Lab 2 Impedance Matching PDF

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UEEA1253 – CIRCUITS, SIGNALSAND SYSTEMSLAB 2IMPEDANCE MATCHINGNAME : Tan Joe Yan ID : 1903709 GROUP : P COURSE : BI DATE OF SUBMISSION : 22/3/ EMAIL : joeyantan@1utarContribution Details and Percentage Introduction (10%) Results and Discussion (50%) Conclusions (20%) References (5%) Report presentat...

Description

UEEA1253 – CIRCUITS, SIGNALS AND SYSTEMS LAB 2 IMPEDANCE MATCHING NAME

:

Tan Joe Yan

ID

:

1903709

GROUP

:

P6

COURSE

:

BI

DATE OF SUBMISSION EMAIL

:

: [email protected]

Contribution Details Percentage

and

Introduction (10%) Results and Discussion (50%) Conclusions (20%) References (5%) Report presentation format and proper English (15%) PENALTY Total

22/3/2021

Introduction

The objectives of this experiment are to measure the power transfer coefficient of a circuit and to show how impedance matching can improve the power transfer to the load for a narrowband about the frequency of interest. Maximum power transfer theorem is an important and useful circuit analysis method to ensure that the DC voltage source will deliver maximum power to the variable load resistor only when the load resistance is equal to the source resistance. Also, the theorem also states that the AC voltage source will deliver maximum power to the variable complex load only when the load impedance is exactly equal to the complex conjugate of the source impedance (Maximum Power Transfer Theorem - Tutorialspoint, no date). Therefore, in other words, when the load resistance is equal to the Thevenin or Norton resistance of the network supplying the power, the amount of power dissipated by load resistance will be maximum. In contrast, if the load resistance is unequal to the Thevenin or Norton resistance, either higher or lower, the power dissipated by load resistance will be lesser than the maximum (Maximum Power Transfer Theorem | DC Network Analysis | Electronics Textbook, no date). Consider the Thevenin’s circuit below:

As stated, the load resistance will dissipate maximum power when it has equivalent value to the Thevenin source resistance, which is RL=RS. Some of the applications of maximum power transfer theorem are radio transmitter at the final amplifier stage design, a grid-tied inverter loading a solar array and an electric vehicle design. Most of these applications involves seeking to maximize the power delivered to the drive motor using maximum power transfer theorem. However, the maximum power dissipated is not equal to

the maximum efficiency. (Maximum Power Transfer Theorem | DC Network Analysis | Electronics Textbook, no date). Impedance matching is essential to match the load impedance to the source or internal impedance of a driving source. The graph below showing the load power versus load resistance shows that a matching load with source impedance will achieve maximum power dissipated.

Based on the plot above, when the load matches the source, the amount of power delivered to the load equal to the power dissipated in the source. So, transfer of maximum power is only 50% efficient (Frenzel, 2011). When RL=RS it is called a “matched condition”. If there is improper impedance matching, it may lead to excessive power loss and heat dissipation(Maximum Power Transfer Theorem in DC Theory, no date).

Results and Discussion

Experiment 3.1: Measurement of the Power Transfer Coefficient

a. Set a sine wave f = 3k Hz and vs = 16 Vpp . vs (peak value) = 8 Vp vs (rms) =

vs (peak value) 8 √2

√2

= 5.66V

b. By assuming vL (peak value) = 6.5V and R L = 200ꭥ

RL vL (peak value) = v (peak value) R g + RL s 6.5 =

200 8 R g + 200

R g = 46.15 ꭥ

c. Maximum power available: Pmax=

vs 2 (𝑅𝑀𝑆)

No.

