Black Box
Apparatus and materials
(1) A double beam oscilloscope.
(2) A function generator capable to generate sine, triangle and square waves over the 0.02 Hz to 2 MHz range.
(3) A “Black Box” with two groups of connectors: the ABCD group and A’B’C’D’ group. Besides, there are also two connectors for the standard resistor $R_{n}$ = 5 kΩ, which is isolated from the two groups.

Question

Black Box
Apparatus and materials
(1) A double beam oscilloscope.
(2) A function generator capable to generate sine, triangle and square waves over the 0.02 Hz to 2 MHz range.
(3) A "Black Box" with two groups of connectors: the ABCD group and A'B'C'D' group. Besides, there are also two connectors for the standard resistor [latex]R_{n}[/latex] = 5 kΩ, which is isolated from the two groups.

(4) Conductors of negligible resistance.
(5) Graph paper.
Warning: You are not allowed to open the black box.

Experiment

In the black box, there are two groups of passive elements (that are elements of the types: resistor R, capacitor C or inductor (induction coil) L). The first group consists of three elements [latex]Z_{1}[/latex] , [latex]Z_{2}[/latex],[latex]Z_{3}[/latex]connected in a star circuit as shown in Fig. 5 -12. The elements are led out to the connectors A, B, C and D, with A - the common connector of the ABCD group. The second group consists of three elements [latex]Z_{1}^{\prime}[/latex] , [latex]Z_{2}^{\prime}[/latex],[latex]Z_{3}^{\prime}[/latex] connected in the same manner to connectors A', B', C' and D' , with A' - the common connector of the A'B' C'D' group (see Fig. 5 - 13).

(1) By using the oscilloscope and the function generator, determine the type and the parameter (that is resistance of R, capacity of C, inductivity of L) of each of the elements [latex]Z_{1}[/latex] , [latex]Z_{2}[/latex],[latex]Z_{3}[/latex] and [latex]Z_{1}^{\prime}[/latex] , [latex]Z_{2}^{\prime}[/latex],[latex]Z_{3}^{\prime}[/latex].
(2) Connect five points B, C, B', C' and D' together. We obtain a new black box with terminals DD' A' (called DD' A').
(a) Draw the electric circuit of this black box.
(b) Apply a sine wave from the generator to connectors D and A'.

Plot a graph of the ratio of the voltage amplitudes [latex]K={\frac{U_{D^{\prime} A^{\prime}}}{U_{D A^{\prime}}}}[/latex] and the phase shift φ between these voltages as functions of the frequency [latex]f_{0}[/latex] of the signal.
(c) The graphs possess a particular point at a certain frequency [latex]f_{0}[/latex]

Determine the value of the frequency [latex]f_{0}[/latex], the ratio[latex]K={\frac{U_{D A^{\prime}}}{U_{D A^{\prime}}}}[/latex]
and the phase shift φ at this frequency.

(d) Derive the relation between [latex]f_{0}[/latex] and the parameters of the elements in the black box and calculate the values of [latex]f_{0}[/latex].

Appendix

Instruction to instruments

1. Oscilloscope

(1) Power ON/OFF Button

(2) CH2 or Y IN connector For applying an input signal to vertical

amplifier CH2, or the Y-axis (Vertical)

amplifier during X - Y operation.

(3) CH1 or X IN connector For applying an input signal to vertical

amplifier CH1, or the X-axis (Horizontal)

amplifier during X - Y operation.

(4) CH2 VOLTS/DIV switch To select the calibrated deflection factor of

the input signal fed to CH2 vertical amplifier.

(5) (7) VARIABLE controls Provide continuosly variable adjustement of

deflection factor between steps of the VOLTS/

DIV switches.

VOLTS/DIV calibration are accurate only

when the VARIABLE controls are click

stopped in their fully clockwise position.

(6) CH1 VOLTS/DIV switch To select the calibrated deflection factor of

the input signal fed to CH1 vertical

amplifier.

(8) FOCUS control To obtain maximum trace sharpness.

(9) INTEN control Adjusts the brightness of the CRT display.

Clockwise rotation increases brightness.

(10) × 5 MAG switch The sensibility of vertical axis will become

5 times if the switch selected at x 5 MAG.

That's to say, the measuring voltage will be

1/5 of indicator value of VOLTS/DIV.

