Differential Equation of a High Pass Filter

[benbingham]

Greetings,

I'm struggling with a signal analysis electrical engineering college assignment. We need to a derive differential equation for a low-pass filter, high-pass filter and a band pass filter (made by connecting the output of a low pass filter to the input of a high pass filter). Here are diagrams for reference:

Low pass filter:


High pass filter:


Band pass filter (buffer can be removed):


The equations I derived are:
- low pass filter: Vo=Vi-RC(dVi/dt)
- high pass filter: Vo=RC(dVi/dt)
- band pass filter: Vo=R2C2((1-C1-R1)-R1C1(d^2Vi/dt^2))

The equation for the band pass filter I found by making the input of the high pass filter the output of the low pass filter. So I derived the equation of the low pass filter with respect to time and got dVo/dt = dVi/dt - R1*dVi/dt - C1*dVi/dt - R1C1*(d^Vi/dt^2). I substituted in this rate of change of voltage into the equation of the high pass filter to get the equation I derived for the band pass filter.

 

Could anyone please tell me if what I have done is valid and if not, please explain to me what the correct strategy is. Thank you kindly.

[studiot]

In the first two, the resistor and capacitor are in series

 

So the total voltage is the sum of the resistor voltage plus the capacitor voltage and the same current I(t) passes through both.

 

V_T = V_C + V_R

 

The equation for this circuit relates the circuit variables (current and voltage) to each other through the circuit constants (resistance and capacitance)

[math]{V_T} = \frac{1}{C}\left( {\int {Idt} } \right) + RI[/math]
This is an integral equation. We obtain the differential equation by differentiating it. This can be done term by term to obtain

[math]\frac{{d({V_T})}}{{dt}} = \frac{I}{C} + R\frac{{dI}}{{dt}}[/math]

The total voltage in V_in; the output voltage is across the capacitor in the first and across the resistor in the second.

 

Can you substitute for V_in and V_out to obtain the differential equations you need?

Edited April 15, 2014 by studiot