I am trying to determine how the common-mode and differential input impedances of difference amplifiers are calculated. I am having a worse time finding out how these specifications are defined.
For example, if I look at the INA133, the common-mode and differential input impedance are both 50k. All 4 internal resistors are 25k. The INA132 seems to follow suit where common-mode and differential input impedances are 80k, and all 4 resistors are 40k. Seems rather straight forward although I am not sure how these were really calculated.
What throws me is the INA106 and the INA143. Both difference amplifiers have the same set of resistors. The input resistors on both devices are 10k and the feedback resistors are 100k. Yet the INA143 has a differential input impedance of 20k, and common-mode input impedance of 55k, and the INA106 has a differential input impedance of 10k and differential input impedance of 110k.
What is the difference (no pun intended) and how are these values calculated/defined? Thanks in advance.
These specs make no sense to me. Assuming ideal opamps, to simplify the math, the basic assumption is that the inputs to the opamp have infinite impedance and the voltages of the two inputs match perfectly.
So the non inverting input just has a pair of resistors in series to the reference voltage (let's say that is 0 V). So the impedance of that input must be 10 k plus 100k = 110 k. This impedance is independent of whether the applied signal is in common to both inputs or part of a differential voltage.
The impedance of the inverting input is 10 k to a voltage source that has a value of 100 k / (10 k + 110 k) = 90.9% of the voltage applied to the inverting input (since the feedback forces the - input of the opamp to match the voltage applied to the + input of the opamp.
So the problem is, how do you replace these two actual equivalent circuits with a common mode and differential impedance.
If you tie the two inputs together, they look like 110k in parallel with 10 k / (1 - 100/110) = 110k. So the common mode impedance is 110 k /2 = 55 k. Perhaps the INA106 sheet shows the common mode impedance of each input, individually, while the INA143 sheet shows their parallel combination.
Differentially, things are messier, since the individual impedances do not match.
If we apply a voltage to the non inverting input and the inverse of that voltage to the inverting input, this would be a purely differential signal with an amplitude of the difference of the two voltages, but the two currents are very different. So how do you define a single differential impedance that involves two different currents, each driven by half of the voltage? One manufacturer might spec one half and one might spec the other.
One way to define the differential impedance would be to apply a completely floating voltage source between the two inputs. With the uneven differential impedances, that voltage will be converted to some combination of common mode voltage and differential voltage. The common mode voltage should see the 55 k common mode impedance calculated, above, and the differential component would be loaded with the effective differential impedance.
So lets say we apply a 1 volt floating source to the two inputs, with the positive side on the non inverting input. By virtue of the float, the current into one input must also be the current from the other. I'll call the voltage applied to the non inverting input V+ and the current into that input I+ and the voltage applied to the inverting input V- and the current into I-.
I+ = - I- and V+ - V- = 1
But I+ = V+ / 110k
so I- = - (V+ / 110k) (the same current in the other direction)
but from the equivalent circuit of the inverting input,
I- = (V- - (V+ * 100/110))/10k
so we can combine these two equations to find out how the floating input voltage divides between V+ and V-.
V+ / 110k = ((V+ * 100/110) - V-) / 10k
so V+ = V- * 11/9
or equivalently, V- = V+ * 9/11
Note that both V+ and V- have the same sign, rather than splitting across zero.
But their difference must be 1 volt, so
1 = V+ - (V+ * 9/11)
So V+ = 11/2 and V- = 9/2
Checking, the differential voltage is 1 volt or 2/2. The common mode voltage is 10/2.
So each input should produce a common mode current of (10/2)/110k and a differential current of 1/Rdif.
so I+ = (10/2)/110k + 1/Rdif
but also, I+ = V+ / 110k
therefore, (11/2)/110k = (10/2)/110k + 1/Rdif
So Rdif = 220k
Which matches neither data sheet, so I am satisfied. ;-)
A more useful way to define the input impedances might be to hold one input at zero and apply voltage to the other, and define the input impedance under those conditions.
As always, the non inverting input impedance is 110k.
The inverting impedance under this condition is 10k.
How do you get this? The second page of the INA133 data sheet says the differential impedance = 50k and the common mode impedance = 25k.
Yes, that's what the data sheet says. Seems inconsistent with what the INA133 sheet says.
It's too bad they don't say what circuit connections were used to determine those impedances. I assume that figure 1 of each data sheet would be the configuration to use.
I would also assume that the common mode impedance is measured with the two inputs tied together and driven together.
Let Ri be the two input resistors, and Ro be the two output resistors. My nodal analysis indicates that if the amplifier gain is very high, the common mode input impedance is (Ri + Ro)/2 and the differential input impedance is 2 * Ri.
This is for the circuit of Figure 1 on each data sheet.
The INA133 and INA143 data sheets are consistent with this analysis, but the other two aren't. I wonder why.
For the INA143, they don't say whether the impedances are for a gain of 10 or .1; my analysis indicates that the common mode impedance would be the same, but the differential impedance for the .1 gain configuration would be
Ah, you are correct. I read far too many datasheets last night trying to correlate. My mistake. I am waiting for a response from TI, but I will look at your explanation later as well. Thanks for the reply.
A quick update. The information I received was not defintive; however, it seems that the impedances were derived from Spice analysis during testing. The other explanation given is what you guys have already explained. It depends on how the inputs are driven. So I guess in that manner, everyone is correct. :^). It is a shame that there are no figures in the datasheet showing the test circuits. Thanks for the help.