Practical Electronics/Parallel RC

Parallel RC edit

 

Circuit Impedance edit

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikibooks.org/v1/":): {\displaystyle \frac{1}{Z} = \frac{1}{Z_R} + \frac{1}{Z_C}}
 
 

Circuit Response edit

 
 
 

Parallel RL edit

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Circuit Impedance edit

 
 
 

Circuit Response edit

 
 
 

Parallel LC edit

 

Circuit Impedance edit

 
 
 

Circuit response edit

 
 

Parallel RLC edit

 

Circuit Impedance edit

 
 
 
 

Circuit response edit

 
 
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikibooks.org/v1/":): {\displaystyle I = \frac{V}{R} + \frac{1}{L} \int V dt + C \frac{dV}{dt}}
 

Natural Respond edit

 

Forced Respond edit

 

Second ordered equation that has two roots

ω = -α ±  

Where

 
 

The current of the network is given by

A eω1 t + B eω2 t

From above

When  , there is only one real root
ω = -α
When  , there are two real roots
ω = -α ±  
When  , there are two complex roots
ω = -α ± j 

Resonance Response edit

At resonance, the impedance of the frequency dependent components cancel out . Therefore the net voltage of the circui is zero

  and  

 
 
 
 

At Resonance Frequency

  .
  . Current is at its maximum value

Further analyse the circuit

At ω = 0, Capacitor Opened circuit . Therefore, I = 0 .
At ω = 00, Inductor Opened circuit . Therefore, I = 0 .


With the values of Current at three ω = 0 ,   , 00 we have the plot of I versus ω . From the plot If current is reduced to halved of the value of peak current   , this current value is stable over a Frequency Band ω1 - ω2 where ω1 = ωo - Δω, ω2 = ωo + Δω


  • In RLC series, it is possible to have a band of frequencies where current is stable, ie. current does not change with frequency . For a wide band of frequencies respond, current must be reduced from it's peak value . The more current is reduced, the wider the bandwidth . Therefore, this network can be used as Tuned Selected Band Pass Filter . If tune either L or C to the resonance frequency   . Current is at its maximum value   . Then, adjust the value of R to have a value less than the peak current   by increasing R to have a desired frequency band .


  • If R is increased from R to 2R then the current now is   which is stable over a band of frequency
ω1 - ω2 where
ω1 = ωo - Δω
ω2 = ωo + Δω

For value of I <   . The circuit respond to Wide Band of frequencies . For value of   < I >   . The circuit respond to Narrow Band of frequencies

Summary edit

Circuit Symbol Series Parallel
RC
 
 
A parallel RC Circuit
Impedance Z    
Frequency    
 
 
 
 
 
Voltage V    
Current I    
Phase Angle Tan θ = 1/2πf RC
f = 1/2π Tan CR
t = 2π Tan CR
Tan θ = 1/2πf RC
f = 1/2π Tan CR
t = 2π Tan CR