AC Capacitive Voltage Divider Ratio Versus Resistive

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HighVoltage!

Joined Apr 28, 2014
142
What is the mathematical reasoning as to why an AC capacitive voltage divider ratio is inversed (RATIO=C2/C1) and a resistive divider ratio is not (RATIO=R1/R2)? Thanks.
 
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WBahn

Joined Mar 31, 2012
25,557
Thanks for the response. However, why is the impedance of a capacitor inversely proportional to its value?
Resistance, capacitance, and inductance are all just proportionality constants in the constitutive equations for those three components. As such, the constant could have been put on either side of the equation. It's all in how they are defined.

The constitutive equation for a resistor:

V = I·R

R = V/I

The constitutive equation for a capacitor

Q = C·V = di/dt

C = Q/V = (1/V) di/dt

Notice that the resistance is defined such that, for the same current, it is directly proportional to the voltage. But the capacitance is defined such that it is inversely proportional to the voltage.

If you were to use conductance instead of resistance, then the resistive voltage divider would have the same form as the capacitive one. Similarly, if they had defined capacitance (using a different name, most likely) as a constant on the same side of the equation as Q, then the capacitive voltage divider would have the same form as the resistive one.
 

OBW0549

Joined Mar 2, 2015
3,304
Similarly, if they had defined capacitance (using a different name, most likely) as a constant on the same side of the equation as Q, then the capacitive voltage divider would have the same form as the resistive one.
Oddly enough, there actually is a name for that: elastance, the reciprocal of capacitance. Its unit of measurement is the daraf.
 

WBahn

Joined Mar 31, 2012
25,557
Oddly enough, there actually is a name for that: elastance, the reciprocal of capacitance. Its unit of measurement is the daraf.
Interesting -- I've never run across that, but it doesn't surprise me at all. I can definitely see the line of reasoning as elasticity is the inverse of compliance and capacitance is very analogous to compliance.

I've actually used "darafs" before, but I just made it up in class on the fly to show how we can make units up to get them out of the denominator if it makes our lives easier and that, just like the "mho" is reciprocal ohms, we could define the "daraf" to be reciprocal farads and the "yrneh" to be reciprocal henries. It was a throw-away point in a lecture decades ago and I never gave it a second thought. Out of curiosity I just Googled "yrneh" and, sure enough, it popped up as a seldom-used unit for reciprocal henries. I did some looking for the term (not the unit) for reciprocal inductance and wasn't able to find anything. I'll bet that there is (was?) such a term, though.
 

OBW0549

Joined Mar 2, 2015
3,304
I've actually used "darafs" before, but I just made it up in class on the fly to show how we can make units up to get them out of the denominator if it makes our lives easier and that, just like the "mho" is reciprocal ohms, we could define the "daraf" to be reciprocal farads and the "yrneh" to be reciprocal henries.
I actually made use of elastance years ago while working on signal conditioning circuitry for a particular class of moving-plate differential-capacitor displacement sensors; expressing the value of the capacitors formed by the parallel plates in terms of elastance, rather than capacitance, gave me a quantity that varied in direct proportion to the separation between the plates, instead of inversely proportional. Since the circuits were designed to output a voltage that varied directly with plate separation, working in darafs (actually teradarafs, given the small size of the plates) made life a lot easier.

"Yrneh," however is just... just... just wrong. It's unnatural. It should be outlawed. :eek:
 

WBahn

Joined Mar 31, 2012
25,557
I actually made use of elastance years ago while working on signal conditioning circuitry for a particular class of moving-plate differential-capacitor displacement sensors; expressing the value of the capacitors formed by the parallel plates in terms of elastance, rather than capacitance, gave me a quantity that varied in direct proportion to the separation between the plates, instead of inversely proportional. Since the circuits were designed to output a voltage that varied directly with plate separation, working in darafs (actually teradarafs, given the small size of the plates) made life a lot easier.

"Yrneh," however is just... just... just wrong. It's unnatural. It should be outlawed. :eek:
It's actually not as bad when spoken as it is when written. You just imagine some guy from some obscure country being immortalized for their contribution and it suddenly becomes not as bad as some of the other units we've gotten saddled with. :D
 

MrAl

Joined Jun 17, 2014
7,491
Hi,

Yeah and it is interesting that the unit "mho" is deprecated now although it is still used. Most modern stuff uses Siemens now.

The derivation of the capacitor circuit goes like this...

zC1=1/(s*C1)
zC2=1/(s*C2)

and with C1 connected to the input source, we have:
Vout/Vin=zC2/(zC1+zC2)=C1/(C1+C2)

C1 ends up in the numerator instead of C2 because the impedances have the capacitor values in the denominator. For resistors we have;
zR1=R1
zR2=R2
Vout/Vin=zR2/(zR1+zR2)=R2/(R1+R2)

and R2 appears in the numerator because the values were always in the numerators of the impedances of R1 and R2.
 
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WBahn

Joined Mar 31, 2012
25,557
Hi,

Yeah and it is interesting that the unit "mho" is deprecated now although it is still used. Most modern stuff uses Siemens now.
I think there are several reasons for this. The obvious one is that they really like naming units to honor people (I suspect they'd love to rename the meter, the gram, and the second if they thought they could get away with it). But there are some practical issues as well. There are many languages (such as Chinese) in which the notion of "spell the unit backwards" just has no meaning.
 

MrAl

Joined Jun 17, 2014
7,491
Hi again,

Yeah it's amazing how things like this evolve over time. It's also amazing how these 'groups' take so lightly the impact on consumers and even industry.

The USB standard was made so complicated that some companies still cant produce software that works right with it.
"Daylight savings time" was altered so much that many clocks became obsolete because they were programmed to use the old standard.
 
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