter junction .transistor amplifier provides 180° phase shift,and tuned circuit provides another 180° so total 360° phase shift is obtained , ie there is a positive feedback

Working
  
       When     V_{cc} is applied a transient current is developed in tuned L-c circuit -this transient current start natural oscillations in the tank circuit . these natural oscillation induce sonic voltage into L_{1} due to mutual induction which causes corresponding variation in base current these variation are amplified ß times and appear in the collector circuit .      

    \frac{1}{2π\sqrt{LC}}  is the frequencies oscillation 












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Monday, 11 December 2017

Crystal oscillator






Crystal oscillator


Crystal oscillator is used for high frequency application .It is basically a tuned circuit using a pizzo electric crystals as the resonant tank circuits.
A piezzo electrical crystal displays piezo electronic property.It is the ability to transform mechanical deformation into electricals charge and vice versa .Whwn the crystal is squazed it develope voltage and if a voltage is applied across  it a change in materials dimension results.
The most common piezzo electric materials are Rochelle salt and quartz
the resonant frequency and Q of a crystal depend upon the crystal dimension ,ie how the surface are oriented with respect to it axis and how the devices are m=mounted thinner and more frogile crystal are needed for high frequency oscillations
 
If the  resistance R is regulated the impedance of the crystal is purely reluctance

iX=\frac{(jωL+\frac{1}{jωc} ) ( \frac{1}{jωc} )}{jωL+\frac{1}{jωc}+\frac{1}{jωc^`}}

=\frac{1-ω^2LC}{jωc^`-jω^3Lcc^`+jωc)}
=\frac{1-ω^2LC}{jω(c^`+c)-jω^3Lcc^`}

let the reactance be zero at ω=ω_s where ω_s is series resonant frequency

1-ω^2LC=0 =>

ω^2_s=\frac{1}{LC}
ω_s=\sqrt{\frac{1}{LC}}
f_s=\frac{1}{2π\sqrt{LC}}
let the reactance become at ω=ω_p where
ω_p=paralel resonant frequency
at ω=ω_p

 =jω_p(c^`+c)=jω^3Lcc^`
c^`+c=jω^2_pLcc^`
ω^2_p=\frac{Lcc^`}{c^`+c}
ω_p=\sqrt{\frac{Lcc^`}{c^`+c}}   
f_p=\frac{1}{2π\sqrt{\frac{Lcc^`}{c^`+c}}}  
 The circuits can be oscillate at a frequency between ω_s and ω_p .The oscillate frequency is determined by the crystal and not by rest of the circuit

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Wednesday, 6 December 2017

RC wein bridge oscillator


RC wein bridge oscillator

RC network does not produce any phase shift .therefore to obtain total shift of 0° or 360° ,a two stage CE amplifier is required
R_1,C_1 and R_2,C_2 acts as forward network ,voltage across the parallel combination of R_2,C_2 is fed to the input of the amplifier .The frequency of oscillation is determined by R_1,C_2 and R_2,C_2 .The desired frequency of oscillation can be obtained by varying 2 capacitors and resistors
The feedback network provides the positive feed back .In addition to this the resistors R_3 and R_4 provide negative feedback.Hence this oscillator has better amplitude stability R_4 is often a temperature sensitive resistor with positive temperature co-efficient
If the amplitude of oscillation increases the resistance R_2 increases .This reduces the negative feedback which reduces the amplitude of the gain and the amplitude of oscillation is restored to stable value

Advantages
low distortion
better stability
adjustable frequency

MathJax example
Disadvantages
Costlier
used only in low frequency

V_f=V_o\frac{R_2||X_{c2}}{R_2||X_{c2}+R_1+X_{c1}}

let R_1=R_2=R

V_f=V_o\frac{\frac{R\frac{1}{jωc}}{R+\frac{1}{jωc}}}{\frac{R\frac{1}{jωc}}{R+\frac{1}{jωc}}+R+\frac{1}{jωc}}
=V_o\frac{\frac{R\frac{1}{jωc}}{R+\frac{1}{jωc}}}{\frac{R\frac{1}{jωc}+(R+\frac{1}{jωc})(R+\frac{1}{jωc})}{(R+\frac{1}{jωc})}}=V_o\frac{R\frac{1}{jωc}}{R\frac{1}{jωc}+(R+\frac{1}{jωc})(R+\frac{1}{jωc})}       
 =V_o\frac{\frac{R}{jωc}}{\frac{R}{jωc}+(R+\frac{1}{jωc})^2}=V_o\frac{\frac{R}{jωc}}{\frac{R}{jωc}+\frac{(Rjωc+1)}{(jωc)^2}^2}
=V_o\frac{R}{R+\frac{(Rjωc+1)}{jωc}^2}=\frac{V_oRjωc}{Rjωc+(Rjωc+1)^2} 
=\frac{V_oRjωc}{Rjωc-R^2ω^2c^2+2Rjωc+1}=\frac{V_oRjωc}{3Rjωc-R^2jω^2c^2+1}

