12. the impedance matching in the audio signal processing (part vii)
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The document analyzes impedance matching in audio signal processing circuits. It presents: 1) Analytical calculations of circuit parameters like Thevenin voltage, voltages, and currents for DC and AC signals. 2) Simulation results that match the analytical work with differences of 18-21% due to modeling approximations. 3) The analysis helps achieve maximum power transfer by equalizing source and load impedances using techniques like emitter followers that provide impedance matching.
![Electronusa Mechanical System [Research Center for Electronic and Mechanical]
The Impedance Matching in The Audio Signal Processing
Umar Sidik.BEng.MSc*
Director of Engineering
Electronusa Mechanical System (CTRONICS)
*umar.sidik@engineer.com
1. Introduction
Commonly, impedance is obstruction to transfer energy in the electronic circuit. Therefore, the
impedance matching is required to achieve the maximum power transfer. Furthermore, the
impedance matching equalizes the source impedance and load impedance. In other hand, the
emitter-follower (common-collector) provides the impedance matching delivered from the base
(input) to the emitter (output). The emitter-follower has high input resistance and low output
resistance. In the emitter-follower, the input resistance depends on the load resistance, while the
output resistance depends on the source resistance. In addition, this study implements the radial
electrolytic capacitor 10ߤ. ܸ53⁄ ܨ
2. Analytical Work
In this study, ܴଵ and ܴଶ form the Thevenin voltage, while ܥଵ and ܥଶ deliver ac signal as ݒ and
ݒ௨௧ (figure 1).
(a) (b)
Figure 1. (a). The concept of circuit analyzed in the study
(b). The equivalent circuit
2.1 Analysis of dc
First step, we have to calculate the Thevenin’s voltage in figure 1:
ܴଶ
்ܸு ൌ ൈ ܸ
ܴଵ ܴଶ
For this circuit, ܸ is 5ܸ, then:
24݇Ω
்ܸு ൌ ൈ 5ܸ
10݇Ω 24݇Ω
24݇Ω
்ܸு ൈ 5ܸ
34݇Ω
்ܸு ൌ ሺ0.71ሻ ൈ 5ܸ
்ܸு ൌ 3.55ܸ
1|Page](https://image.slidesharecdn.com/12-theimpedancematchingintheaudiosignalprocessingpartvii-130221095241-phpapp01/85/12-the-impedance-matching-in-the-audio-signal-processing-part-vii-1-320.jpg)
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Actually, in this circuit ்ܸு ൌ ܸ , so ܸ ൌ 3.55ܸ.
The second step, we have to calculate ܸா :
ܸா ൌ ܸ െ ܸா
ܸா ൌ 3.55ܸ െ 0.7ܸ
ܸா ൌ 2.85ܸ
The third step, we have to calculate ܫா :
ܸா
ܫா ൌ
ܴா
2.85ܸ
ܫா ൌ
150Ω
ܫா ൌ 19݉ܣ
2.2 Analysis of ac
In the analysis of ac, we involve the capacitor to pass the ac signal and we also involve the internal
resistance of emitter known as ݎ (figure 2).
(a) (b)
Figure 2. (a). The ac circuit
(b). The equivalent circuit for ac analysis
The first step, we have to calculate ݎ in the figure 2:
25݉ݒ
ݎ ൌ
ܫா
25ܸ݉
ݎ ൌ
19݉ܣ
ݎ ൌ 1.32Ω
The second step, we have to calculate ݎሺ௦ሻ :
ݎሺ௦ሻ ൌ ሺߚ 1ሻ൫ሺܴଷ ܴସ ሻԡݎ ൯
ݎሺ௦ሻ ൌ ሺ200 1ሻ൫ሺ150Ω 8.2Ωሻԡ1.32Ω൯
2|Page](https://image.slidesharecdn.com/12-theimpedancematchingintheaudiosignalprocessingpartvii-130221095241-phpapp01/85/12-the-impedance-matching-in-the-audio-signal-processing-part-vii-2-320.jpg)
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ݎሺ௦ሻ ൌ ሺ201ሻ൫ሺ158.2Ωሻԡ1.32Ω൯
1 1
ݎሺ௦ሻ ൌ ሺ201ሻ ൬ ൰
158.2Ω 1.32Ω
1.32 158.2
ݎሺ௦ሻ ൌ ሺ201ሻ ൬ ൰
208.824Ω 208.824Ω
159.52
ݎሺ௦ሻ ൌ ሺ201ሻ ൬ ൰
208.824Ω
ݎሺ௦ሻ ൌ ሺ201ሻሺ0.764Ωሻ
ݎሺ௦ሻ ൌ 153.564Ω
The third step is to calculate ݅ :
ݒ
݅ ൌ
ݎሺ௦ሻ
1ܸ݉
݅ ൌ
153.564Ω
݅ ൌ 0.0065݉ܣ
݅ ൌ 6.5ߤܣ
The fourth step is to calculate ݅ :
݅ ൌ ߚ݅
݅ ൌ ሺ200ሻሺ0.0065݉ܣሻ
݅ ൌ 1.3݉ܣ
The last step is to calculate ݒ௨௧ :
ݒ௨௧ ൌ ݅ ݎ௨௧
ݒ௨௧ ൌ ሺ1.3݉ܣሻሺ0.764Ωሻ
ݒ௨௧ ൌ 0.9932ܸ݉
ݒ௨௧ ൌ 993.2ߤܸ
3. Simulation Work
The simulation work can be classified into the dc analysis and the ac analysis.
