Transfer Functions

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In summary, the conversation discusses the process of writing differential equations for first and second order systems in an engineering circuits class. The instructor has shown the use of transfer functions and complex impedances to derive solutions, but the homework only covers the differential equation method. The conversation also mentions the use of Laplace transforms and the need for more examples to better understand the process. The main question is how to write the differential equations for a circuit with a resistor, capacitor, and voltage source in parallel.
  • #1
Will
This is probably a basic question for al you EEs out their:


I am in an engineering circuits class, and now we are writing the differential equations for first and second order systems. Our instructor skips around, and showed us how to derive solutions using tranfer functions and complex impedances.( Z-cap=1/(Cs), Z-ind = Ls, similar to frequency repspones expressions).
The chapter our HW came from only shows the DFQ method, writing the system for either voltage across the capacitor or current through the inductor. One of the problems was to find the expression for voltage for t>0 across a resistor, not one of the reactive devices. My question is, how do you write the DFQ for this, as it is not one of the two above mentioned situations? Any sites with examples? This circuit had a voltage source(DC), with one resistor in series with two branches in parallel. On branch had a capacitor and resistor in series, the other branch just a resistor. I know how to write and solve the DFQ for voltage through the capacitor, but how about for the voltage across the resistor branch?
Our teacher was not even sure how to do this using DFQs, he said use the tranfer function method Vout/Vin and set the characteristioc equation, (the denominator for of tranfer function ) equal to zero and solve for s to get the time constant for the natural response, and set s= 0 to find the forced response. There must be a way to find the DFQ, because the text doesn't even mention tranfer functions until a later chapter, although we have done that because the teacher skips around.
I know that this method is the result of taking the Laplace transform of these equations, which makes things a lot easier by tranforming the DFQ into an algebreic expression. I remember doing all this in my Elementary DFQ class, but our circuits teacher has sort of skipped a lot of steps. I would like to see more examples of writing the equations, and then doing the Laplace Transform on them, this would help me to understand better, but I don't know how to derive the DFQs for any thing but the two situations, and the texts only show examples of such.
 
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  • #2
Will,

I'm asusming DFQ stands for differential equation. Never seen that abbreviaton.

To solve a circuit for current and voltage means to calculate the current and voltage through every component. For passive components (e.g., L, C and R) you use the relation between between current and voltage (called the i-v characteristic) that the physics of that component forces it to have. Some of these relations (as you pointed out) are in the form of a diff eq. Some are algebraic eqs. But they'll all get combined into one equation that sets the current into a node or the voltage around a loop to zero.

So, what's the relation between v and i in a resistor? Just use that wherever you come to an R in the circuit.
 
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  • #3


Transfer functions are an important tool in the analysis and design of electrical circuits. They are used to represent the relationship between the input and output of a system in the frequency domain. In simple terms, a transfer function is a mathematical expression that describes how a system responds to different frequencies of input signals.

The transfer function of a circuit can be derived by taking the Laplace transform of the differential equations that describe the circuit. This transforms the differential equations into algebraic equations, making it easier to analyze the system's behavior. The transfer function is typically expressed as a ratio of the output to the input, and it can be used to determine the system's frequency response and stability.

In the case of your specific circuit, it sounds like your instructor is using the transfer function method to find the response of the circuit. This involves setting the characteristic equation (denominator of the transfer function) to zero and solving for the complex variable s. This will give you the time constant for the natural response of the system. Setting s=0 will give you the forced response of the system.

As for finding the differential equation for the voltage across the resistor branch, you can use Kirchhoff's voltage law to write an equation for the voltage drop across the resistor. This can then be used to derive the differential equation for the voltage across the resistor branch.

In terms of finding more examples and resources for understanding transfer functions, there are many online tutorials and textbooks available that provide step-by-step explanations and examples. It may also be helpful to consult with your instructor or a tutor for further clarification and guidance.

Overall, transfer functions are a powerful tool in the analysis and design of electrical circuits, and understanding how to derive and use them is essential for any electrical engineer. Keep practicing and seeking out resources, and you will gain a better understanding of this important concept.
 

1. What is a transfer function?

A transfer function is a mathematical representation of the relationship between the input and output of a system. It describes how the system responds to different inputs and is used to analyze and design control systems.

2. How is a transfer function different from a frequency response?

A transfer function is a mathematical function, while a frequency response is a plot of the system's output amplitude and phase as a function of frequency. The transfer function can be derived from the frequency response, but they are not the same thing.

3. What is the use of transfer functions in control systems?

Transfer functions are used in control systems to model the behavior of the system and design controllers to achieve desired performance. They help to understand how the system responds to different inputs and how to manipulate it to achieve a desired output.

4. How are transfer functions derived?

Transfer functions can be derived using mathematical techniques such as Laplace transforms or by experimental methods. They are typically derived by representing the system as a set of differential equations and solving for the output in terms of the input.

5. Can a transfer function be used for nonlinear systems?

Transfer functions are typically used for linear systems, but they can also be used for some nonlinear systems by linearizing the equations around a certain operating point. However, for highly nonlinear systems, other methods such as state-space modeling may be more appropriate.

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