 \frac{V_o}{V_f}=\frac{3Rjωc-R^2jω^2c^2+1}{Rjωc}
MathJax example



 \frac{V_o}{V_f}=\frac{3Rjωc-R^2ω^2c^2+1}{Rjωc}=3+\frac{1-R^2ω^2c^2}{Rjωc}

Equating imaginary part 

 \frac{1-R^2ω^2c^2}{Rjωc}=0
 =\frac{1-R^2ω^2c^2}{Rωc}=0
 1=R^2ω^2c^2
ω=\frac{1}{Rc}

 ω=2πf

f=\frac{1}{2πRc}
Now from real part
 \frac{V_o}{V_f}=3=> ß=\frac{1}{3}
|Aß|=1=>
A=3
therfore gain of 2 amplifier required is 3
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Tuesday, 5 December 2017

RC phase shift oscillator

RC phase shift oscillator

In this case the CE amplifiers is followed by a frequency determining network ,R_1,R_2 combination provides dc potential divides bias and R-E,C_E provides temperature stability and provids AC signal degeneration RC-RC-RC network may be used as positive feedback between input and output.The frequency determining network provides 180°  ie each RC produces 60°,amplifier produces another 180° phase shift.thus the total phase shift is 180+180=360° and satisfy the bark criterion ,.When V_CC is applied ,the current through R_L increases beacuase of biasing .this change capacitor C an induces voltages across R_1

Advantages
1) not bulky and expensive
2) pure sine wave output
Disadvantages
1) suited for low frequency
Replacing transistor by its approximate small signal model

writing loop equations
loop 1
ßI_3R_L+I_1R_L+\frac{I_1}{jωc}+(I_1-I_2)R=0
    \bbox[5px,border:2px solid black]{     +I_1(R_L+\frac{1}{jωc}+R)-I_2R+ßI_3R_L=0}
Loop 2
(I_2-I_1)R+\frac{I_2}{jωc}+(I_2-I_3)R=0

-I_1R+I_2(R+\frac{1}{jωc}+R)-I_3R=0
\bbox[5px,border:2px solid black]{ -I_1R+I_2(\frac{1}{jωc}+2R)-I_3R=0}
loop 3

(I_3-I_2)R+\frac{I_3}{jωc}+I_3R=0

-I_1R+I_2(R+\frac{1}{jωc}+R)-I_3R=0
-I_2R+I_3(R+\frac{1}{jωc}+R)=0
\bbox[5px,border:2px solid black]{ 0-I_2R+I_3(\frac{1}{jωc}+2R)=0 }

this can be put in determinent form by

\left[\begin{array}{ccc}   (R_L+\frac{1}{jωc}+R)&-R&ßR_L\\   -R&(\frac{1}{jωc}+2R)&-R\\ 0&-R&(\frac{1}{jωc}+2R) \end{array}\right]\left[\begin{array}{ccc}   I_1\\   I_2\\ I_3 \end{array}\right]=0
loop current    I_1, I_2,I_3 cannot be zero

\left[\begin{array}{ccc}   (R_L+\frac{1}{jωc}+R)&-R&ßR_L\\   -R&(\frac{1}{jωc}+2R)&-R\\ 0&-R&(\frac{1}{jωc}+2R) \end{array}\right]=0
=>

(R_L+\frac{1}{jωc}+R)\left[(\frac{1}{jωc}+2R)^2-R^2 \right]+R\left[-R(\frac{1}{jωc}+2R)+0 \right]+ßR_LR^2=0


(R_L+\frac{1}{jωc}+R)\left[\frac{1}{j^2ω^2c^2}+4R\frac{1}{jωc} +4R^2-R^2 \right]-\frac{R^2}{jωc}-2R^3+ßR_LR^2=0
-------------------------------------------------------------
(R_L+\frac{1}{jωc}+R)\left[\frac{1}{j^2ω^2c^2}+4R\frac{1}{jωc} +3R^2 \right]-\frac{R^2}{jωc}-2R^3+ßR_LR^2=0


