Problem 11 - Analysis 1: Ein Lehr- und Arbeitsbuch für Studienanfänger

Question

(Einschaltvorgang) in elnen Gleichstrowkreis sind ein Ohscher widerstand \(R\), cine Spule mit der Induktivität L sowie eine Spannungsquelle wit der kon= stanten spannung \(U_{0}\) geschaltet. nach dem Ohmschen Gesetz. zu erwartenden Nert \(\frac{U_{O}}{R}\) an. Sie wird Deschrieben ?urch die Funktion $$ i(t) ;=\frac{v_{0}}{R}\left(1-e-\frac{R(t-t))}{L}\right), t \geq t_{o^{*}} $$ Hierbei sind \(z .8 . U_{0}\) in Volt, \(R\) in ohm, \(L\) in Henry = Voltsec/Amp und t in Sekunden zu messen. a) Diskutieren sie die Funktion i fiur \(t=t_{0}\), fulr \(\mathrm{t}>t_{0^{\prime}}\) fïr "sehr grose" \(t)\) D) Berechnen Sie die Anderung der stromstärke wathrend dos ginschaltvorganges, \(d . h \cdot \frac{d i(t)}{d t}, t \geq t_{0}\). Wann ist die Anderung der stromst?rke am gröliten? c) zeigen sie: Für alle \(t \partial t_{0}\) gilt \(L \cdot \frac{d i(t)}{d t}+R \cdot i(t)=v_{0}\). d) Geben Sie eine physikalische Intorprotation der Gleichung in c).

Step-by-Step Solution

Answer

Initially, current is zero and grows to \( \frac{U_{0}}{R} \). The rate of change is maximal at \( t=t_0 \). The equation balances induced and Ohmic forces.

Step 1: Initial Analysis at t = t_0

The function for the current is given by \( i(t) = \frac{U_{0}}{R}\left(1-e^{-\frac{R(t-t_{0})}{L}}\right) \). At \( t = t_{0} \), the exponential term simplifies to \( e^{0} = 1 \), making the function \( i(t_{0}) = \frac{U_{0}}{R}(1-1) = 0 \). Hence, at \( t = t_0 \) the initial current is zero.

Step 2: Behavior for t > t_0

For \( t > t_{0} \), the exponential term decreases towards zero as \( t \) becomes large. Therefore, \( i(t) \) approaches \( \frac{U_{0}}{R} \). This indicates that the steady-state current is reached and depends only on the resistance and the voltage source.

Step 3: Limit for very large t

As \( t \rightarrow \infty \), \( e^{-\frac{R(t-t_{0})}{L}} \rightarrow 0 \), thus \( i(t) \rightarrow \frac{U_{0}}{R} \), indicating that the system reaches equilibrium with current \( \frac{U_{0}}{R} \).

Step 4: Calculate the rate of change of current

To find \( \frac{di(t)}{dt} \), we differentiate the function: \[ \frac{di(t)}{dt} = \frac{U_{0}}{R} \cdot \frac{R}{L} e^{-\frac{R(t-t_{0})}{L}} = \frac{U_{0}}{L} e^{-\frac{R(t-t_{0})}{L}} \]. The rate of change is maximal at \( t = t_{0} \) since exponentials decay with increasing \( t \).

Step 5: Verification of L\( \cdot \frac{di(t)}{dt} + R \cdot i(t) = U_{0} \)

Substitute \( i(t) \) and its derivative into \( L \cdot \frac{di(t)}{dt} + R \cdot i(t) = U_{0} \):\[ L \cdot \frac{U_{0}}{L} e^{-\frac{R(t-t_{0})}{L}} + R \cdot \frac{U_{0}}{R} \left(1-e^{-\frac{R(t-t_{0})}{L}}\right) = U_{0} \]. This simplifies to \( U_{0} e^{-\frac{R(t-t_{0})}{L}} + U_{0} - U_{0} e^{-\frac{R(t-t_{0})}{L}} = U_{0} \), confirming the equation.

Step 6: Interpretation of Differential Equation

The equation \( L \cdot \frac{di(t)}{dt} + R \cdot i(t) = U_{0} \) represents the balance between the induced electromotive force (from the change in magnetic field, captured by \( L \cdot \frac{di(t)}{dt} \)) and the Ohmic voltage drop (\( R \cdot i(t) \)). This ensures that the sum is equal to the applied voltage \( U_{0} \).

Key Concepts

Ohm's Law

Ohm's Law is a fundamental principle used to understand electrical circuits. It relates the current flowing through a conductor to the voltage across it and the resistance of the conductor. The formula is given by \( V = IR \), where \( V \) represents the voltage in volts, \( I \) is the current in amperes, and \( R \) is the resistance in ohms. This simple equation is the backbone of circuit analysis. It helps us predict how electrical currents will behave in various conditions.
Applying Ohm's Law allows us to determine unknown values within an electrical circuit. If two of the values (voltage, current, or resistance) are known, the third can easily be calculated. In the original exercise, we see this principle at work in the determination of the steady-state current \( \frac{U_{0}}{R} \). This means that eventually, the current stabilizes at this value once the inductor's effect diminishes. This aspect of Ohm's Law is essential in predicting circuit behavior.

Inductive Reactance

Inductive reactance is a property of an inductor in an AC circuit that resists the change in current. It is represented by the symbol \( X_L \) and calculated using the formula \( X_L = 2\pi fL \), where \( f \) is the frequency of the AC signal and \( L \) is the inductance in henrys. However, in DC circuits, like the one in the exercise, inductive reactance affects the circuit transiently.The inductance, \( L \), causes a delay in reaching the steady state. This is because when current changes, an opposing voltage (back EMF) is induced. This induces a temporary resistance to changes in current. As time progresses, \( t \rightarrow \infty \), the inductor's effect decreases, and the current can reach its maximum steady state determined by \( \frac{U_{0}}{R} \). Understanding inductive reactance helps explain the initial lag in current flow seen right after switch-on in a circuit.

Transient Analysis

Transient analysis involves studying how electrical circuits respond to changes, such as the moment a switch is closed or opened. It focuses on these short-lived states before the circuit reaches a steady state.In the exercise, the function \( i(t) = \frac{U_{0}}{R} \left(1 - e^{-\frac{R(t-t_{0})}{L}}\right) \) represents the transient response of the current after the circuit is activated. Initially, at \( t = t_0 \), no current flows since the exponential term equals one, giving a zero result. As time progresses, the exponential term decreases, allowing the current to increase gradually toward \( \frac{U_{0}}{R} \).Transient analysis is crucial for understanding how components like inductors and capacitors react when circuits are switched on/off. It ensures engineers can predict and mitigate unwanted effects in electronic devices.

Differential Equations

Differential equations are mathematical equations that relate a function with its derivatives, expressing how a quantity changes over time. In electrical engineering, they're used to model the behavior of circuits with reactive components like inductors and capacitors.In this exercise, the differential equation \( L \cdot \frac{di(t)}{dt} + R \cdot i(t) = U_{0} \) describes the interaction between the inductance, resistance, and voltage source in the circuit. It explains how the rate of change of the current and the current itself combine to balance the applied voltage \( U_{0} \).The solution of this differential equation gives us the transient current function. Solving such equations involves calculating both the transient response and the long-term steady state of the system. Understanding these concepts allows for the precise design and analysis of electronic circuits that behave predictably over time.