3.1 Analysis of dc
In the simulation, ்ܸு is 3ܸ (figure 3), while in the analytical work ்ܸு is 3.55ܸ.
The different of the analytical work and the simulation work is:
்ܸுሺ௬௧ሻ െ ்ܸுሺ௦௨௧ሻ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100%
்ܸுሺ௬௧ሻ
3.55ܸ െ 3ܸ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100%
3.55ܸ
3|Page](https://image.slidesharecdn.com/12-theimpedancematchingintheaudiosignalprocessingpartvii-130221095241-phpapp01/85/12-the-impedance-matching-in-the-audio-signal-processing-part-vii-3-320.jpg)
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0.55ܸ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100%
3.55ܸ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ 18.33%
Figure 3. ்ܸு in the simulation
In the simulation, ܸா is 2.25ܸ (figure 4), while in the analytical work ܸா is 2.85ܸ. The different of the
analytical work and the simulation work is:
ܸாሺ௬௧ሻ െ ܸாሺ௦௨௧ሻ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100%
ܸாሺ௬௧ሻ
2.85ܸ െ 2.25ܸ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100%
2.85ܸ
0.6ܸ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100%
2.85ܸ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ 21.05%
Figure 4. ܸா in the simulation
In the simulation, ܫா is 15݉( ܣfigure 5), while in the analytical work ܫா is 19݉ .ܣThe difference is:
ܫாሺ௬௧ሻ െ ܫாሺ௦௨௧ሻ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100%
ܫாሺ௬௧ሻ
19݉ ܣെ 15݉ܣ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100%
19݉ܣ
4݉ܣ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100%
19݉ܣ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ 21.05%
4|Page](https://image.slidesharecdn.com/12-theimpedancematchingintheaudiosignalprocessingpartvii-130221095241-phpapp01/85/12-the-impedance-matching-in-the-audio-signal-processing-part-vii-4-320.jpg)
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Figure 5. ܫா in the simulation
3.2 Analysis of ac
In the analytical ݅ is 6.5ߤܣ݉5600.0( ܣሻ, while in the simulation ݅ is 0.07݉( ܣfigure 6). The
difference is:
݅ሺ௦௨௧ሻ െ ݅ሺ௬௧ሻ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100%
݅ሺ௦௨௧ሻ
0.07݉ ܣെ 0.0065݉ܣ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100%
0.07݉ܣ
0.0635
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100%
0.07
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ 90.71%
(a) (b) (c)
(d) (e)
Figure 6. (a). ݅ in the simulation at 1Hz
(b). ݅ in the simulation at 10Hz
(c). ݅ in the simulation at 100Hz
(d). ݅ in the simulation at 1kHz
(e). ݅ in the simulation at 10kHz
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In the simulation, ݅ is 14.9݉( ܣfigure 7), while in the analytical ݅ is 1.3݉ .ܣThe difference is:
݅ሺ௦௨௧ሻ െ ݅ሺ௬௧ሻ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100%
݅ሺ௦௨௧ሻ
14.9݉ ܣെ 1.3݉ܣ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100%
14.9݉ܣ
13.6݉ܣ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100%
14.9݉ܣ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ 91.275%
(a) (b) (c)
(d) (e)
Figure 7. (a). ݅ in the simulation at 1Hz
(b). ݅ in the simulation at 10Hz
(c). ݅ in the simulation at 100Hz
(d). ݅ in the simulation at 1kHz
(e). ݅ in the simulation at 10kHz
In the simulation, ݅௨௧ is 0.01ߤ ܣat 1Hz, is 0.42ߤ ܣat 10Hz, is 4.37ߤ ܣat 100Hz, is 39.0ߤ ܣat 1kHz,
83.3ߤ ܣat 10kHz, and 84.2ߤ ܣat 16kHz (figure 8). The difference is:
For 1Hz,
݅௨௧ሺ௬௧ሻ െ ݅௨௧ሺ௦௨௧ሻ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100%
