Book Creator
 Add this page to your book
Book Creator
 Remove this page from your book  

 Manage book(0 page(s))
 Help

This is an old revision of the document!

Experiment 4: Alternating Voltage

Objectives of the Experiment

Getting to know

  • sinusoidal quantities, active (real), reactive, and apparent resistance/impedance, phasor diagrams
  • behavior of an RC and RL series circuit with different passive components at constant frequency
  • low-pass filter (behavior of an RC circuit at different frequencies)
  • RLC series resonant circuit: voltage across the resistor and impedances at different frequencies

Applying

  • voltage analysis in the time domain using a simulation program

Alternating Current Circuits: Capacitors and Inductors

Sinusoidal quantities


figure 1 shows the waveforms of a sinusoidal voltage $u(t) = \hat{U} \cdot \sin(\omega t + \varphi_{\rm u})$ and a sinusoidal current $i(t) = \hat{I} \cdot \sin(\omega t + \varphi_{\rm i})$:
lab_electrical_engineering:fig-1_v4-sinusoidal-quantities.svg

Fig. 1: Sinusoidal quantities


  • $\omega = 2\pi f$ is defined as the angular frequency and has the unit $1/s$
  • $\hat{I}$ is referred to as the peak value or amplitude of the current
  • $I$ is the RMS (effective) value of the current


In alternating current systems, the RMS value is defined as the value that produces the same amount of heat in a purely resistive load as a direct current of the same magnitude. The crest factor is defined as the ratio of the peak value to the RMS value:
$$c = \frac{\hat{I}}{I} = \frac{\hat{U}}{U}$$
For sinusoidal quantities, $c = \sqrt{2}$.
$\varphi = \varphi_{\rm u} - \varphi_{\rm i}$ is called the phase displacement angle or simply the phase shift.
Given a two-terminal network (a) with the corresponding phasor diagram (b), s. figure 2.
Sketch the time variation of current and voltage, clearly indicating the phase shift.
lab_electrical_engineering:fig-2_v4-two-terminal-network.svg

Fig. 2: Two terminal network








In alternating current circuits, a two-terminal network is characterized by its impedance or complex resistance $Z$.
$Z = R + jX$, where $R$ is the real (active) resistance and $X$ is the reactive resistance. The magnitude of the impedance (apparent resistance) is $Z = \sqrt{(R^2 + X^2)}$, s. figure 3.

lab_electrical_engineering:fig-3_v4-reative-resistance.svg

Fig. 3: Resistance characterization


The following relationships hold:
$$R = Z\cdot \cos\varphi, ~~~ X = Z\cdot \sin\varphi, ~~~ \varphi = \arctan\frac{X}{R}$$
Similar to Ohm’s law, for sinusoidal quantities the following applies: $$Z = \frac{U}{I}$$








Experiment 1: Series Circuit of a Resistor and a Capacitor

Assemble the measurement circuit shown in figure 4:

lab_electrical_engineering:fig-4_v4-series-circuit-r_c.svg

Fig. 4: Series Circuit of a Resistor and a Capacitor


The corresponding phasor diagram is also illustrated in figure 4. The apparent resistance (impedance) of a capacitor depends on the frequency. The capacitive reactance is given by: $$X_{\rm C} = \frac {1}{ωC}$$
For a series connection of a resistor and a capacitor, the magnitude of the impedance is calculated as:
$$Z = \sqrt{R_{\rm 1}^2 + \frac{1}{(ωC_{\rm 1})^2}}$$
Adjust the function generator to output a sinusoidal voltage with a frequency of 6 kHz and an amplitude of 6 V. Measure both the frequency and the amplitude using the oscilloscope.