RL (ꭥ)

VL(RMS), V

1 2 3 4 5 6 7 8 9

10 33 56 68 100 220 330 470 680

1.004 2.350 3.091 3.359 3.858 4.661 4.947 5.134 5.280

4Rg

=

5.662

4(46.15)

𝑷𝑳 =

= 0.1735𝑊

𝒗𝑳 𝟐 (𝑹𝑴𝑺) ,𝑾 𝑹𝑳 0.1008 01673 0.1706 0.1659 0.1488 0.0987 0.0742 0.0561 0.0410

𝑷𝑳 𝑷𝒎𝒂𝒙 0.5810 0.9643 0.9833 0.9562 0.8576 0.5689 0.4277 0.3233 0.2363

𝒕=

Graph of t against RL Power transfer coefficient, t

1.2 1 0.8 0.6 0.4 0.2 0 0

100

200

300

400

500

600

700

800

Load resistance, RL (ꭥ)

Chart 1: Graph of t against RL of the experimental results Confirmation of Experimental Results (Theoretical Results)

Given:

R g = 46.15 ꭥ

vs (rms) = 5.66V 𝑃𝑚𝑎𝑥= 0.1735W

Let

R L = 10 ꭥ RL 10 v𝐿(𝑅𝑀𝑆) = vs(rms)= (5.66) = 1.008𝑉 R g + RL 46.15 + 10

Using voltage divider rule:

𝑃𝐿 =

𝑣𝐿 2 (𝑅𝑀𝑆) 1.0082 = 0.1016𝑊 = 10 𝑅𝐿

𝑡=

𝑃𝐿 0.1016 = 0.5856 = 𝑃𝑚𝑎𝑥 0.1735

The calculations are repeated for other load resistance values and are the answers are tabulated below.

No.

RL (ꭥ)

VL(RMS), V

1 2 3 4 5 6 7 8 9

10 33 56 68 100 220 330 470 680

1.008 2.360 3.103 3.372 3.873 4.679 4.966 5.154 5.300

𝑷𝑳 =

𝒗𝑳 𝟐 (𝑹𝑴𝑺) ,𝑾 𝑹𝑳 0.1016 0.1688 0.1719 0.1672 0.1500 0.0995 0.0747 0.0565 0.0413

𝑷𝑳 𝑷𝒎𝒂𝒙 0.5856 0.9729 0.9908 0.9637 0.8646 0.5735 0.4305 0.3256 0.2380

𝒕=

Chart of t against RL Power transfer coeeficient, t

1.2 1 0.8 0.6 0.4 0.2 0 0

100

200

300

400

500

600

700

800

Load resistance, RL(ꭥ) Experimental Results

Theoretical Results

Chart 2: Chart of t against RL of the theoretical and experimental results in comparison. Percentage difference of t between the experimental and theoretical result at maximum point: t max(theoretical) -tmax(experimental) Percentage difference = × 100% t max(theoretical) =

Discussions

0.9908-0.9833 × 100% = 0.76% 0.9908

After completing experiment 3.1, it is found that the theoretical and the experimental

values slightly differ, with a percentage difference of 0.76% using values of t max. This is due

to the neglection of internal wire resistance in the circuit when calculating the theoretical results using the LTspice simulation software. It is observed that the power transfer coefficient (t) is maximum when load resistance is 56ꭥ. When the load resistance is around 68ꭥ, the slope starts to fall. A geometry symmetry can be seen at the top part of the graph, which is around the maximum point. This is due to the load resistance is almost equal to the Rg, which is 46.15ꭥ. This explains the maximum power transfer theorem. Which is when the load resistance is equal to the source resistance, the power dissipated by load resistance will be at its maximum.

Experiment 3.2: Impedance Matching for Maximum Power Transfer

Suppose Rg is as measured and the load resistor RL=680ꭥ, hence a mismatching. The power transfer ratio:

t=

Given L =

1

2𝜋f0

4R g R L 4(46.15)(680) = = 0.2381 2 (R g + R L ) (46.15 + 680)2

√R g (R L -R g ) and C =2𝜋f

1

0 RL

√

RL -Rg Rg

,

L = 10mH, Rg = 46.15ꭥ, RL = 680ꭥ (ideal)

Substitute in the equation: f0 =

1

2𝜋L

C = 2𝜋f 1

√R g (R L -R g ) =2𝜋(0.01) √46.15(680 -46.15) = 2722 Hz

0 RL

√

RL -Rg Rg

=C=

1

1

2𝜋(2722)(680)

√46,15 = 319nF 680-46.15

Assume internal resistance of inductor=59.6ꭥ.