݅௨௧ሺ௬௧ሻ
1.3݉ ܣെ 0.01ߤܣ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100%
1.3݉ܣ
1.3000݉ ܣെ 0.00001݉ܣ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100%
1.3000݉ܣ
1.29999݉ܣ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100%
1.3000݉ܣ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ 99.99%
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For 10Hz,
݅௨௧ሺ௬௧ሻ െ ݅௨௧ሺ௦௨௧ሻ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100%
݅௨௧ሺ௬௧ሻ
1.3݉ ܣെ 0.42ߤܣ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100%
1.3݉ܣ
1.3000݉ ܣെ 0.00042݉ܣ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100%
1.3000݉ܣ
1.29958݉ܣ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100%
1.3000݉ܣ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ 99.967%
For 100Hz,
݅௨௧ሺ௬௧ሻ െ ݅௨௧ሺ௦௨௧ሻ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100%
݅௨௧ሺ௬௧ሻ
1.3݉ ܣെ 4.37ߤܣ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100%
1.3݉ܣ
1.3000݉ ܣെ 0.00437݉ܣ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100%
1.3000݉ܣ
1.29563݉ܣ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100%
1.3000݉ܣ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ 99.66%
For 1kHz,
݅௨௧ሺ௬௧ሻ െ ݅௨௧ሺ௦௨௧ሻ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100%
݅௨௧ሺ௬௧ሻ
1.3݉ ܣെ 39.0ߤܣ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100%
1.3݉ܣ
1.3000݉ ܣെ 0.039݉ܣ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100%
1.3000݉ܣ
1.261݉ܣ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100%
1.3000݉ܣ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ 97.0%
For 10kHz,
݅௨௧ሺ௬௧ሻ െ ݅௨௧ሺ௦௨௧ሻ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100%
݅௨௧ሺ௬௧ሻ
1.3݉ ܣെ 83.3ߤܣ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100%
1.3݉ܣ
1.3000݉ ܣെ 0.0833݉ܣ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100%
1.3000݉ܣ
1.2167݉ܣ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100%
1.3000݉ܣ
7|Page](https://image.slidesharecdn.com/12-theimpedancematchingintheaudiosignalprocessingpartvii-130221095241-phpapp01/85/12-the-impedance-matching-in-the-audio-signal-processing-part-vii-7-320.jpg)
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ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ 93.592%
For 16kHz,
݅௨௧ሺ௬௧ሻ െ ݅௨௧ሺ௦௨௧ሻ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100%
݅௨௧ሺ௬௧ሻ
1.3݉ ܣെ 84.2ߤܣ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100%
1.3݉ܣ
1.3000݉ ܣെ 0.0842݉ܣ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100%
1.3000݉ܣ
1.2158݉ܣ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100%
1.3000݉ܣ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ 93.52%
(a) (b) (c)
(d) (e) (f)
Figure 8. (a). ݅௨௧ in the simulation at 1Hz
(b). ݅௨௧ in the simulation at 10Hz
(c). ݅௨௧ in the simulation at 100Hz
(d). ݅௨௧ in the simulation at 1kHz
(e). ݅௨௧ in the simulation at 10kHz
(f). ݅௨௧ in the simulation at 16kHz
In the simulation, ݒ௨௧ is 0.1ߤܸ at 1Hz, is 2.37ߤܸ at 10Hz, is 24.7ߤܸ at 100Hz, is 219ߤܸ at 1kHz, is
466ߤܸ at 10kHz, and 472ߤܸ at 16kHz (figure 9). The difference is:
For 1Hz,
ݒ௨௧ሺ௬௧ሻ െ ݒ௨௧ሺ௦௨௧ሻ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100%
ݒ௨௧ሺ௬௧ሻ
993.2ߤܸ െ 0.1ߤܸ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100%
993.2ߤܸ
993.19ߤܸ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100%
993.2ߤܸ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ 99.99%