Sketch the time-domain waveforms of the voltages $u_{\rm F}$ and $u_{\rm C}$, clearly indicating the phase shift between them.

lab_electrical_engineering:fig-00_v4-screen-image_.svg Fig. 5

Channel 1: $ \frac{V}{\rm DIV} = $

Channel 2: $ \frac{V}{\rm DIV} = $

Time basis: $ \frac{T}{\rm DIV} = $


Using the measured values, draw the phasor diagram. Based on the given component values $R_{\rm 1}$, $C_{\rm 1}$, and the source voltage amplitude $\hat{U}_{\rm F}$, calculate the impedance $Z$, the current amplitude $\hat{I}$, and the voltage amplitudes $\hat{U}_{\rm R}$ and $\hat{U}_{\rm C}$. Compare the calculated results with the measured values.
Repeat this experiment using a capacitor with a capacitance of 1 µF

lab_electrical_engineering:fig-00_v4-screen-image_.svg Fig. 6

Channel 1: $ \frac{V}{\rm DIV} = $

Channel 2: $ \frac{V}{\rm DIV} = $

Time basis: $ \frac{T}{\rm DIV} = $


Draw the corresponding phasor diagram using the newly measured values:











Experiment 2: Series Circuit of a Resistor and an Inductor

Assemble the measurement circuit shown in figure 7:

lab_electrical_engineering:fig-5_v4-series-circuit-r_l.svg

Fig. 7: Series Circuit of a Resistor and an inductor


The corresponding phasor diagram is also illustrated in figure 7. The apparent resistance of an inductor is frequency-dependent. The inductive reactance is given by: $$X_{\rm L} = \omega \cdot L$$

For a resistor and an inductor connected in series, the impedance magnitude is calculated as: $$Z = \sqrt{R_{\rm 1}^2 + (ωL_{\rm 1})^2}$$

Set the function generator to produce a sinusoidal voltage with a frequency of 6 kHz and an amplitude of 6 V. Measure the amplitude and frequency using the oscilloscope.

Sketch the time-domain waveforms of the voltages $u_{\rm F}$ and $u_{\rm L}$, clearly indicating the phase shift between them.
lab_electrical_engineering:fig-00_v4-screen-image_.svg Fig. 8

Channel 1: $ \frac{V}{\rm DIV} = $

Channel 2: $ \frac{V}{\rm DIV} = $

Time basis: $ \frac{T}{\rm DIV} = $


Draw the phasor diagram using the measured values. From the known component values $R_{\rm 1}$, $L_{\rm 1}$, and the source voltage amplitude $\hat{U}_{\rm F}$, calculate the impedance $Z$, the current amplitude $\hat{I}$, and the voltage amplitudes $\hat{U}_{\rm R}$ and $\hat{U}_{\rm L}$. Compare the calculated results with the measured values.











Repeat this using an inductor with an inductance of 10 mH

lab_electrical_engineering:fig-00_v4-screen-image_.svg Fig. 9

Channel 1: $ \frac{V}{\rm DIV} = $

Channel 2: $ \frac{V}{\rm DIV} = $

Time basis: $ \frac{T}{\rm DIV} = $

Draw the corresponding phasor diagram based on the measured data.



















Experiment 3: R–C Circuit as a Low-Pass Filter

Assemble the R–C measurement circuit shown in figure 10:

lab_electrical_engineering:fig-6_v4-series-circuit-r-c-meas-circ.svg

Fig. 10: R–C Circuit as a Low-Pass Filter


The circuit can be interpreted as a voltage divider. With increasing frequency, the capacitive reactance of capacitor $C_{\rm 1}$ decreases, which leads to a reduction in the amplitude of the output voltage $u_{\rm 2}$. The ratio of the output voltage to the input voltage as a function of frequency is called the transfer function: $$F(j\omega) = \frac {U_{\rm 2}}{U_{\rm 1}}$$

Applying the voltage divider rule yields the following expression for the transfer function:
$$F(j\omega) = \frac {U_{\rm 2}}{U_{\rm 1}} = \frac{\frac{1}{j\omega C_{\rm 1}}}{R_{\rm 1}+\frac{1}{j\omega C_{\rm 1}}} = \frac{1}{1+j\omega C_{\rm 1}R_{\rm 1}}$$

