Wechselspannungsgenerator (E15) - Alternating Current Generator

Goal of the Experiment (E15: Alternating Current Generator)

• This experiment aims to verify that a time-varying magnetic flux, caused by a change in the area permeated by the flux, induces a voltage.

• It also demonstrates that the power of a voltage source depends on the ratio of its internal resistance to the load resistance.

Theoretical Background

• Induced Voltage in a Rotating Coil:

• A coil with $N$ turns and cross-sectional area $A$ (area normal $\vec{n}$, area vector $\vec{A} = A\vec{n}$) rotates with frequency $\nu$ in a homogeneous, time-constant magnetic field of flux density $\vec{B}$.

• The axis of rotation is perpendicular to $\vec{B}$.

• According to Faraday's Law of Induction, the induced voltage at the ends of the coil is given by: $U_{ind} = -N\dot{\phi} = -N\frac{d\phi}{dt} = -N \frac{d}{dt} (\vec{B} \cdot \vec{A}) = -N \frac{d}{dt} (BA \cos\theta)$

  • Since $\vec{B}$ and $A$ are constant, the equation simplifies to:
    $U_{ind} = -NBA \frac{d}{dt} (\cos\theta)\quad (1)$

• Angle of Rotation:

  • $\theta$ is the angle between the area vector $\vec{A}$ and the magnetic field $\vec{B}$.

  • For rotation with constant frequency $\nu$ or constant angular velocity $\omega$, this angle is a linear function of time:
    $\theta = \omega t = 2\pi\nu t\quad (2)$

• Alternating Current (AC) Induction Voltage:

  • Substituting $\theta$ into Equation (1), the induction voltage becomes:
    $U_{ind} = -NBA \frac{d}{dt} (\cos(\omega t)) = NBA\omega \sin(\omega t)\quad (3)$

  • This shows that an alternating voltage is induced in the rotating coil, and the system acts as a generator.

• Source Voltage vs. Terminal Voltage:

• Source Voltage ($U<em>{ind}$): Also known as electromotive force (EMF), this is the constant voltage of a source, independent of the external circuit. In this experiment, $U</em>{ind}$ is identical to the induced voltage.

• Terminal Voltage ($U_K$): This is the voltage actually available to the consumer. It depends on the current $I$ drawn by the consumer.

• Voltage Drop: When a current is drawn (loaded source), the terminal voltage decreases proportionally to the current due to the source's internal resistance:
  $U<em>K = U</em>{ind} - IR_i$

• Internal Resistance ($R_i$):

• The quotient of the voltage drop ($U<em>{ind} - U</em>K$) and the current ($I$) defines the internal resistance of the voltage source:
  $R<em>i = \frac{U</em>{ind} - U_K}{I}\quad (4)$

• Current Limitation: The source is current-limited. In a short-circuit ($U<em>K = 0$), the maximum current that can be drawn is: $I</em>{max} = \frac{U<em>{ind}}{R</em>i}$

• When no current is drawn (unloaded generator, $I=0$), $U<em>K = U</em>{ind}$ (open circuit).

• Equivalent Circuit Diagram (Figure 1):

• Figure 1a: Unloaded Generator: Represents the ideal source voltage ($U_{ind}$) across its terminals.

• Figure 1b: Loaded Generator: Represents a real generator as an equivalent circuit consisting of:

  • An ideal voltage source ($U_{ind}$).

  • An ohmic resistor ($R<em>S$) in series, representing the internal resistance ($R</em>i$).

  • An ideal (resistanceless) inductor ($L_S$) in series, representing the coil's inductance.

• This series circuit describes the electrical behavior of the real generator coil, simplifying complex elements for calculation.

• The load resistance ($R_V$) represents a potentially much more complex real-world load, though in this experiment, a simple resistance decade is used.

• Approximation for Low Frequencies:

• When the generator is loaded, only part of the source voltage $U<em>{ind}$ drops across the load resistance $R</em>V$. The remainder drops across the internal resistance $R<em>S$ and the inductance $L</em>S$ (due to AC).

• For this experiment, the inductance ($L<em>S$) and the frequency ($\nu$) of the alternating voltage are considered very small. Therefore, the inductive reactance ($\omega L</em>S$) can be neglected compared to the ohmic resistances ($R<em>S$ and $R</em>V$).

• In this approximation, effective values of currents and voltages are used, denoted as $U<em>K$, $U</em>{ind}$, and $I$ for simplicity. (Note: The effective value for a sinusoidal voltage is $U<em>{eff} = U</em>{peak} / \sqrt{2}$.).

• The terminal voltage $U<em>K$ is given by: $U</em>K = R<em>V I = U</em>{ind} - R_S I\quad (5)$

• Rearranging Equation (5) yields the terminal voltage and current:
  $U<em>K = \frac{R</em>V U<em>{ind}}{R</em>V + R<em>S}, \quad I = \frac{U</em>{ind}}{R<em>V + R</em>S}\quad (6)$

• The maximum current flows when $R<em>V \rightarrow 0$ (short circuit), at which point $U</em>K$ drops close to zero.

