# Logarithm of the voltage of the inductor in RL circuit experiment is nonlinear with time

I performed an experiment about the RL circuit, with following circuit: (I used the PASCO RLC circuit lab.)

By Kirchhoff's voltage law, (denoting the emf of the signal generator $\mathscr E$, inductance of the inductor $L$, the resistance of the resistor $R$, and the current of the circuit $i$) $$\mathscr E - iR - L \frac{\mathrm d i}{\mathrm d t}=0$$ with initial current $i_0 = -\frac{\mathscr E}{R}$ (almost), hence $$i = \frac {\mathscr E}{R} - 2 \frac{\mathscr E}{R} e^{-Rt/L}$$ and then $$\Delta v_L = -L \frac{\mathrm d i}{\mathrm dt} = 2 \mathscr E e^{-Rt/L}.$$ So the natural logarithm of the potential of the inductor must be linear. However, the data of the experiment was strange: (The blue graph: the emf $\mathscr E$ of the signal generator; the orange graph: $\ln (-\Delta v_L / \mathscr E)$.)

Is this just an error of the experiment? Or is there another reason for this graph?

+) The aim of the experiment is finding the time constant of the circuit, with the gradient of the (linear) graph. The inductance of the coil is $$L = \mu N^2 \frac{A}{\ell} \propto \mu$$ where $\mu$ is the permeability of the space in the coil, $N$ is the number of turns, $A$ is the cross-sectional area and $\ell$ is the length of the coil. The time constant of the other experiments, which has the linear log graphs, are as follows:

$R = 10.0 \; \Omega$, $\mu = \mu_\text{air}$: $\tau=0.400 \; \mathrm{ms}$, $L=4.00 \; \mathrm{mH}$

$R = 10.0 \; \Omega$, with iron core: $\tau=1.06 \; \mathrm{ms}$, $L=10.6 \; \mathrm{mH}$

$R = 300. \; \Omega$, $\mu = \mu_\text{air}$: $\tau=0.0149 \; \mathrm{ms}$, $L=4.47 \; \mathrm{mH}$

If we take the slope of the tangent (that is, the average slope with very short time interval) as of the graph, then we have $L_\text{iron core} = 9.91 \;\text{mH},$ which is similar to the case of $R = 10.0 \; \Omega.$ However if we take the average slope, we have $L_\text{iron core} = 3.11 \;\text{mH}.$

• I would suggest you to edit the title. At the moment it is a bit cumbersome and it is not mentioned nonlinear with what? (i.e. time). I can suggest something like "Apparently non-constant with time R/L in RL circuit" – Ivan Madan Apr 10 '16 at 11:24
• Are you sure the EMF of the source is truly zero in its off-state? If it deviates from zero, it would not be surprising the voltage does not drop exponentially. – dominecf Apr 10 '16 at 12:52
• @dominecf There is no off-states, but the voltage alternates. As the formula above, I think the voltage of the inductor should be $2\mathscr E e^{-\frac R L t}$. Is there wrong points on my formulae? – Kanu Kim Apr 10 '16 at 12:59
• If I understand it correctly, the very premise of U dropping exponentially is wrong whenever there is nonzero EMF in the circuit. What do you get if you set the source to have zero voltage in its off state? – dominecf Apr 10 '16 at 13:02
• Unfortunately I cannot do experiment now. Could you explain why the voltage of the inductor does not drop exponentially? – Kanu Kim Apr 10 '16 at 13:10