-\frac{R_L}{ω^2c^2}+4R\frac{R_L}{jωc} +3R^2R_L-\frac{1}{jω^3c^3}-4R\frac{1}{ω^2c^2} +3R^2\frac{1}{jωc}-\frac{R}{ω^2c^2}+4R^2\frac{1}{jωc} +3R^3-\frac{R^2}{jωc}-2R^3+ßR_LR^2=0


 
ßR_LR^2+3R^2R_L +3R^3-2R^3+4R\frac{R_L}{jωc}+3R^2\frac{1}{jωc}+4R^2\frac{1}{jωc}- \frac{R^2}{jωc}-\frac{R_L}{ω^2c^2}-4R\frac{1}{ω^2c^2} -\frac{R}{ω^2c^2} -\frac{1}{jω^3c^3}=0
ßR_LR^2+3R^2R_L +R^3+\frac{4RR_L+3R^2+4R^2-R^2}{jωc}+\frac{-R_L-4R-R}{ω^2c^2} -\frac{1}{jω^3c^3}=0

ßR_LR^2+3R^2R_L +R^3+\frac{4RR_L+6R^2}{jωc}-\frac{(R_L+5R)}{ω^2c^2} -\frac{1}{jω^3c^3}=0
\bbox[5px,border:2px solid red]{    ßR_LR^2+3R^2R_L +R^3+\frac{4RR_L+6R^2}{jωc}-\frac{(R_L+5R)}{ω^2c^2} -\frac{1}{jω^3c^3}=0          }      
to make total phase shift around loop zero 
Equating imaginary part to zero

\frac{4RR_L+6R^2}{jωc}-\frac{1}{jω^3c^3}=0         

\frac{4RR_L+6R^2}{jωc}=\frac{1}{jω^3c^3}  
         
4RR_L+6R^2=\frac{1}{ω^2c^2}

\frac{1}{4RR_L+6R^2}=ω^2c^2
ωc=\sqrt{\frac{1}{4RR_L+6R^2}}
2πfc=\sqrt{\frac{1}{4RR_L+6R^2}}
\bbox[5px,border:2px solid black]{  f=\frac{1}{2πc\sqrt{4RR_L+6R^2}}}
is the expression for frequency of oscillation
Equating real part
ßR_LR^2+3R^2R_L +R^3-\frac{(R_L+5R)}{ω^2c^2} =0
here we have→
ω^2c^2=\frac{1}{4RR_L+6R^2}
ßR_LR^2+3R^2R_L +R^3-(R_L+5R)(4RR_L+6R^2) =0
ßR_LR^2+3R^2R_L +R^3-4RR_L^2-20R^2R_L-6R_LR^2-30R^3 =0  
ßR_LR^2 +R^3-30R^3-4RR_L^2-20R^2R_L-6R_LR^2+3R^2R_L =0  
ßR_LR^2 -29R^3-4RR_L^2-23R^2R_L =0    
ßR_LR^2  =29R^3-4RR_L^2+23R^2R_L
ß  =\frac{29R^3}{R_LR^2}+\frac{4RR_L^2}{R_LR^2}+\frac{23R^2R_L}{R_LR^2}
  ß  =\frac{29R}{R_L}+\frac{4R_L}{R}+23
ß  =\frac{29R}{R_L}+\frac{4R_L}{R}+23
\bbox[5px,border:2px solid black]{ let  k=\frac{R_L}{R} }
ß  =\frac{29}{k}+4k+23
the min value of k is obtained by differentiating with respect to k  
 \frac{dß}{dk}=4-\frac{29}{k^2}
Equating to zero
4-\frac{29}{k^2}=0
     
4=\frac{29}{k^2}
k=\sqrt{\frac{29}{4}}=2.69
The min value of ß required by the transistor can be obtained by

ß  =\frac{29}{2.69}+4*2.69+23
ß  =44.5
The phase sift oscillator using BJT will not oscillate if transmitter is chosen whose ß<44.5

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Monday, 4 December 2017

RC and LC oscillator

RC and LC oscillator
Based on the feedback circuits used ,oscillator are classified into RC and & LC ocillator .The time period of RC and LC oscillator are 2ΠRC and 2Π\sqrt{LC} respecyively.For low frequency or Audio Frequency (AF range) the time periods are large .Since larger values of L&C are bulky and rare ,we use RC oscillator for AF. For high frequency or radion frequencies ,the time periods are smaller , The R &C become compatible with the internal parameter of transistor and hence LC are used for high frequency RF oscillator
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Class A Power amplifiers(series fed)

