Time domain vs Frequency domain Analysis: What, When and Why’s
For a control engineer, these terms are very frequent. There’s always confusion regarding what, when and why the system should be subjected to time domain or frequency domain analysis.
Let’s briefly look at what a system analysis is. It’s a process of defining the working or the performance of a system. There are two methods of analysing a system: Qualitative and Quantitative.
The qualitative analysis is a descriptive explanation of the performance of the system. It’s very impossible for the qualitative analysis be unambiguous. It’s very much difficult when you try to compare systems qualitatively.
On the other side, it’s very easy to compare the systems quantitatively. However, identifying the parameters that define the systems quantitatively is very difficult issue. But once these parameters are correctly identified, then it’s self explanatory and comparing systems is quite simple.
Let’s look at some examples to clarify these points. Assume that you’re asked to compare two cars before deciding to purchase. You take test drives and then try to explain your experiences. Qualitatively, you can say the car’s pick up is good. Its smooth while driving. The speed is good. And so on. But these words cannot convince others about the experiences of the narrator.
Instead, if the experiences are measured like
- the time it takes to reach a speed of 100kmph from rest, or
- the maximum speed of the car
it is quite easy to compare the performance of the two cars. That’s the power of quantitative data.
There are two types of quantitative analysis of the system depending upon your response time:
- Time domain
- Frequency domain
What is Time domain analysis?
Time domain analysis is preferred when the system varies slowly when compared to your response time. This allows you to measure the parameters from time to time. This time-series of the data measured will help us to understand the system behaviour.
e.g. the time the car takes to pick up to 100kmph speed from rest is the time domain analysis. Our response time is in milli seconds whereas the experimentation time is about 10-15 seconds. We can track the change in speed and record for further analysis.
Imagine you’re watching a train move from some distance. You can keep noting its position at different instances of time and record for analysis. The train takes minutes to cross from your vision and noting its position is matter of seconds.
But the same is not possible to track the position of a point in the wheel of the train. Hence time domain analysis can’t be used for such cases.
What is frequency domain analysis?
When the system is faster than your response time, and if the periodic experimentation can be performed, then frequency domain analysis is preferred. Here the experiments over different frequencies are performed and then analysed.
e.g. the experiments conducted to measure the mileage of the car while running with different speeds is a frequency domain analysis. Here the car is run for a fixed distance at different speeds and each time the fuel consumed is measured and thereby the mileage. The mileage for different speeds are recorded and analysed.
Study of rotation of moon (time per revolution) is a time domain analysis while the rotation of wheel of a car (revolutions per unit time) is a frequency domain analysis.
Time domain analysis of a system:
In this case, a known form of input is applied to the system and the system response is measured for analysing. The simplest form of input is just to start the system working. Mathematically, it’s called step function. Before t=0, the system was off and from t=0 onwards, the system is on. Now the parameters, like speed, of the system are measured as they change after switching on. This change is quantized by appropriate parameters as shown is the figure below.
- Initial value
- Final value
- Rise time
- Peak time
- Maximum overshoot
- Settling time
The system can be represented without any ambiguity by these parameters listed above.
Assume that you’re driving a car and aiming to maintain a speed of 80kmph. Depending on your driving style there are three possibilities:
- Arrogant driver: In this case, the car is suddenly accelerated and before the driver realises, the speed has crossed the limit and hence he may suddenly shows down. It may now go below the set speed which makes him to accelerate a little and cross the limit again. This up and down speeding continues till the steady speed is reached. This is the case of underdamped condition (damping factor<1). In this case, both the rise and settling times are lowest. In the negative side, it produces overshoot and oscillatory behaviour.
- Very careful driver: You slowly accelerate the car and makes sure not to cross the speed limit. In this situating, the car takes a long time to reach almost the set speed but never reaches the exact speed. This is called overdamped condition (damping factor>1). No overshoot, no oscillations but very high rise time and almost infinite settling time.
- Perfect driver: You know the art of driving perfectly and make sure to use appropriate acceleration from time to time and ensure quickly reaching the set speed and maintaining the speed constant there. This is critically damped condition (dampingfactor=1). No overshoot, moderate rise and settling times.
The system representation can further be simplified by knowing the damping condition.
System control in time domain
When the step input is applied to the system, then ideally a step output is expected. Zero overshoot, zero rise time and zero settling time. But none of the damping value of the system can attain this. The optimal system must have least possible rise and settling times and zero overshoot. This can be made possible by using an underdamped system with closed loop and a proper controller. The underdamped system has very low rise and settling times but has overshoot. This overshoot must be minimized through the controller.
Controller
There are three types of basic controllers.
- Proportional(P): This will vary the magnitude of the output depending upon the error — the difference between the actual output and the intended output.
- Differential(D): In this case, the rate of change of error will control the output. This is useful whenever there is sudden changes in the error. This limits the sudden changes in the error.
- Integral(I): This will vary the output such that more stability is attained earlier. Here the average of the error over a period will control the output.
Depending upon the requirements, the controller can be any of the above-mentioned controllers or their combinations like PI,PD, PID. Usually combined controllers perform better for any physical system.
Frequency domain analysis of a system:
When the system behaviour is harmonic and the period is very small compared to our response time, then frequency domain analysis is the only option to study the system. In this set-up, the system is subjected to sinusoidal input of different frequencies and the system response such as the magnitude and the phase of the output is measured. The variation of the behaviour of the system to different frequencies can be studied to understand the system and these measurements can further be used in comparing the performance of two systems.
The above plot shows the mileage of the car at different speeds. This experiment was conducted by running a car with some fixed amount of fuel at different speeds. It can be seen that the car runs efficiently between 30 to 60 mph. This experiment will be helpful in designing the cars. The controlling mechanisms in the car can be trained to improve this efficiency for any given range of speeds.
System control under frequency domain
The major problem we encounter with feedback system is of sustained oscillations. This is possible when for some frequency the gain around the loop is unity and the overall phase lag is integral multiples of 360⁰. These two must be simultaneously true for a particular frequency. Usually, the gain will be less than unity for the frequency at which the phase lag is 360⁰. The amount of gain that can be increased at the frequency where the phase lag is 360⁰is called the gain margin. Similarly, the phase that can be delayed by to make the phase lag of 360⁰ at the frequency where the gain is unity is called the phase margin. These two are the better representatives of the system in frequency domain analysis. These two parameters help in understanding the system’s behaviour and useful in comparing the performance of two systems. With negative feedback, the phase lag required is 180⁰ as the other 180⁰ phase delay is provided by the feedback itself.
Why such analysis is required?
Whenever the system doesn’t satisfy the requirements, then it’s subjected to feedback and controller. The analysis, either in time domain or in frequency domain, performed will help us in identifying the shortcomings of the system as compared to the requirements. This analysis will further provides the means to develop the controller and achieve the required performance.
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