# Engineering Transimpedance Operational Amplifier Simulation circuit help

#### CNC101

1. Homework Statement

(a) Derive the relationship
between the output voltage V and the input current I; i.e. if V = kI
find k in terms of R1, R2 and Rf.

(b) Calculate the current I if Rf = 10 MΩ, R1 = 90 kΩ, R2 = 10 kΩ
and V –0.1 V.

(c) Model the circuit using PSPICE [using an ideal opamp]
and use the model to confirm the calculation made in (b).
2. Homework Equations

3. The Attempt at a Solution
(a) V= KI, K= R2/ R2+ (Rf*R1/Rf+R1)

(b) I= V/K , I= -0.1/(10k/10k+(90k*10M/90k+10M), I= -0.992A (too big a current?)

(c)

Im not sure how to complete this schematic, I need to check current calculation in (b). I dont kow what "Vin" is. Any help is greatly appreciated as I have no training on this software.

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#### gneill

Mentor
In part (a) the syntax in your algebraic expression for K is ambiguous: the order of operations is not clear. Use parentheses to make the order of operations unambiguous. It also seems unlikely that the output voltage would be positive if $I$ is positive. Shouldn't K have a negative value? The input current $I$ should impose a potential drop across the feedback resistor, pushing the junction of R1 and R2 negative.

Can you show your derivation of K?

For part (b), yes, that current looks to be too big and of the wrong sign. Note that the given output voltage is negative.

For part (c) you should be able to use a current source at the input. Set it to the value you calculate in part (b) and check the resulting output voltage V.

#### CNC101

Ok, for (a) my derivation of "K" is that Rf (feedback resistor) is in parallel with R2, so I have (Rf*R2)/(Rf+R2) = (10*10^6*10*10^3)/(10*10^6+10*10^3) = 1. Using the voltage divider rule I have -(R1/(R1+1)) = K (where 1= (Rf*R2)/(Rf+R2)). Thats my relationship for K, for (b) I have V=KI, I=V/K therefore -0.1/-(90*10^3/(90*10^3+1)) =0.1A which is positive but still too large considering it is a transimpedance circuit that measures very small currents.

#### gneill

Mentor
You should avoid plugging in numbers and rounding before the very end when you have values where small differences can be important to the behavior of the system. A purely symbolic derivation is safer. Also, I think you've lost several orders of magnitude in calculating the value of Rf || R2. It should be closer to 10 kΩ, not 1 Ω .

By using the voltage divider rule you're ignoring the current $I$ that's injected into the junction of R1 and R2.

I get a different value for K, one that is negative and quite large.

Instead, try KCL node analysis at the junction of R1 and R2, and note that the voltage there is fixed by the potential drop across Rf (v1 in the image below).

#### CNC101

KCL node analysis at the junction of R1 and R2, I have (V1-Vout)/R1 for I1 and for I2 I have (Vout-V1)/R2.

#### NascentOxygen

Mentor
KCL node analysis at the junction of R1 and R2, I have (V1-Vout)/R1 for I1 and for I2 I have (Vout-V1)/R2.
And what's your related expression for I, the current through Rf?

#### LvW

.......and for I2 I have (Vout-V1)/R2.
Ohms law for R2 results in I2=-V1/R2.

#### CNC101

Well according to KCL, I1+I2=I so V1/R1-V1/R2= I

#### gneill

Mentor
Well according to KCL, I1+I2=I so V1/R1-V1/R2= I
Nope and nope.

Both I and i2 are shown flowing into the node, while only i1 flows out. So i1 = i2 + I. That means i1 - i2 = I. Also, R1 and R2 terminate at different nodes at different potentials (R1 terminates at the output node that has potential V), so you can't write V1/R1 - V1/R2 = I. You need to write the node equation that takes into account the nodes where the branches terminate.

#### CNC101

If R1 terminates at the output node then thats Vout/R1 and R2 terminates at the central node so thats -V1/R2, and for Rf is V1/Rf. Since I1=I+I2 then Vout/R1=-V1/R2+V1/Rf.

