Zero input response and zero state response

These two responses are introduced in detail in the university’s circuit. The purpose of learning is to understand the decomposability of responses from different angles, and to recognize zero-input linearity and zero-state linearity. This part of the content is the key point in the learning process. , It is easy to be overlooked, because it plays a key role in the analysis of the LTI system in the signal and system.
Therefore, I personally think that before realizing the concepts and calculations of zero-input response and zero-state response, it is necessary to have mathematics (calculus solving method, the ability to solve homogeneous and non-homogeneous equations), circuits (that can be built at low frequencies) in advance. The circuit model under), college physics.

1. Zero input response

definition:

When there is no external excitation, only the response caused by the non-zero initial state at t = 0. Depending on the initial state and circuit characteristics, this response decays exponentially over time.

It can be directly known from the definition that the zero input response itself is the characteristic of the circuit. At t=0, the change is actually due to its own characteristic and the existing state, which should not be caused by the influence of the system excitation. When the system is linear and its characteristics can be expressed by linear differential equations, the form of zero input response is the sum of several exponential functions.
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2. Zero status response

definition:

In a dynamic circuit, when the initial stored energy of the dynamic element is zero (ie, zero initial state), only the response caused by the input (stimulus) of the circuit.

In fact, this can be said to be the opposite of the zero-input response. You can directly refer to the zero-input response and then take the right.
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In fact, it can be known from the above formula that if you need to find the solution of the zero-state response according to the requirements, you should solve the inhomogeneous differential equation, and we know that simply solving the inhomogeneous differential equation of a signal is a very complicated matter. Therefore, the convolution integral method is introduced to solve the problem. The principle is:

The zero-state response of the system = the convolution of the excitation and the impulse response of the system, as shown in the following formula:
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Through the calculation process of the convolution integral, the complete response of the system can be directly calculated, namely:

Free response
Zero input response
Homogeneous solution for zero state response

The final result should be the addition of these three parts.

3. The difference between the two responses

Zero state response: the response is 0 before time 0 (that is, the initial state is 0), and the system response depends on the signal f(t) added from time 0;
Zero input response: There is no signal input from time 0 (or the input signal is 0), and the response depends on the initial energy storage before time 0.

4. Two methods of judging response

If there is power excitation, and the component itself has no voltage or current, it is zero state. On the contrary, without power excitation, only the initial value voltage and current of the component itself is zero input response.

5. Two methods to solve the response

Zero input response: the response generated by the initial energy storage without external excitation, which is part of the homogeneous solution;
Zero-state response: the initial state is zero, and the response generated by the external excitation. It can be solved by convolution integral. The zero state response is equal to the convolution of the unit sample value and the excitation. Among them, the unit sample value corresponds to the reverse pull transformation or z transformation of the system function.

6. The connection between the two responses

There are two factors that cause the circuit to respond. One is the excitation of the circuit, but the initial energy stored by the dynamic components. When the excitation is zero, the response caused only by the initial energy stored by the dynamic element is called zero input response; when the initial energy stored by the dynamic element is zero, the response caused only by the excitation is called the zero-state response; two simultaneous responses are called full response.