(11) CH1 Position control For vertically positioning the CH1 trace on

the CRT screen, clockwise rotation moves

the trace upward, counterclockwise rotation

moves the trace down.

(12) V MODE switch To select the vertical amplifier display

mode. CH1 position displays only the CH1

input signal on the CRT screen.

CH2 position displays only the CH2 input

signal on the CRT screen.

DUAL position displays the CH1 and CH2

input signal on the CRT screen.

If trigger Source is selected CH1, CHOP

mode: TIME/DIV O. 2 s~l ms

ALT mode: TIME/DIV 0. 5 ms~0.2 μS

If Trigger Source is selected VERT,

ALT mode: TIME/DIV O. 2 s~0.2 μs

ADD position displays the algebraic sum of

CH1 and CH2 signal.

(13) CH2 INV switch Select switch at INV the signal added to

CH2 will be turned over.

(14) CH2 Position control For vertically positioning the CH2 trace on

the CRT screen, clockwise rotation moves

the trace upward, counterclockwise rotation

moves the trace down.

(15) Trigger LEVEL control To select the trigger signal amplitude at which

triggering occurs. When rotated clockwise,

the trigger point moves toward the positive

peak of the trigger signal. When this

control is rotated counterclockwise, the

trigger point moves toward the negative

peak of the trigger signal.

(16) Trigger Slope switch To select the positive or negative slope of

the trigger signal (on LEVEL control) for

initiating sweep. Pulled in, the switch

selects the positive (+) slope. When

pushed, this switch selects the negative (-)

slope.

(17)Horizontal Position control To adjust the horizontal position of the

traces displayed on control the CRT.

Clockwise rotation moves the traces to the

right, counterclockwise rotation moves the

traces to the left.

(18) × 10 MAG switch Placing the switch on × 10 MAG sweep

time will be expanded to 10 times and in

this instance sweep time becomes 1/10 of

TIME/DIV indicator value.

(19) VARIABLE control Provides continuously variable adjustment

of sweep rate between steps of the TIME/

DIV switch. TIME/DIV calibrations are

accurate only when the VARIABLE control is

click-stopped fully clockwise.

(20) VARIABLE control

(21) Trigger MODE switch To select the sweep triggering mode.

AUTO position selects free-running sweep

where a baseline is displayed in the absence

of a signal

This condition automatically reverts to

triggered sweep when a trigger signal of 25

Hz or higher is received and other trigger

controls are properly set.

NORM position produces sweep only when

a trigger signal is received and other controls

are properly set. No trace is visible when

the signal frequency is 25 Hz or lower.

TV - V and TV - H positions are used for

observing composite video signals. They

are not used in our case.

(22) TIME/DIV switch To select either the calibrated sweep rate

of the main time base , the delay time range

for delayed sweep operation or X - Yoperation.

(23) Trigger Source switch To conveniently select the trigger source.

In our case, set it to CH1 .

2. Function generator

• The Power button may be pressed for "ON" and pressed again for "OFF".

• Select the Frequency Range and press the proper button.

• The frequency is shown on the digital display.

• Use the Frequency adjusting knob to tune to the proper frequency.

• Select the Waveform (sine wave, triangle wave or square wave) by pressing the appropriate button.

Accepted Answer

(1) The type of the elements Z.

Adjust the oscilloscope to obtain the same gain for the two channels. Use the sine wave from the generator.

Use the circuit in Fig. 5 - 16 to determine the type of the elements. We find finally:

[latex]Z_{3}[/latex], [latex]Z_{1}^{\prime}[/latex] and [latex]Z_{2}^{\prime}[/latex] are resistors and [latex]Z_{1}^{\prime}[/latex] = [latex]Z_{2}^{\prime}=2Z_{3}=(10.0\pm0.5)\ \mathrm{k\Omega},[/latex]

[latex]Z_{1}[/latex], [latex]Z_{2}[/latex] and [latex]Z_{3}^{\prime}[/latex] are capacitors, and [latex]C_{1}=C_{2}=\frac{1}{2}C^{\prime}{}_{3}=\ (47\pm2)\ \mathrm{nF}.[/latex]

(2) (a) The electric circuit of the black box DD' A' is shown In Fig. 5 -17.

