Circuit Theory/RLC Circuits

Series RLC CircuitEdit

Second Order Differential EquationEdit

 
 

The characteristic equation is

 
 

Where

 
 

When  

 
The equation only has one real root .  
The solution for  
The I - t curve would look like

When  

 
The equation has two real root .  
The solution for  
The I - t curve would look like

When  

 
The equation has two complex root .  
The solution for  
The I - t curve would look like

Damping FactorEdit

The damping factor is the amount by which the oscillations of a circuit gradually decrease over time. We define the damping ratio to be:

Circuit Type Series RLC Parallel RLC
Damping Factor    
Resonance Frequency    

Compare The Damping factor with The Resonance Frequency give rise to different types of circuits: Overdamped, Underdamped, and Critically Damped.

BandwidthEdit


[Bandwidth]

 

For series RLC circuit:

 

For Parallel RLC circuit:

 

Quality FactorEdit


[Quality Factor]

 

For Series RLC circuit:

 

For Parallel RLC circuit:

 

StabilityEdit

Because inductors and capacitors act differently to different inputs, there is some potential for the circuit response to approach infinity when subjected to certain types and amplitudes of inputs. When the output of a circuit approaches infinity, the circuit is said to be unstable. Unstable circuits can actually be dangerous, as unstable elements overheat, and potentially rupture.

A circuit is considered to be stable when a "well-behaved" input produces a "well-behaved" output response. We use the term "Well-Behaved" differently for each application, but generally, we mean "Well-Behaved" to mean a finite and controllable quantity.

ResonanceEdit

With R = 0Edit

When R = 0 , the circuit reduces to a series LC circuit. When the circuit is in resonance, the circuit will vibrate at the resonant frequency.

 
 
 
 

The circuit vibrates and may produce a standing wave, depending on the frequency of the driver, the wavelength of the oscillating wave and the geometry of the circuit.

With R ≠ 0Edit

When R ≠ 0 and the circuit operates in resonance .

The frequency dependent components L , C cancel out ie ZL - ZC = 0 so that the total impedance of the circuit is  
The current of the circuit is  
The Operating Frequency is  

If the current is halved by doubling the value of resistance then

 
Circuit will be stable over the range of frquencies from  

The circuit has the capability to select bandwidth where the circuit is stable . Therefore, it is best suited for Tuned Resonance Select Bandwidth Filter

Once using L or C to tune circuit into resonance at resonance frequency   The current is at its maximum value   . Reduce current above   circuit will respond to narrower bandwidth than  . Reduce current below   circuit will respond to wider bandwidth than  .

ConclusionEdit

Circuit General Series RLC Parallel RLC
Circuit
Impedance Z    
Roots λ λ =   λ =  
I(t) Aeλ1t + Beλ2t Aeλ1t + Beλ2t Aeλ1t + Beλ2t
Damping Factor      
Resonant Frequency      
Band Width      
Quality factor