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

$f(t)\,$ $y(t)=y(t_{0})+\int _{t_{0}}^{t}f(\tau )d\tau .$

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.

$y(t)=\underbrace {y(t_{0})} _{Zero-input\ response}+\underbrace {\int _{t_{0}}^{t}f(\tau )d\tau } _{Zero-state\ response}.$

The contributions of $y(t_{0})\,$ and $f(t)\,$ to output $y(t)\,$ are additive and each contribution $y(t_{0})\,$ and $\int _{t_{0}}^{t}f(\tau )d\tau$ vanishes with vanishing $y(t_{0})\,$ and $f(t).\,$

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

The Zero State Response $\int _{t_{0}}^{t}f(\tau )d\tau$ represents the system output $y(t)\,$ when $y(t_{0})=0.\,$

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

$y(t)=\int _{t_{0}}^{t}f(\tau )d\tau .\,$

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:

$f_{1}(t)\,$$y_{1}(t)\,$

and

$f_{2}(t)\,$$y_{2}(t).\,$

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

$Kf_{1}(t)+Kf_{2}(t)\,$$Ky_{1}(t)+Ky_{2}(t).\,$

Verification of Zero State Response in Integrator and Differentiator Circuits

Simple Integrator Circuit

The circuit to the right acts as a simple integrator circuit and will be used to verify the equation $y(t)=\int _{t_{0}}^{t}f(\tau )d\tau \,$ as the zero state response of an integrator circuit.

Capacitors have the current-voltage relation $i(t)=C{\frac {dv}{dt}}$ 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

$\int _{a}^{b}i(t)dt=\int _{a}^{b}C{\frac {dv}{dt}}dt.$

By integrating the right side

$\int _{a}^{b}i(t)dt=C[v(b)-v(a)].$

Distribute and subtract $Cv(a)\,$ to get

$Cv(b)=Cv(a)+\int _{a}^{b}i(t)dt.$

Divide by $C\,$ to achieve

$v(b)=v(a)+{\frac {1}{C}}\int _{a}^{b}i(t)dt.$

By substituting $t\,$ for $b\,$ and $t_{o}\,$ for $a\,$ and by using the dummy variable $\tau \,$ as the variable of integration the general equation

$v(t)=v(t_{0})+{\frac {1}{C}}\int _{t_{0}}^{t}i(\tau )d\tau$

is found.

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

$v(t)=v(t_{0})+\int _{t_{0}}^{t}i(\tau )d\tau ,$

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

To verify its zero state linearity, set $v(t_{0})=0\,$ to get

$v(t)=\int _{t_{0}}^{t}i(\tau )d\tau .\,$

By putting two different inputs into the integrator circuit, $i_{1}(t)\,$ and $i_{2}(t)\,$, the two different outputs

$v_{1}(t)=\int _{t_{0}}^{t}i_{1}(\tau )d\tau \,$

and

$v_{2}(t)=\int _{t_{0}}^{t}i_{2}(\tau )d\tau \,$

are found respectively.

By using the superposition principle the inputs $i_{1}(t)\,$ and $i_{2}(t)\,$ can be combined to get a new input

$i_{3}(t)=K_{1}i_{1}(t)+K_{1}i_{2}(t)\,$

and a new output

$v_{3}(t)=\int _{t_{0}}^{t}(K_{1}i_{1}(\tau )+K_{1}i_{2}(\tau ))d\tau .$

By integrating the right side of

$v_{3}(t)=K_{1}\int _{t_{0}}^{t}i_{1}(\tau )d\tau +K_{2}\int _{t_{0}}^{t}i_{2}(\tau ))d\tau ,$

$v_{3}(t)=K_{1}v_{1}(t)+K_{2}v_{2}(t)\,$

is found, which infers the system is linear at Zero State, $v(t_{0})=0\,$.

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

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.

Zero State Response Links

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

Zero State Response References

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Zero Input Response in Integrator and Differentiator Circuits

The Zero Input Response $y(t_{0})$ represents the system output $y(t)\,$ when $\int _{t_{0}}^{t}f(\tau )d\tau =0$

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

$y(t)=y(t_{0})\,$

This usually results in a decaying output.

Also note, that the Zero Input Response $y(t_{0})\,$ can still be non zero due to previously charged components.