# Analysis of a modified Howland Current Pump

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.

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.