Set vs = 7 Vrms sine wave, f=100Hz and RL=680ꭥ, Pmax=

vs 2 (𝑅𝑀𝑆) 4R g

72 = 0.2654𝑊 = 4(46.15)

No

f, Hz

vL(RMS) ,V

1 2 3 4 5 6 7 8 9 10 11 12

100 500 1000 2000 2500 3000 3500 4000 5000 6000 7000 8000

6.029 6.133 6.460 7.666 8.054 7.418 5.993 4.536 2.684 1.733 1.259 0.915

𝑷𝑳 =

𝑽𝑳(𝑹𝑴𝑺)𝟐 ,𝑾 𝑹𝑳 0.0535 0.0553 0.0614 0.0864 0.0954 0.0809 0.0528 0.0303 0.0106 4.417m 2.331m 1.231m

𝒕=

0.2016 0.2084 0.2313 0.3255 0.3595 0.3048 0.1989 0.1114 0.0400 0.0166 8.783m 4.638m

Graph of t against f 0.4 Power transfer coefficient, t

0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 -0.05

0

2

4

6 8 Frequency, f (Hz)

10

12

Chart 3: Chart of t against f of experimental result. Confirmation of Experimental Results (Theoretical Results)

𝑷𝑳 𝑷𝒎𝒂𝒙

14

VS -VL

V RLL = V sC + V -V S S RL + sL Rg + l 1) = VS (𝑠𝐶 + R L VS R g + R l + sL 1 1 = VL (sC + + R + R + sL) g l R g + R l + sL RL sC(R L )(R g + R l + sL) + R g + R l + sL + R L VS = VL ( ) R g + R l + sL R L (R g + R l + sL) sCR L R g + sCR L R l + s2 LCR L + R g + R l + sL + R L VL ( ) = VS RL RL ) VL = VS ( 2 sCR L R g + sCR L R l + s LCR L + R g + R l + sL + R L RL ) VL = VS ( s(CR L R g + CR L R l + L) + s2 LCR L + R g + R l + R L 𝑠𝑖𝑛𝑐𝑒 𝑠 = 𝑗𝜔, RL ) VL = VS ( 𝑗𝜔(CR L R g + CR L R l + L) + (𝑗𝜔)2 LCR L + R g + R l + R L RL ) VL = VS ( 𝑗𝜔(CR L R g + CR L R l + L) + 𝜔 2 LCR L + R g + R l + R L

VL = VS

RL

√𝜔 2 (CR L R g + CR L R l + L)2 + (-𝜔 2 LCR L + R g + R l + R L )2 ( )

Assume that the inductor is ideal, Rl=0. Given,

𝜔 = 2𝜋f 𝐶 = 319𝑛𝐹 L = 10mH R g = 46.15ꭥ R L = 680ꭥ VS = 7V Pmax = 0.2654𝑊

4760 VL = ( ) √0.015809f 2 + ((-85.637𝜇)f 2 + 726.15)2

Simplifying the equation:

VL = 6.562V

Let f=100 Hz,

Repeat using the same formula to calculate the other values of f and tabulate the data below: No

f, Hz

vL(RMS) ,V

1

100

6.562

𝑷𝑳 =

𝑽𝑳(𝑹𝑴𝑺)𝟐 ,𝑾 𝑹𝑳 0.0633

𝒕=

𝑷𝑳

𝑷𝒎𝒂𝒙 0.2385

2 3 4 5 6 7 8 9 10 11 12

500 1000 2000 2500 3000 3500 4000 5000 6000 7000 8000

6.728 7.292 10.378 12.943 12.532 8.721 5.825 3.075 1.924 1.330 0.9795

0.0666 0.0782 0.1584 0.2464 0.2310 0.1118 0.0499 0.0139 5.444m 2.601m 1.411m

0.2509 0.2946 0.5968 0.9284 0.8704 0.4213 0.1880 0.0524 0.0205 9.800m 5.317m

Assume that the inductor is not ideal, Rl=59.6ꭥ. Given, 𝜔 = 2𝜋f 𝐶 = 319𝑛𝐹 L = 10mH R g = 46.15ꭥ R L = 680ꭥ VS = 7V Pmax = 0.2654𝑊 4760 VL = ( ) 2 √0.04283f + ((-85.637𝜇)f 2 + 785.75)2

