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 through it. The resulting equation also demonstrates the zero state and zero input responses to the integrator circuit. First, by integrating both sides of the above equation Second, by integrating the right side Third, distribute and subtract to get Fourth, 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. To arrive at circuit specific example The general equation can then be used to further demonstrate this verification by using the conditions of the simple integrator circuit above. By using the capacitance of 1 farad as shown in the integrator circuit above which is the equation containing the zero input and zero state response seen at the top of the page. To verify zero state linearity To verify its zero state linearity set the voltage around the capacitor at time 0 equal to 0, or , meaning that there is no initial voltage. This eliminates the first term forming the equation . In accordance with the methods of linear time-invariant systems, 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 implies the system is linear at zero state, . This entire 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 software, such as SPICE, support the method in one form or another. References External links Electrical engineering