8|Page](https://image.slidesharecdn.com/12-theimpedancematchingintheaudiosignalprocessingpartvii-130221095241-phpapp01/85/12-the-impedance-matching-in-the-audio-signal-processing-part-vii-8-320.jpg)
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For 10Hz,
ݒ௨௧ሺ௬௧ሻ െ ݒ௨௧ሺ௦௨௧ሻ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100%
ݒ௨௧ሺ௬௧ሻ
993.2ߤܸ െ 2.37ߤܸ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100%
993.2ߤܸ
990.63ߤܸ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100%
993.2ߤܸ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ 99.741%
For 100Hz,
ݒ௨௧ሺ௬௧ሻ െ ݒ௨௧ሺ௦௨௧ሻ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100%
ݒ௨௧ሺ௬௧ሻ
993.2ߤܸ െ 24.7ߤܸ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100%
993.2ߤܸ
968.5ߤܸ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100%
993.2ߤܸ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ 97.51%
For 1kHz,
ݒ௨௧ሺ௬௧ሻ െ ݒ௨௧ሺ௦௨௧ሻ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100%
ݒ௨௧ሺ௬௧ሻ
993.2ߤܸ െ 219ߤܸ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100%
993.2ߤܸ
774.2ߤܸ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100%
993.2ߤܸ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ 77.96%
For 10kHz,
ݒ௨௧ሺ௬௧ሻ െ ݒ௨௧ሺ௦௨௧ሻ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100%
ݒ௨௧ሺ௬௧ሻ
993.2ߤܸ െ 466ߤܸ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100%
993.2ߤܸ
527,2ߤܸ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100%
993.2ߤܸ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ 53.081%
For 16kHz,
ݒ௨௧ሺ௬௧ሻ െ ݒ௨௧ሺ௦௨௧ሻ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100%
ݒ௨௧ሺ௬௧ሻ
993.2ߤܸ െ 472ߤܸ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100%
993.2ߤܸ
521.2ߤܸ
ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100%
993.2ߤܸ
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ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ 52.477%
In this study, the simulation shows that the ݅௨௧ and ݒ௨௧ became stable started at 1 kHz.
(a) (b) (c)
(d) (e) (f)
Figure 9. (a). ݒ௨௧ in the simulation at 1Hz
(b). ݒ௨௧ in the simulation at 10Hz
(c). ݒ௨௧ in the simulation at 100Hz
(d). ݒ௨௧ in the simulation at 1kHz
(e). ݒ௨௧ in the simulation at 10kHz
(f). ݒ௨௧ in the simulation at 16kHz
10 | P a g e](https://image.slidesharecdn.com/12-theimpedancematchingintheaudiosignalprocessingpartvii-130221095241-phpapp01/85/12-the-impedance-matching-in-the-audio-signal-processing-part-vii-10-320.jpg)
12. the impedance matching in the audio signal processing (part vii)
- 1. Electronusa Mechanical System [Research Center for Electronic and Mechanical] The Impedance Matching in The Audio Signal Processing Umar Sidik.BEng.MSc* Director of Engineering Electronusa Mechanical System (CTRONICS) *umar.sidik@engineer.com 1. Introduction Commonly, impedance is obstruction to transfer energy in the electronic circuit. Therefore, the impedance matching is required to achieve the maximum power transfer. Furthermore, the impedance matching equalizes the source impedance and load impedance. In other hand, the emitter-follower (common-collector) provides the impedance matching delivered from the base (input) to the emitter (output). The emitter-follower has high input resistance and low output resistance. In the emitter-follower, the input resistance depends on the load resistance, while the output resistance depends on the source resistance. In addition, this study implements the radial electrolytic capacitor 10ߤ. ܸ53⁄ ܨ 2. Analytical Work In this study, ܴଵ and ܴଶ form the Thevenin voltage, while ܥଵ and ܥଶ deliver ac signal as ݒ and ݒ௨௧ (figure 1). (a) (b) Figure 1. (a). The concept of circuit analyzed in the study (b). The equivalent circuit 2.1 Analysis of dc First step, we have to calculate the Thevenin’s voltage in figure 1: ܴଶ ்ܸு ൌ ൈ ܸ ܴଵ ܴଶ For this circuit, ܸ is 5ܸ, then: 24݇Ω ்ܸு ൌ ൈ 5ܸ 10݇Ω 24݇Ω 24݇Ω ்ܸு ൈ 5ܸ 34݇Ω ்ܸு ൌ ሺ0.71ሻ ൈ 5ܸ ்ܸு ൌ 3.55ܸ 1|Page