Simplifying the equation:

VL = 6.062V Repeat using the same formula to calculate the other values of f and tabulate the data below:

Let f=100 Hz,

No

f, Hz

vL(RMS) ,V

1 2 3 4 5 6 7 8 9 10 11 12

100 500 1000 2000 2500 3000 3500 4000 5000 6000 7000 8000

6.062 6.171 6.520 7.849 8.280 7.665 6.176 4.697 2.792 1.823 1.285 0.956

𝑷𝑳 =

𝑽𝑳(𝑹𝑴𝑺)𝟐 ,𝑾 𝑹𝑳 0.0540 0.0560 0.0625 0.0906 0.1008 0.0864 0.0561 0.0324 0.0115 4.887m 2.417m 1.344m

𝒕=

𝑷𝑳 𝑷𝒎𝒂𝒙

0.2035 0.2184 0.2438 0.3414 0.3798 0.3255 0.2114 0.1221 0.0433 0.0184 9.107m 5.064m

Chart of t against f Power Transfer Theorem, t

1.2 1 0.8 0.6 0.4 0.2 0 -0.2

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

Frequency, f (kHz) Experimental Results Theoretical Value with Ideal Internal Resistance Theoretical results with nonideal internal resistance

Percentage difference of t between the experimental and theoretical result at maximum point: t max(theoretical,ideal)-t max(experimental) Percentage difference = × 100% t max(theoretical) 0.9284-0.3595 × 100% = 61.28% 0.9284 t max(theoretical,non- ideal) -t max(experimental) × 100% Percentage difference = t max(theoretical) 0.3798-0.3595 × 100% = 5.34% = 0.3798 =

Discussion In experiment part 3.2, the load resistance of 680ꭥ is fixed and the frequency of signal generator is manipulating instead. When the case is assumed to be ideal with internal resistance of 0ꭥ, the highest power transfer coefficient is 0.9284 at 2500 Hz. When the case is assumed to be non-ideal with internal resistance of 59.6ꭥ, the highest power transfer coefficient is 0.3789 at 2500 Hz. Both experimental and theoretical results, no matter with ideal or non-ideal internal resistance, the highest power transfer coefficient is met at 2500 Hz. The percentage error of this part of the experiment is quite large, with 61.28% when comparing the experimental and theoretical result with an ideal internal resistance. This proves that the internal resistance should not be neglected because it makes a huge

difference. Therefore, it can be seen that the theoretical value will be closer to the experimental value if the internal resistance was taken into account.

Conclusion From experiment part 3.1, the power transfer coefficient is conducted by varying the value of load resistance. This is to identify which load resistance gives the maximum t. From experiment part 3.2, it is proven theoretically that calculating and neglecting the internal resistance causes a huge difference during the experiment. For other cases, the percentage error of theoretical and experimental results is almost the same with or without the internal resistor. This proves that the internal resistance should be taken into account in order to get a higher accuracy during theoretical calculation. The objective of this experiment is achieved. In this experiment, the power coefficient of a circuit is calculated and how the impedance matching improved the power transfer has been proven.

References: Frenzel, L. (2011) Back to Basics: Impedance Matching (Part 1) | Electronic Design. Available at: https://www.electronicdesign.com/technologies/communications/article/21796367/backto-basics-impedance-matching-part-1 (Accessed: 22 March 2021). Maximum Power Transfer Theorem - Tutorialspoint (no date). Available at: https://www.tutorialspoint.com/network_theory/network_theory_maximum_power_transf er_theorem.htm (Accessed: 22 March 2021). Maximum Power Transfer Theorem | DC Network Analysis | Electronics Textbook (no date). Available at: https://www.allaboutcircuits.com/textbook/direct-current/chpt-10/maximumpower-transfer-theorem/ (Accessed: 22 March 2021). Maximum Power Transfer Theorem in DC Theory (no date). Available at: https://www.electronics-tutorials.ws/dccircuits/dcp_9.html (Accessed: 22 March 2021)....