(b) Apply a sine wave signal to connectors D and A'. Connect D and A' to channel 1 (CH1) and D' and A' to channel 2 (CH2).

The ratio [latex]K={\frac{U_{D^{\prime} A^{\prime}}}{U_{\mathrm{DA^{\prime}}}}}[/latex] is determined from the amplitudes of the signals [latex]U_{D^{\prime} A^{\prime}}[/latex]

vá [latex]U_{D A^{\prime}}[/latex]. The phase shift φ can be determined directly from the traces of the signals (or from the Lissajous patterns).

Tabulating K and φ versus f, we get for example:

Table--1--

The experimental error of f is ± 1 Hz, of φ is ± 5° and of K is ± 0.02. Here are the plots of K and φ as functions of f.

(c) The graphs possess a particular point at [latex]f_{0}[/latex] = 330 ± 1 Hz, at which φ equals zero and K has a maximum of K = 0.33 ± 0.01. φ changes its sign from positive to negative with increasing of the frequency f across [latex]f_{0}[/latex].
The value of [latex]f_{0}[/latex] may vary from 325 Hz to 335 Hz depending on the set of experiment, due to the deviation in the value of the resistance and capacitance in the set.
(d) The phasor diagram for the circuit is shown in Fig. 5 - 20, where [latex]u_{1}[/latex] is the instantaneous voltage between D and D' , and [latex]U_{1}[/latex] - its amplitude.

We have tan [latex]\alpha_{1}=\frac{1}{\omega C R}\mathrm{~and}\,U_{1}=I_{1}\sqrt{\Bigl(\frac{R}{2}\Bigr)^{2}+\frac{1}{4C^{2}\omega^{2}}}.[/latex]

For the D' A' parallel cicuit, [latex]I_{1}[/latex] = [latex]I_{2}[/latex] + [latex]I_{3}[/latex], and the phasor diagram is shown in Fig. 5 - 21. [latex]U_{2}[/latex] is in phase with [latex]I_{3}[/latex] •

tan [latex]\alpha_{2}=\frac{I_{2}}{I_{1}}=\omega\cdot 2C\cdot\frac{R}{2}=\omega C R.[/latex] Let [latex]u_{2}[/latex] the voltage between D' and A', we have [latex]I_{3}=\frac{U_{2}}{\frac{R}{2}};~I_{2}=U_{2}\cdot 2C\cdot\omega.[/latex] Hence

[latex]U_{2}=I_{1}\cdot \frac{1}{\sqrt{\frac{1}{\left(\frac{R}{2} ight)^{2}}+4C^{2}\omega^{2}}}.[/latex]

By combining Fig. 5 - 20 and Fig. 5 - 21, we obtain Fig. 5 - 22, with U = [latex]U_{1}[/latex] + [latex]U_{2}[/latex] being the instantaneous voltage between D and A' .

For [latex]\omega=2\pi f=\frac{1}{C R},\mathrm{~tan~}\alpha_{1}=\frac{1}{\omega C R}=\mathrm{tan~}\alpha_{2}=\omega C R.[/latex] In this condition, [latex]U_{1}[/latex] ,[latex]U_{2}[/latex] and U are in phase, so that φ=[latex]{\alpha}_{2}-{\alpha}_{1}\,=\,0.[/latex] Hence [latex]K=\frac{U_{D^{\prime} A^{\prime}}}{U_{D A^{\prime}}}\,=[/latex] [latex]{\frac{U_{2}}{U_{1}+U_{2}}}.[/latex] Substituing [latex]\omega={\frac{1}{C R}}[/latex] into [latex]U_{1}[/latex] and [latex]U_{2}[/latex] , we obtain [latex]K={\frac{1}{3}}.[/latex]

That is, for [latex]f_{0}\,=\,{\frac{\omega}{2\pi}}={\frac{1}{2\pi R C}}={\frac{1}{2\pi\cdot 10^{4}\cdot 48\cdot 10^{-9}}}\,=\,331{\mathrm{~Hz}},\;K[/latex] [latex]{\frac{1}{3}}=0.33,\;\varphi=0\,,[/latex] which is observed in the experiment.

For [latex]\omega eq{\frac{1}{C R}},[/latex] [latex]U_{1}[/latex] and [latex]U_{2}[/latex] are out of phase, and K has values smaller than [latex]{\frac{1}{3}}.[/latex]

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