- 2. Electronusa Mechanical System [Research Center for Electronic and Mechanical] Actually, in this circuit ்ܸு ൌ ܸ , so ܸ ൌ 3.55ܸ. The second step, we have to calculate ܸா : ܸா ൌ ܸ െ ܸா ܸா ൌ 3.55ܸ െ 0.7ܸ ܸா ൌ 2.85ܸ The third step, we have to calculate ܫா : ܸா ܫா ൌ ܴா 2.85ܸ ܫா ൌ 150Ω ܫா ൌ 19݉ܣ 2.2 Analysis of ac In the analysis of ac, we involve the capacitor to pass the ac signal and we also involve the internal resistance of emitter known as ݎ (figure 2). (a) (b) Figure 2. (a). The ac circuit (b). The equivalent circuit for ac analysis The first step, we have to calculate ݎ in the figure 2: 25݉ݒ ݎ ൌ ܫா 25ܸ݉ ݎ ൌ 19݉ܣ ݎ ൌ 1.32Ω The second step, we have to calculate ݎሺ௦ሻ : ݎሺ௦ሻ ൌ ሺߚ 1ሻ൫ሺܴଷ ܴସ ሻԡݎ ൯ ݎሺ௦ሻ ൌ ሺ200 1ሻ൫ሺ150Ω 8.2Ωሻԡ1.32Ω൯ 2|Page
- 3. Electronusa Mechanical System [Research Center for Electronic and Mechanical] ݎሺ௦ሻ ൌ ሺ201ሻ൫ሺ158.2Ωሻԡ1.32Ω൯ 1 1 ݎሺ௦ሻ ൌ ሺ201ሻ ൬ ൰ 158.2Ω 1.32Ω 1.32 158.2 ݎሺ௦ሻ ൌ ሺ201ሻ ൬ ൰ 208.824Ω 208.824Ω 159.52 ݎሺ௦ሻ ൌ ሺ201ሻ ൬ ൰ 208.824Ω ݎሺ௦ሻ ൌ ሺ201ሻሺ0.764Ωሻ ݎሺ௦ሻ ൌ 153.564Ω The third step is to calculate ݅ : ݒ ݅ ൌ ݎሺ௦ሻ 1ܸ݉ ݅ ൌ 153.564Ω ݅ ൌ 0.0065݉ܣ ݅ ൌ 6.5ߤܣ The fourth step is to calculate ݅ : ݅ ൌ ߚ݅ ݅ ൌ ሺ200ሻሺ0.0065݉ܣሻ ݅ ൌ 1.3݉ܣ The last step is to calculate ݒ௨௧ : ݒ௨௧ ൌ ݅ ݎ௨௧ ݒ௨௧ ൌ ሺ1.3݉ܣሻሺ0.764Ωሻ ݒ௨௧ ൌ 0.9932ܸ݉ ݒ௨௧ ൌ 993.2ߤܸ 3. Simulation Work The simulation work can be classified into the dc analysis and the ac analysis. 3.1 Analysis of dc In the simulation, ்ܸு is 3ܸ (figure 3), while in the analytical work ்ܸு is 3.55ܸ. The different of the analytical work and the simulation work is: ்ܸுሺ௬௧ሻ െ ்ܸுሺ௦௨௧ሻ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100% ்ܸுሺ௬௧ሻ 3.55ܸ െ 3ܸ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100% 3.55ܸ 3|Page
- 4. Electronusa Mechanical System [Research Center for Electronic and Mechanical] 0.55ܸ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100% 3.55ܸ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ 18.33% Figure 3. ்ܸு in the simulation In the simulation, ܸா is 2.25ܸ (figure 4), while in the analytical work ܸா is 2.85ܸ. The different of the analytical work and the simulation work is: ܸாሺ௬௧ሻ െ ܸாሺ௦௨௧ሻ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100% ܸாሺ௬௧ሻ 2.85ܸ െ 2.25ܸ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100% 2.85ܸ 0.6ܸ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100% 2.85ܸ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ 21.05% Figure 4. ܸா in the simulation In the simulation, ܫா is 15݉( ܣfigure 5), while in the analytical work ܫா is 19݉ .ܣThe difference is: ܫாሺ௬௧ሻ െ ܫாሺ௦௨௧ሻ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100% ܫாሺ௬௧ሻ 19݉ ܣെ 15݉ܣ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100% 19݉ܣ 4݉ܣ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100% 19݉ܣ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ 21.05% 4|Page
