How Do You Write a Differential Equation for Resistor Voltage in a Circuit?

[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.

 

[jdavel]

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.