The transfer function depends on the angular frequency ($\omega = 2\pi \cdot f$) and can be expressed in terms of magnitude and phase.
The magnitude $|\frac{U_{\rm 2}}{U_{\rm 1}}| = \frac{1}{\sqrt 1+(\omega R_{\rm 1}C_{\rm 1})^2}$ is referred to as the amplitude response.
The phase angle $\varphi$ between the input voltage $u_{\rm 1}$ and the output voltage $u_{\rm 2}$ is called the phase response. The following relation applies: $$\varphi_{\rm U_{\rm 2}}=−\arctan\omega R_{\rm 1}C_{\rm 1}$$
The cutoff angular frequency $\omega_{\rm G}$ is defined as the frequency at which the following conditions apply to the amplitude ratio $\frac{U_{\rm 2}}{U_{\rm 1}}$ and the phase angle $\varphi$:

$$\frac{U_{\rm 2}}{U_{\rm 1}}=\frac{1}{\sqrt2}, ~~~ \varphi=−45°$$
Calculate the cutoff angular frequency for the given filter.

$\omega_{\rm G}$ = ……………………………..

From this, the cutoff frequency follows as

$f_{\rm G}$ = ………………………


Measure the amplitude response of the low-pass filter.
The sinusoidal input voltage generated by the function generator shall have an amplitude of 6 V. The voltages $u_{\rm 1}$ and $u_{\rm 2}$ to be measured are connected to the channels of the oscilloscope.
Use the measured and calculated values to complete table 1.
lab_electrical_engineering:table-1-r-c-low-pass_v4.svg

Tab. 1: Amplitude response of the low-pass filter


Draw the amplitude response on the provided logarithmic graph paper.

Experiment 4: R–L–C Series Resonant Circuit


“Resonance describes the oscillatory response of a system when it is excited by a frequency that matches or is close to the system’s natural frequency.”
Set up the following measurement circuit:

lab_electrical_engineering:fig-7_v4-r_l_c-series-resonant-circuit.svg

Fig. 11: R–L–C Series Resonant Circuit


The corresponding phasor diagram is shown in figure 11.
The function generator shall be set to produce a sinusoidal signal, and the amplitude of the output voltage shall be 6 V. For the circuit shown above, the magnitude of the impedance is given by: $$Z = \sqrt{R^2 + (\omega L − \frac{1}{\omega C})^2}$$
As can be seen from figure 11, the inductive and capacitive voltage components partially cancel each other. There exists a specific frequency at which these components cancel completely, causing the circuit to behave as a purely resistive circuit. This frequency is known as the resonance frequency.
The resonance frequency is given by: $$f_{\rm r} = \frac{1}{2\pi \sqrt{LC}}\text{, or} ~~~ \omega_{\rm r} = \frac{1}{\sqrt{LC}}$$
At resonance, the current is given by: $$I = \frac{U}{R}$$
Calculate the resonance frequency for the assembled measurement circuit.
(The capacitance of the capacitor is determined using a measuring instrument)

$f_{\rm r}~=~ ..........................................................$

Adjust the frequency of the function generator so that the ratio $\frac{f}{f_{\rm r}}$ corresponds to the values listed in table 2. Measure the RMS voltage $U_{\rm R}$ using the oscilloscope and record the measured values in table 2.

lab_electrical_engineering:table-2-r_l_c_resonant_v4.svg

Tab. 2: R-L-C-circuit RMS voltage


Plot the current as a function of the ratio $\frac{f}{f_{\rm r}}$ on a logarithmic scale.















Preparation for the Oral Short Quiz

Fig. 12: Simplified representation of an antenna

For this experiment you should

  1. be able to apply and explain the following concepts:
    1. phasor representation
      1. of harmonic signals
      2. of impedance quantities
    2. complex impedance operators (e.g. $j \cdot X_L$)
    3. cutoff frequency
    4. series resonant circuit
      1. different voltages $U_R$, $U_C$, $U_L$ for $f\rightarrow 0$ and $f\rightarrow \infty$
      2. graphical and computational determination of the complex impedance $\underline{Z}$ of the circuit
      3. behavior of $\underline{Z}$ at different frequencies
      4. definition, behavior, and calculation of the resonance frequency
    5. relationship between resonant circuit and antenna (see figure 12)
  2. consider the circuit when, instead of the voltage across the resistor $U_R$, the output voltage across the capacitor $U_C$ is measured.
    1. What is this circuit called?
    2. What is it used for?
    3. What output voltage results for $f\rightarrow 0$ and $f\rightarrow \infty$?