- 5. Electronusa Mechanical System [Research Center for Electronic and Mechanical] Figure 5. ܫா in the simulation 3.2 Analysis of ac In the analytical ݅ is 6.5ߤܣ݉5600.0( ܣሻ, while in the simulation ݅ is 0.07݉( ܣfigure 6). The difference is: ݅ሺ௦௨௧ሻ െ ݅ሺ௬௧ሻ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100% ݅ሺ௦௨௧ሻ 0.07݉ ܣെ 0.0065݉ܣ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100% 0.07݉ܣ 0.0635 ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100% 0.07 ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ 90.71% (a) (b) (c) (d) (e) Figure 6. (a). ݅ in the simulation at 1Hz (b). ݅ in the simulation at 10Hz (c). ݅ in the simulation at 100Hz (d). ݅ in the simulation at 1kHz (e). ݅ in the simulation at 10kHz 5|Page
- 6. Electronusa Mechanical System [Research Center for Electronic and Mechanical] In the simulation, ݅ is 14.9݉( ܣfigure 7), while in the analytical ݅ is 1.3݉ .ܣThe difference is: ݅ሺ௦௨௧ሻ െ ݅ሺ௬௧ሻ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100% ݅ሺ௦௨௧ሻ 14.9݉ ܣെ 1.3݉ܣ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100% 14.9݉ܣ 13.6݉ܣ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100% 14.9݉ܣ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ 91.275% (a) (b) (c) (d) (e) Figure 7. (a). ݅ in the simulation at 1Hz (b). ݅ in the simulation at 10Hz (c). ݅ in the simulation at 100Hz (d). ݅ in the simulation at 1kHz (e). ݅ in the simulation at 10kHz In the simulation, ݅௨௧ is 0.01ߤ ܣat 1Hz, is 0.42ߤ ܣat 10Hz, is 4.37ߤ ܣat 100Hz, is 39.0ߤ ܣat 1kHz, 83.3ߤ ܣat 10kHz, and 84.2ߤ ܣat 16kHz (figure 8). The difference is: For 1Hz, ݅௨௧ሺ௬௧ሻ െ ݅௨௧ሺ௦௨௧ሻ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100% ݅௨௧ሺ௬௧ሻ 1.3݉ ܣെ 0.01ߤܣ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100% 1.3݉ܣ 1.3000݉ ܣെ 0.00001݉ܣ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100% 1.3000݉ܣ 1.29999݉ܣ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100% 1.3000݉ܣ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ 99.99% 6|Page
- 7. Electronusa Mechanical System [Research Center for Electronic and Mechanical] For 10Hz, ݅௨௧ሺ௬௧ሻ െ ݅௨௧ሺ௦௨௧ሻ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100% ݅௨௧ሺ௬௧ሻ 1.3݉ ܣെ 0.42ߤܣ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100% 1.3݉ܣ 1.3000݉ ܣെ 0.00042݉ܣ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100% 1.3000݉ܣ 1.29958݉ܣ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100% 1.3000݉ܣ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ 99.967% For 100Hz, ݅௨௧ሺ௬௧ሻ െ ݅௨௧ሺ௦௨௧ሻ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100% ݅௨௧ሺ௬௧ሻ 1.3݉ ܣെ 4.37ߤܣ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100% 1.3݉ܣ 1.3000݉ ܣെ 0.00437݉ܣ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100% 1.3000݉ܣ 1.29563݉ܣ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100% 1.3000݉ܣ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ 99.66% For 1kHz, ݅௨௧ሺ௬௧ሻ െ ݅௨௧ሺ௦௨௧ሻ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100% ݅௨௧ሺ௬௧ሻ 1.3݉ ܣെ 39.0ߤܣ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100% 1.3݉ܣ 1.3000݉ ܣെ 0.039݉ܣ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100% 1.3000݉ܣ 1.261݉ܣ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100% 1.3000݉ܣ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ 97.0% For 10kHz, ݅௨௧ሺ௬௧ሻ െ ݅௨௧ሺ௦௨௧ሻ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100% ݅௨௧ሺ௬௧ሻ 1.3݉ ܣെ 83.3ߤܣ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100% 1.3݉ܣ 1.3000݉ ܣെ 0.0833݉ܣ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100% 1.3000݉ܣ 1.2167݉ܣ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100% 1.3000݉ܣ 7|Page
