From ringing in a transmission line to switching in digital circuits, transients in any circuit describe the behavior of a circuit as it makes the transition between two steady equilibria.
Gaining an understanding of this behavior can help you determine whether your device will function as you intended and whether you should take additional measures in your circuit to ensure a smooth transition between different free-running states in your system.
Analyzing poles and zeros provides an easy way to examine the behavior of your circuits as they switch between different free-running states. The poles and zeros of your system describe this behavior nicely. 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. Note that the above equation is defined where the initial conditions are zero. Because we are dealing with a purely linear circuit, one of the initial conditions can always be set to zero by applying a translational transformation to the response in the circuit.
The remaining initial condition is a real constant that is independent of s, thus it is lumped into the pole terms in the transfer function and only determines the real part of a pole.
Let us take a simple transfer function as an example:. Generally, the number of Poles is equal or greater than Zeros. When s approached a pole the value of denominator becomes Zero making the value of transfer function reach infinity. To determine the response, a system the location of Poles is analyze along with the values of real and imaginary parts of each pole.
Real part determines the exponential and imaginary part determines sinusoidal values. Similar to Poles, Zeros are the roots of nominator of a transfer function. The number of Zeros is lesser or equal to the Poles. Zeros mean that the output at those frequencies is zero. Let us have a look at the differences between Poles and Zeros and their effects for a given function:.
The frequencies that turn nominator or denominator zero are called zero and poles of a transfer function respectively. Poles and Zeros of a transfer function are the frequencies for which the value of the denominator and numerator of transfer function becomes zero respectively. The values of the poles and the zeros of a system determine whether the system is stable, and how well the system performs. Control systems, in the most simple sense, can be designed simply by assigning specific values to the poles and zeros of the system.
Physically realizable control systems must have a number of poles greater than the number of zeros. Systems that satisfy this relationship are called Proper. We will elaborate on this below. Now, we can use partial fraction expansion to separate out the transfer function:. Using the inverse transform on each of these component fractions looking up the transforms in our table , we get the following:.
If we just look at the first term:. Using Euler's Equation on the imaginary exponent, we get:. If a complex pole is present it is always accomponied by another pole that is its complex conjugate.
The imaginary parts of their time domain representations thus cancel and we are left with 2 of the same real parts. Assuming that the complex conjugate pole of the first term is present, we can take 2 times the real part of this equation and we are left with our final result:. We can see from this equation that every pole will have an exponential part, and a sinusoidal part to its response. We can also go about constructing some rules:.
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