Pho.rs: Hysteresis in a direct current circuit

Условие

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Hysteresis is a common phenomenon in the study of real physical systems. It refers to situations in which the current state of a system depends not only on the present conditions to which it is subjected, but also on its past history — i.e., on the “trajectory” by which the system arrived at its current state.

Although direct-current (DC) circuits typically do not exhibit hysteresis, in this problem we consider a circuit that does possess hysteretic behavior due to the presence of an component $X$, which models the behavior of a Shockley diode.

Part A. Component X. (3.5 points)

Component $X$ exhibits an $I$–$V$ characteristic of an unusual shape, combining a parabolic and a hyperbolic branches. The transition between the two operating regimes occurs at the current $I = I_{\mathrm{c}}$.

If the current $I$ through component $X$ is less than $I_{\mathrm{c}}$, its $I$–$V$ characteristic is described by
\[
U = 4U_0\,\frac{I}{I_0}\left(1 - \frac{I}{I_0}\right). \tag{1}
\]

If the current $I$ through component $X$ exceeds $I_{\mathrm{c}}$, its $I$–$V$ characteristic follows the hyperbolic relation
\[
U = \frac{A}{I}. \tag{2}
\]

The two operating regimes merge smoothly, so that the resulting $I(U)$ curve is differentiable.

Assume that the values $U_0$ and $I_0$ are known.

A1 ^0.50 Write the equation for the $I$ with the solutions corresponding to the intersection points of curves (1) and (2).

A2 ^0.50 Determine the slope $\mathrm{d}U/\mathrm{d}I$ of curve (2) at a given current $I$. Express your answer in terms of $A$ and $I$.

A3 ^1.00 Determine the slope $\mathrm{d}U/\mathrm{d}I$ of curve (1) at a given current $I$. Express your answer in terms of $I_0$, $U_0$, and $I$.

A4 ^1.50 Determine the value of $A$ for which curve (1) transitions smoothly into curve (2) as illustrated above (i.e., the resulting $I(U)$ curve is differentiable at the junction). Find the corresponding junction current $I_{\mathrm{c}}$. Express both answers in terms of $I_0$ and $U_0$.

Part B. Load line. (6.5 points)

Consider component $X$ connected to a source with an internal resistance $R_0 = U_0/I_0$. At the initial moment, the source is turned off, i.e., $\mathcal{E} = 0$.

The voltage generated by the source at its terminals depends on the current $I$ flowing through it according to the relation
\begin{equation}
U = \mathcal{E} - I R_0. \tag{3}
\end{equation}

Here, the voltage $U$ is also the voltage across component $X$. Therefore, to determine the current that will establish in the circuit, one needs to plot line (3) on top of the $I$–$V$ characteristic of component $X$ and find the intersection point. Line (3) is called the load line and is used for the graphical analysis of the behavior of nonlinear elements.

The voltage $\mathcal{E}$ is gradually increasing up to the value $U_0$.

B1 ^1.50 Write an expression for the current $I$ in the circuit as a function of $\mathcal{E}$. Express your answer in terms of $I_0$ and $U_0$.

The voltage $\mathcal{E}$ continues to be gradually increased until the current in the circuit suddenly jumps when $\mathcal{E}$ reaches a certain value $\mathcal{E} = \mathcal{E}_\uparrow > U_0$.

B2 ^2.00 Determine the value of $\mathcal{E}_\uparrow$. Express your answer in terms of $I_0$ and $U_0$.

After that, the voltage $\mathcal{E}$ is gradually decreased until the current in the circuit suddenly drops when $\mathcal{E}$ reaches a certain value $\mathcal{E} = \mathcal{E}_\downarrow < \mathcal{E}_\uparrow$.

B3 ^2.00 Determine the value of $\mathcal{E}_\downarrow$. Express your answer in terms of $I_0$ and $U_0$.

B4 ^1.00 Determine the internal resistance of the source $R_0$ for which, during a gradual increase and subsequent decrease of $\mathcal{E}$, no sudden jumps in the current occur?