Zero State Response and Zero Input Response in Integrator and Differentiator Circuits

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On linear time-invariant (LTI) systems an output can be characterized by a superposition or sum of the Zero Input Response and the Zero State Response.

 

The contributions of   and   to output   are additive and each contribution   and   vanishes with vanishing   and  

This behavior constitutes a linear system. A linear system has an output that is a sum of distinct zero-input and zero-state components, each varying linearly, with the initial state of the system and the input of the system respectively.

The zero input response and zero state response are independent of each other and therefore each component can be computed independently of the other.

Zero State Response in Integrator and Differentiator Circuits

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The Zero State Response   represents the system output   when  


When there is no influence from internal voltages or currents due to previously charged components

 

Zero state response varies with the system input and under zero-state conditions we could say that two independent inputs results in two independent outputs:

   

and

   

Because of linearity we can then apply the principles of superposition to achieve

   


Verification of Zero State Response in Integrator and Differentiator Circuits

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Simple Integrator Circuit

The circuit to the right acts as a simple integrator circuit and will be used to verify the equation   as the zero state response of an integrator circuit.

Capacitors have the current-voltage relation   where C is the capacitance, measured in Farads, of the capacitor.

By manipulating the above equation the capacitor can be shown to effectively integrate the current running through it. The resulting equation also demonstrates the zero state and zero input responses to the integrator circuit.

By integrating both sides of the above equation

 

By integrating the right side

 

Distribute and subtract   to get

 

Divide by   to achieve

 

By substituting   for   and   for   and by using the dummy variable   as the variable of integration the general equation

 

is found.

By using the capacitance of 1 Farad as shown in the integrator circuit

 

which is the equation containing the zero input and zero state response seen above.

To verify its zero state linearity, set   to get

 

By putting two different inputs into the integrator circuit,   and  , the two different outputs

 

and

 

are found respectively.

By using the superposition principle the inputs   and   can be combined to get a new input

 

and a new output

 

By integrating the right side of

 

 

is found, which infers the system is linear at Zero State,  .


This verification example could also have been done with a voltage source in place of the current source and an inductor in place of the capacitor. We would have then been solving for a current instead of a voltage.

Zero State Response Industry Uses

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The circuit analysis method of breaking a system output down into a Zero State and Zero Input response is used industry wide including circuits, control systems, signal processing, and electromagnetics. Also most circuit simulation softwares, such as SPICE, support the method in one form or another.

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http://en.wikibooks.org/wiki/Circuits - Provides basic understanding of electronic circuits

Zero State Response References

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Seperate Article




Zero Input Response in Integrator and Differentiator Circuits

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The Zero Input Response   represents the system output   when  

In other words, when there is no external influence on the circuit

 

This usually results in a decaying output.

Also note, that the Zero Input Response   can still be non zero due to previously charged components.


Verification of Zero Input Response in Integrator and Differentiator Circuits

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Zero Input Response Industry Uses

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Zero Input Response References

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