Consider the circuit shown below. Assume that the op-amp is ideal.
fa2014_fn_diode_opamp2.png
Figure 3-1
The diode has the following I-V characteristics: iD=Is*(e^(vD/vth)-1)
where iD is the current through the diode,( Is ) is the reverse bias saturation current Vd is the voltage across the diode, and
Vth is the thermal voltage.
Derive the expression for the output voltage, Vout
, in terms of one or more of the variables Vin,R1 ,R2 , Is, and Vth .
.Vout=
so far I got =Vth ln(Id/Is +1)
Id cannot be right.
To derive the expression for the output voltage, Vout, in terms of the given variables, we need to analyze the circuit and understand the behavior of the components.
In the given circuit, assuming the op-amp is ideal, we can assume that the input terminals of the op-amp have virtually no current flowing into them. Therefore, the current passing through resistor R1 and diode D1 is equal to the current passing through resistor R2.
Let's denote the voltage at the non-inverting terminal of the op-amp as Vin+ and the voltage at the inverting terminal as Vin-. The voltage at the node connecting R1 and D1 will also be Vin+.
Now, let's analyze the diode behavior. According to the given I-V characteristics, the current passing through the diode, iD, is given by:
iD = Is * (e^(vD/Vth) - 1)
where Is is the reverse bias saturation current, vD is the voltage across the diode, and Vth is the thermal voltage.
Since the voltage at the node connecting R1 and D1 is Vin+, the voltage across the diode (vD) can be expressed as:
vD = Vin+ - Vout
Now, we can write the current passing through diode D1 in terms of Vin+ and Vout:
iD = Is * (e^((Vin+ - Vout)/Vth) - 1)
Since the current passing through R1 is equal to the current passing through R2, we can write:
(Vin+ - Vout) / R1 = Vout / R2
Simplifying this equation for Vout, we get:
Vout = (R2 / (R1 + R2)) * Vin+ - ((R1 * Is * (e^((Vin+ - Vout) / Vth) - 1)) / (R1 + R2))
Therefore, the expression for the output voltage, Vout, in terms of Vin, R1, R2, Is, and Vth is:
Vout = (R2 / (R1 + R2)) * Vin+ - ((R1 * Is * (e^((Vin+ - Vout) / Vth) - 1)) / (R1 + R2))
It's important to note that this expression for Vout is an implicit equation because Vout appears on both sides of the equation. So, in order to solve for Vout, iterative methods may be required.