### Author Topic: Help Deriving an OpAmp Transfer Function  (Read 3538 times)

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#### dreaquil

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##### Help Deriving an OpAmp Transfer Function
« on: October 16, 2015, 10:24:07 pm »
Hi,

I'm working on a project which uses an inverting low pass filter and I've introduced some parallel input capacitance to my system so it can be modelled as shown below. Can anyone help me by deriving the transfer function (Vout/Vin) for this schematic? I've gotten the following and wanted to know if I'm correct. If so, I may need help making sense of it too

Vo/Vi=-(Rf/Ri)((1+jwRiCi)/(1+jwRfCf))

#### Phoenix

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##### Re: Help Deriving an OpAmp Transfer Function
« Reply #1 on: October 17, 2015, 12:29:02 am »
The transfer function you've dervied looks good to me. What part don't you understand?

#### dreaquil

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##### Re: Help Deriving an OpAmp Transfer Function
« Reply #2 on: October 17, 2015, 07:29:40 am »
Well the thing is I'm applying a step input for Vi. 0V to 40.96V using a microcontroller. I'm working with extremely high resistances. Here are the values:

Ri=10Tohm, Rf=10Gohm, Ci=3nF, Cf=47pF. The output of my opamp is taking a long time to settle because of that input capacitance but I don't have a clear picture in my head of what's the reason. The only thing I am capable of changing is the 47pF but not by much.

#### mikerj

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##### Re: Help Deriving an OpAmp Transfer Function
« Reply #3 on: October 17, 2015, 08:10:30 am »
You can't be using something with Teraohms of impedance, the input bias currents and capacitance of the op-amp itself and leakage currents with the caps and even the PCB will have a huge effect.

#### dreaquil

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##### Re: Help Deriving an OpAmp Transfer Function
« Reply #4 on: October 17, 2015, 08:26:11 am »
I've taken that into consideration, using an LPC662 (2fA input bias current) and using special PCB design techniques.

Edit: and high voltage caps for very low leakage.
« Last Edit: October 17, 2015, 08:29:54 am by dreaquil »

#### Phoenix

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##### Re: Help Deriving an OpAmp Transfer Function
« Reply #5 on: October 17, 2015, 10:04:58 am »
Well the thing is I'm applying a step input for Vi. 0V to 40.96V using a microcontroller. I'm working with extremely high resistances. Here are the values:

Ri=10Tohm, Rf=10Gohm, Ci=3nF, Cf=47pF. The output of my opamp is taking a long time to settle because of that input capacitance but I don't have a clear picture in my head of what's the reason. The only thing I am capable of changing is the 47pF but not by much.

I think you might need to explain better what you're trying to achieve because it doesnt quite make sense to me.

You've setup an amplifier gain of 1000 (using stupidly high impedances)
Your opamp supply rails are max +-15V ??
You're appling a 40V step

Sounds like you're just jamming it from rail to rail, so I'm not surprise it isnt behaving well.

#### SteveLy

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##### Re: Help Deriving an OpAmp Transfer Function
« Reply #6 on: October 17, 2015, 10:06:16 am »
I've taken that into consideration, using an LPC662 (2fA input bias current) and using special PCB design techniques.
That's not enough. Your op-amp can no longer be treated as ideal if you're going to use such extreme values for the components. Even if your external circuit was perfect, the op-amp's non-ideal characteristics: input and output impedances, finite gain and bandwidth; need to be figured into the transfer function calculation.

Consider that the input impedance of a typical op-amp is a few Mohms, so you cannot expect ideal behaviour with G & Tohm components. You're off by many orders of magnitudes. Try using resistances under ~100k and the ideal transfer function should be reasonably close to reality within the BW limit of the device.

#### dreaquil

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##### Re: Help Deriving an OpAmp Transfer Function
« Reply #7 on: October 17, 2015, 10:25:22 am »
My application does not permit me to do so (picoammeter), also the LPC662 has an input impedance of greater than 1Tohm. Apart from that the signals I am dealing with are DC in nature however there is a step which is causing it to take long to settle.

#### JimRemington

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##### Re: Help Deriving an OpAmp Transfer Function
« Reply #8 on: October 17, 2015, 03:48:37 pm »
Quote
there is a step which is causing it to take long to settle.
No surprise there. Calculate the time constants.

#### SteveLy

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##### Re: Help Deriving an OpAmp Transfer Function
« Reply #9 on: October 17, 2015, 04:13:36 pm »

#### Zero999

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• 0999
##### Re: Help Deriving an OpAmp Transfer Function
« Reply #10 on: October 17, 2015, 09:57:25 pm »
My application does not permit me to do so (picoammeter), also the LPC662 has an input impedance of greater than 1Tohm. Apart from that the signals I am dealing with are DC in nature however there is a step which is causing it to take long to settle.
The impedances at each input aren't matched to minimise the effect of bias currents. There should be a 10G resistor from the + input to 0V.

What's the input capacitance of the op-amp? It's not specified but is likely to be significant.

You've setup an amplifier gain of 1000 (using stupidly high impedances)
Your opamp supply rails are max +-15V ??

*snip*
Quote
You're appling a 40V step

Sounds like you're just jamming it from rail to rail, so I'm not surprise it isnt behaving well.
You've got the gain formula upside down. It has a gain of 1/1000 or a loss of 1000.

Yes you're right, at these high impedances, leakage can be a problem.

« Last Edit: October 17, 2015, 10:18:33 pm by Hero999 »

#### Phoenix

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##### Re: Help Deriving an OpAmp Transfer Function
« Reply #11 on: October 18, 2015, 12:06:00 am »
You've got the gain formula upside down. It has a gain of 1/1000 or a loss of 1000.