#### gneill

Mentor
No, in nodal analysis you write the current on a branch between two nodes as the potential difference between the nodes divided by the branch resistance. So for example if there was a resistance R between a node at potential Va and another node at potential Vb, then the current flowing from node A to node B through resistance R would be given by (Va - Vb)/R.

#### CNC101

Well for that I have (V1-Vout)/R1 + (V1-Vout)/R2 + (Vin-V1)/Rf assuming the reference node V1.

#### gneill

Mentor
Well for that I have (V1-Vout)/R1 + (V1-Vout)/R2 + (Vin-V1)/Rf assuming the reference node V1.
R2 connects between v1 and ground (0V), not Vout. You already know the current coming from the Vin node as $I$, so you don't need to know Vin. Nodal analysis uses a sum of current terms (it's KCL at a node), and $I$ is the known current in that particular branch.

#### CNC101

If I have- (V1-Vout)/R1+(0-V1)/R2 =0 for my nodal equation, where does Rf come in? As I need to find K in terms of all 3 resistors.

#### gneill

Mentor
If I have- (V1-Vout)/R1+(0-V1)/R2 =0 for my nodal equation, where does Rf come in? As I need to find K in terms of all 3 resistors.
There are three branches meeting at node v1. You need three currents.... which one's missing in your equation?

#### CNC101

Well its "I" flowing through Rf, Which is V1/Rf, So the three currents therefore are (V1-Vout)/R1+(0-V1)/R2 + V1/Rf =0

#### gneill

Mentor
True, but look at the diagram I posted in post #4. That current is a given quantity (well, a given symbolic quantity anyways).

#### CNC101

Are we talking about representing V1/Rf as simply "I" in our nodal equation? If so how is Rf represented for my derivation of K? Thanks for your patience.

#### gneill

Mentor
Are we talking about representing V1/Rf as simply "I" in our nodal equation? If so how is Rf represented for my derivation of K? Thanks for your patience.
Yes, you need both an "I" and and "Rf" eventually. Since "I" is a given you can use it as though it were a constant. So plugging it into the node equation you have (summing currents leaving the node):

$\frac{v1 - 0}{R_2} + \frac{v1 - Vout}{R_1} - I = 0$

Now since $I$ is a given quantity you can find a value for v1 separately. Just follow the path from the input to the v1 node. What potential drop can you see?

#### CNC101

The p.d is across the Rf resistor from the input to V1 node, and V1 represents the p.d.

#### gneill

Mentor
The p.d is across the Rf resistor from the input to V1 node, and V1 represents the p.d.
Right. So write the corresponding expression for v1. v1 = ?

#### CNC101

Well V1=I*Rf. So the nodal equation is (V1-Vout)/R1+(V1-0)/R2-(I*Rf)

#### gneill

Mentor
Well V1=I*Rf. So the nodal equation is (V1-Vout)/R1+(V1-0)/R2-(I*Rf)
Ah, no and no

What's the direction of the potential drop from the input across the resistor Rf? Is it a positive or negative change in potential? In other words, will the change cause v1 to be positive (above the potential at the input) or negative (below the potential at the input)?

For the node equation, don't substitute a voltage for a current! I*Rf would be a voltage. Leave the $I$ alone! It's a given value, so don't complicate it. What you really want to do is get rid of the v1 in the node expression. That's why we're looking at finding v1 via the current $I$ and the drop across Rf.

#### CNC101

The p.d has a negative potentional, not too sure how to remove V1. In terms of (V1-Vout)/R1+(V1-0)/R2= 0, I get Vout=V1((R1/R2)+1). If its a negative p.d, does V1 in that equation become Vout= -I*Rf(R1/R2+1)

#### gneill

Mentor
You left out the $I$ current in the node equation; There are three terms as I wrote in post #19.

To remove V1 from the node equation you find it a different way and use that to replace it. That's why you want to find the potential V1 using the current $I$ and the resistance $R_f$.