I am trying to do an analysis of the following modified howland current pump based on the circuit description in the datasheet of the LMP7701. This is a voltage controlled current pump that is driven by a sine generator with aroung 10 kHz.
simulate this circuit – Schematic created using CircuitLab
I want to derive the formula for the output current, that should only depend on the resistor \$R_s\$ and the input \$V_{in}\$. The formula is already given in the datasheet of the LMP7701 and in AN-1515:
\$ i_l = \frac{V_{in}}{R_s} \$
In order to derive this, I started with the followng nodal equations:
\$ i_1 = i_2 \Longleftrightarrow \frac{V}{R_1} = \frac{V_{out}-V}{R_2} \Longleftrightarrow V=\frac{R_1}{R_1 + R_2} V_{out} \ \ (1) \$
\$ i_3 = i_4 \Longleftrightarrow \frac{V-V_{in}}{R_3} = \frac{V_{L}-V}{R_4} \Longleftrightarrow V=\frac{R_4 V_{in} + R_3 V_L}{R_3 + R_4} \ \ (2) \$
\$ i_l = i_s \Longleftrightarrow \frac{V_{out}-V_L}{R_s} = \frac{V_L}{R_L} \Longleftrightarrow V_{out}=\frac{R_LV_L + R_sV_L}{R_L} \ \ (3) \$
Now I set \$ (1) = (2) \$ and derived after \$ V_{out} \$ to get
\$ V_{out} = \frac{(R_1 + R_2)(R_4 V_{in} + R_3 V_L)}{R_1 (R3 + R4)} \ \ (4) \$
Now I set \$ (3) = (4) \$ and solve after \$ V_in \$ to get
\$ V_{in} = \frac{R_3 R_s V_L}{R_4 R_L} \ \ (5) \$
Now, assuming that \$ R_1 = R_2 = R_3 = R_4 \$ and setting \$ V_L = \frac{R_L}{i_L} \$ I get that
\$ V_{in} = \frac{i_L * R_s}{R_L^2} \$
Which seems to be wrong, or am Imissing something? Are some of my assumptions wrong or are some things missing? I recalculated many times and tried different approaches but this is not my field and do not really now how to proceed further, would appreciate some help.
Thanks in advance.
Best Answer
Unfortunately, for some reason I made a mistake and assumed that \$ V_L = R_L/i_L\$, which is obviously wrong. Setting \$ V_L = R_L*i_L\$ and doing the calculations again I get now \$ i_L = V_{in}/R_S \$ which is what I wanted to derive.