Class A Power amplifiers(series fed)

An amplifier is of type class A if its output remain in the active region during a complete cycle of sine wave input signal .It is an amplifier under normal condition i.e the output never saturates or cut off.If the input is 360°,then output is also 360° ,i.e distortion is very low

Efficiency

V_{pp}=V_{cc}
I_{pp}=\frac{V_{cc}}{R_c} 
I_c=\frac{V_{cc}}{2R_c}
η =\frac{V_{pp}I_{pp}}{8V_{cc}I_c}
=\frac{V_{cc}\frac{V_{cc}}{R_c}}{8V_{cc}\frac{V_{cc}}{2R_c}} 
=\frac{1}{4}=25%
η is low .So class A is never used as a power amplifier  MathJax example

Transformer coupled(Class A power amplifier)

Transformer coupled(Class A power amplifier)
Instead of resistive coupling transformer coupling is used

R^{,}_{L}=(\frac{N_1}{N_2})^2R_L 

N_2=2N_1
\frac{N_1}{N_2}=\frac{V_1}{V_2}  =>
V_2=2V_1

R^{,}_{L} is the resistance reflected to winding.It acts like R_c in series fed class A
V_{pp}=2V_{cc} 
I_{pp}=\frac{2V_{cc}}{R^{,}_{L}}  
I_c=\frac{V_{cc}}{R^{,}_{L}}
η =\frac{V_{pp}I_{pp}}{8V_{cc}I_c}
=\frac{2V_{cc}\frac{2V_{cc}}{R^{,}_{L}}}{8V_{cc}\frac{V_{cc}}{R^{,}_{L}}}=\frac{1}{2}=50%    MathJax example

Harmonic Distortion/Non-linear /amplitude distortion

 Harmonic Distortion/Non-linear /amplitude distortion
The presence of unwanted frequency components in the output which are harmonics of the input frequency is called harmonic distortion .When a sinusoidal signal is applied to a transistor ,non-linearity occurs.Some portion of the signal is amplified more than the other portion


I_c=K_1I_b(linear circuit)
with harmonic distortion I_c=K_1I_b+K_2I_B^2+K_3I_B^3....  
if I_b is sinusoidal  I_b=I_bcosωt
I_c=K_1I_bcosωt+K_2I_B^2I_bcos^2ωt+K_3I_B^3cos^3ωt....  
=K_1I_bcosωt+K_2I_B^2[\frac{1+cos2ωt}{2}]....  
=K_1I_bcosωt+\frac{1}{2}K_2I_B^2+\frac{1}{2}K_2I_B^2[cos2ωt]....
  =B_1cosωt+B_0+B_2cos2ωt....
D_2=\frac{B_2}{B_1}(2^{nd})      D_3=\frac{B_3}{B_1}(3^{rd})     D_4=\frac{B_4}{B_1}(4^{th})
Total harmonic distortion=\sqrt{D^2_2+D^2_3+D^2_4....} MathJax example

Class A Push-pull power amplifier

Class A Push-pull power amplifier

During positive half cycle Q_1 conducts,So I_{c1} flows
During negative half cycle Q_2 conducts,So I_{c2} flows
  R^{,}_{L}=\left( \frac{\frac{N_1}{2}}{N_2}\right)^2R_L
=\left( \frac{\frac{N_1}{2}}{N_1}\right)^2R_L=\frac{R_L}{4}

V_{cc} is center tapped to N_1 in first half cycle ,only Q_1 conduct.So effective primary winding is \frac{N_1}{2} same for Q_2
 Output current
 I_{c1}=B_0+B_1cosωt+B_2cos2ωt+....  
I_{c2}=B_0+B_1cos(ωt+180°)+B_2cos2(ωt+180°)+....  
I_{c2}=B_0-B_1cos(ωt)+B_2cos(2ωt)+....
Total current I_{c}=k[I_{c1}-I_{c2}]  
=2kB_1cos(ωt)+2B_3cos(3ωt)+....
Thus even harmonics are eliminated MathJax example

Hartley oscillator

Hartley oscillator This is under the category of tuned oscillator or resonant circuit oscillator Operation When V_{cc} is appl...