• Power Dissipation in the Load ($P$):

• The power converted in the load resistance $R<em>V$ is: $P = U</em>K I = R<em>V I^2 = \frac{U</em>K^2}{R_V}\quad (7)$

• Substituting from Equation (6), the power converted in the resistance is:
  $P = R<em>V \left(\frac{U</em>{ind}}{R<em>V + R</em>S}\right)^2 U<em>{ind}^2 = \frac{R</em>V U<em>{ind}^2}{(R</em>V + R_S)^2}\quad (8)$

• Power $P$ depends on $\text{U}<em>{ind}$, $R</em>V$, and the source's internal resistance $R<em>i = R</em>S$.

• Power Matching (Leistungsanpassung):

• Maximum power from the generator is obtained when $\frac{dP}{dR<em>V} = 0$ and \frac{d^2P}{dRV^2} < 0.

• Determining the load resistance $R_V$ for maximum power withdrawal is called power matching.

• Efficiency ($\eta$):

• Efficiency is a crucial metric for optimal utilization of a voltage source.

• It is defined as the ratio of the utilized power ($P$) to the total power supplied ($P<em>{ges}$): $\eta = \frac{P}{P</em>{ges}} = \frac{U<em>K}{U</em>{ind}} = \frac{R<em>V I^2}{U</em>{ind} I} = \frac{R<em>V I}{U</em>{ind}}\quad (9)$

Experimental Setup and Procedure

• Setup:

• A coil mounted on its axis of rotation is placed in a homogeneous magnetic field generated by an electromagnet.

• The current for the electromagnet (and thus the magnetic flux density $\vec{B}$) is kept constant, measured with a Hall probe.

• The coil is rotated by an electric motor.

• Small fluctuations in rotation frequency are unavoidable but generally the frequency is constant.

• An oscilloscope measures the induced voltage, showing its instantaneous voltage-time curve. From this, the peak voltage ($U<em>0$ or $U</em>{ind,0}$) and frequency (via the period) can be determined.

• A resistance decade serves as the load (consumer), allowing R_V to be varied in 10 $\Omega$ steps from 0 to 11100 $\Omega$.

  • Important Note: A minimum load of $R_V \ge 10 \Omega$ is always present to prevent the entire voltage from dropping across the generator in a short circuit, which would heavily load the carbon brushes.

• Measurements:

• First, the amplitude of the source voltage ($U<em>{ind,0}$) and then the amplitude of the terminal voltage ($U</em>{K,0}$) are measured at various load resistances $R_V$ as a function of frequency.

  • The rotation frequency is adjusted via the motor's supply voltage ($U_M$).

  • Frequencies up to a maximum of 50 Hz are recommended.

• Next, $U<em>{K,0}$ is measured as a function of load resistance $R</em>V$ at a constant frequency.

• Considerations during Measurement:

• The rotation frequency can change not only with $U<em>M$ but also with variations in $R</em>V$. $U_M$ may need readjustment to maintain the desired frequency.

• Frequency stability should be monitored using the oscilloscope.

Tasks

1. Maximum Power Calculation:

   • Calculate the load resistance $R_V$ for which the generator delivers maximum power.

   • Calculate the maximum power delivered at this resistance.

   • Neglect the inductive component of the generator's impedance $( \omega^2L<em>S^2 \ll R</em>S^2 )$.

   • Calculate the corresponding efficiency $\eta$ at this point.

   • Determine for which ratios $R<em>S/R</em>V$ the efficiency is maximal or minimal.

2. Internal Resistance Measurement:

   • Determine the ohmic resistance $R_S$ of the generator coil using a multimeter.

3. Voltage Measurement vs. Frequency:

   • Set a fixed current for the stationary, magnetic field-generating coil and measure the flux density $\vec{B}$.

   • Keep $\vec{B}$ constant throughout the experiment.

   • Measure the source voltage $U<em>{ind}$ and the terminal voltages $U</em>K$ for $R_V$ = 10, 100, and 1000 $\Omega$, each as a function of frequency $\nu$.

   • Plot these four series of data points in a graph ($U$ vs. $\omega$) and draw the corresponding best-fit curves.

4. Theoretical vs. Experimental Source Voltage:

   • Calculate theoretical values for the source voltage $U_{ind}$ using Equation (3) and include them in the graph from Task 3.

   • Compare the experimental and theoretical values.

5. Terminal Voltage Measurement vs. Load Resistance:

   • Measure the terminal voltage $U<em>K$ at a constant frequency (e.g., $\nu$ = 50 Hz) as a function of the load resistance $R</em>V$.

   • It is recommended to vary $R<em>V$ from 10 $\Omega$ to 130 $\Omega$ in 10 $\Omega$-steps, and additionally for $R</em>V$ [$\Omega$] = 150, 170, 200, 250, 300, 400, 600, and 1000.

   • Plot the measured points in a graph.

   • Calculate a sufficient number of theoretical values for $U_K$ using Equation (6) and include them in the graph.

   • Discuss the results concerning the validity of Equation (6).

6. Power and Efficiency Analysis:

   • For the measurement series $U<em>K(R</em>V)$ (from Task 5), calculate the corresponding load power $P$ for each data point and plot it against $R_V$.

   • Evaluate if the maximum power occurs at the predicted location.

   • For comparison, plot the functions $y = a/x$ and $y = b \cdot x$.

   • Describe how $P$ can be approximated for $R<em>V \ll R</em>S$ and for $R<em>V \gg R</em>S$.

   • From the curve, determine the value of the maximum load power and the efficiency at this point.