- 8. Electronusa Mechanical System [Research Center for Electronic and Mechanical] ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ 93.592% For 16kHz, ݅௨௧ሺ௬௧ሻ െ ݅௨௧ሺ௦௨௧ሻ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100% ݅௨௧ሺ௬௧ሻ 1.3݉ ܣെ 84.2ߤܣ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100% 1.3݉ܣ 1.3000݉ ܣെ 0.0842݉ܣ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100% 1.3000݉ܣ 1.2158݉ܣ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100% 1.3000݉ܣ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ 93.52% (a) (b) (c) (d) (e) (f) Figure 8. (a). ݅௨௧ in the simulation at 1Hz (b). ݅௨௧ in the simulation at 10Hz (c). ݅௨௧ in the simulation at 100Hz (d). ݅௨௧ in the simulation at 1kHz (e). ݅௨௧ in the simulation at 10kHz (f). ݅௨௧ in the simulation at 16kHz In the simulation, ݒ௨௧ is 0.1ߤܸ at 1Hz, is 2.37ߤܸ at 10Hz, is 24.7ߤܸ at 100Hz, is 219ߤܸ at 1kHz, is 466ߤܸ at 10kHz, and 472ߤܸ at 16kHz (figure 9). The difference is: For 1Hz, ݒ௨௧ሺ௬௧ሻ െ ݒ௨௧ሺ௦௨௧ሻ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100% ݒ௨௧ሺ௬௧ሻ 993.2ߤܸ െ 0.1ߤܸ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100% 993.2ߤܸ 993.19ߤܸ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100% 993.2ߤܸ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ 99.99% 8|Page
- 9. Electronusa Mechanical System [Research Center for Electronic and Mechanical] For 10Hz, ݒ௨௧ሺ௬௧ሻ െ ݒ௨௧ሺ௦௨௧ሻ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100% ݒ௨௧ሺ௬௧ሻ 993.2ߤܸ െ 2.37ߤܸ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100% 993.2ߤܸ 990.63ߤܸ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100% 993.2ߤܸ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ 99.741% For 100Hz, ݒ௨௧ሺ௬௧ሻ െ ݒ௨௧ሺ௦௨௧ሻ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100% ݒ௨௧ሺ௬௧ሻ 993.2ߤܸ െ 24.7ߤܸ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100% 993.2ߤܸ 968.5ߤܸ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100% 993.2ߤܸ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ 97.51% For 1kHz, ݒ௨௧ሺ௬௧ሻ െ ݒ௨௧ሺ௦௨௧ሻ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100% ݒ௨௧ሺ௬௧ሻ 993.2ߤܸ െ 219ߤܸ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100% 993.2ߤܸ 774.2ߤܸ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100% 993.2ߤܸ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ 77.96% For 10kHz, ݒ௨௧ሺ௬௧ሻ െ ݒ௨௧ሺ௦௨௧ሻ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100% ݒ௨௧ሺ௬௧ሻ 993.2ߤܸ െ 466ߤܸ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100% 993.2ߤܸ 527,2ߤܸ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100% 993.2ߤܸ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ 53.081% For 16kHz, ݒ௨௧ሺ௬௧ሻ െ ݒ௨௧ሺ௦௨௧ሻ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100% ݒ௨௧ሺ௬௧ሻ 993.2ߤܸ െ 472ߤܸ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100% 993.2ߤܸ 521.2ߤܸ ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ ൈ 100% 993.2ߤܸ 9|Page
- 10. Electronusa Mechanical System [Research Center for Electronic and Mechanical] ሺ%ሻ݂݂݀݅݁ ݁ܿ݊݁ݎൌ 52.477% In this study, the simulation shows that the ݅௨௧ and ݒ௨௧ became stable started at 1 kHz. (a) (b) (c) (d) (e) (f) Figure 9. (a). ݒ௨௧ in the simulation at 1Hz (b). ݒ௨௧ in the simulation at 10Hz (c). ݒ௨௧ in the simulation at 100Hz (d). ݒ௨௧ in the simulation at 1kHz (e). ݒ௨௧ in the simulation at 10kHz (f). ݒ௨௧ in the simulation at 16kHz 10 | P a g e