The zeros z j of G s do not affect the system stability. However, they do affect the amplitudes of the mode functions in the system response and can block the transmission of certain input signals. In general, to assess the Impact of the zeros on the amplitude of the mode functions, a partial-fraction expansion is performed on the Laplace transform of the output signal, and the residue for each mode function is computed.
However, certain properties of zeros can be readily illustrated in the time domain as shown in the following Example. The fraction may be expanded into two terms:. The second term is the standard second order system as given in. Recalling the Laplace transform property, , and assuming zero initial conditions, we can see that the inverse transform of is the derivative of scaled by.
That is,. This equivalence can be illustrated through the time-domain step responses. These are obtained by first multiplying the transfer functions by , which applies a unit step to the system at , then by taking the inverse Laplace transform. That is, we will show the equivalence of. Beginning with the right-hand-side of this equation,.
Since and are identical, therefore we have shown that the inverse transform of is the time derivative of scaled by. Since the arguments of the sine and cosine functions are identical, we can convert them into a single trigonometric function with a new magnitude and phase offset.
This gives a more intuitive form of the step response. Click and drag the poles or the zero in the S-plane to see the effect on the time domain response on the right. With more complex linear circuits driven with arbitrary waveforms, including linear circuits with feedback, poles and zeros reveal a significant amount of information about stability and the time-domain response of the system. When most designers discuss transfer functions and Bode plots , they are really looking at the steady-state behavior of a circuit.
This tells you how different frequency components in an arbitrary input signal are affected by the circuit after all transient responses have decayed back to zero. This very easily tells you how the phase and amplitude of an input sinusoidal signal are affected by a circuit and what you would measure at the output.
However, a transfer function in the frequency domain does not tell you how the transients in the circuit behave, nor does it tell you the following information:. In other words, working in the frequency domain does not show you how the circuit makes the transition from an undriven state to the driven state after transients have died out. The frequency domain transfer function is still extremely useful as you can easily examine how arbitrary signals such as digital pulses are transformed and distorted by the circuit.
However, the Laplace domain problem is equally important as this tells you something about stability. First, it shows you how the transient response decays or grows as the system approaches the steady state if it even exists. Second, it shows you nicely whether the response of the system is stable in the presence of feedback. One example is in linear control circuits, which require feedback to ensure the system remains controlled in a desired state note that perturbation techniques become important in nonlinear control circuits.
Here, we need to note that transfer function analysis and pole-zero analysis are only applicable for linear circuits. If there are nonlinear circuit elements such as transistors or diodes , then you can only consider the approximate linear response, i.
The voltage or current u t in a linear circuit that is driven with a forcing function F t can be written as an n-th order linear nonhomogeneous differential equation shown below.
Procedure for determining poles and zeros in the Laplace domain. Note that the coefficients in these equations are real numbers. The right hand side in the 2nd step can be expanded as a Taylor series if it is not already a polynomial function. In some cases, the forcing function F t can be written as a solution to its own linear ordinary differential equation and converted into the Laplace domain a simple example is a sinusoid.
In this case, the Laplace transform of the right hand side will always be a polynomial and a Taylor series expansion is unnecessary. You can now define a transfer function in terms of the Laplace variable s.
This is normally determined by factoring the polynomials in the numerator and denominator. This is shown below, where z refers to a zero and p refers to a pole. They influence the stability and the transient behavior of the system. The referenced document is a good start. When dealing with transfer functions it is important to understand that we are usually interested in the stability of a closed loop feedback system. In order for the closed loop system to be stable, the poles have to be located in the left half plane.
The zeros have no importance, since the stability of a linear system is solely determined by the position of the poles. When designing a closed loop system i. Because for the open loop system it is easier to understand how the circuit parameters are going to influence the system behavior.
It can be shown that the position of zeros of the open loop system are important for the stability of the closed loop system. When closing the loop slowly by increasing the feedback while monitoring the poles, it can be seen that the poles are attracted by the zeros. The poles move towards the zeros and if there are zeros in the right half plane, the tendency for the system to become unstable is higher because finally the pole will assume the position of the zero.
Such a system would be called a non-minimum phase system, and they are quite common. Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. Meaning of zeros in transfer function Ask Question.
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