ABSTRACT
A cross-sectional area ሺ݈݁݊݃ݐ݄ଶሻ C capacitance Farad
ܥequivalent capacitance Farad f frequency Hertz I constant current Ampere I phasor current Ampere ܫ௦constant current source (DC) Ampere ܫெ max current Ampere
ܫோெௌ root-mean-square current Ampere ݅ሺݐሻtime-dependent current Ampere ݅ሺ0ሻcurrent at t=0 Ampere
L inductance Henry ܮequivalent inductance Henry
݈ length ሺ݈݁݊݃ݐ݄ሻ P constant power Watt
ܲ௩average power Watt ܲ௧௩ reactive power Watt
ܲ apparent power Watt pሺtሻ time-dependent power Watt
Q constant charge Coulomb qሺtሻ time-dependent charge Coulomb
R resistance Ohm ܴequivalent resistance Ohm ܴresistance at ܶ Ohm ܶ period ሺݐ݅݉݁ሻ
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ܶ temperature ሺݐ݁݉݁ݎܽݐݑݎ݁ሻ ܶreference temperature ሺݐ݁݉݁ݎܽݐݑݎ݁ሻ t time ሺݐ݅݉݁ሻ V constant voltage Volt V phasor voltage Volt ெܸ max voltage Volt
ோܸெௌ root-mean-square voltage Volt vሺtሻ time-dependent voltage Volt ݒሺ0ሻvoltage at t = 0 Volt
W constant energy Joule wሺtሻ time-dependent energy Joule
Z impedance Ohm ܼimpedance of capacitor Ohm ܼequivalent impedance Ohm ܼimpedance of inductor Ohm ܼோimpedance of resistor Ohm
ߙ damping coefficient Radian ሺݐ݅݉݁ሻ⁄ ߙ௧temperature coefficient 1 ሺݐ݁݉݁ݎܽݐݑݎ݁ሻ⁄ ߚ power angle ሺ݈ܽ݊݃݁ሻ ߞ damping factor ‐‐‐‐‐‐‐‐ ߠ voltage phase angle ሺ݈ܽ݊݃݁ሻ ω angular frequency Radian ሺݐ݅݉݁ሻ⁄ ߱natural frequency Radian ሺݐ݅݉݁ሻ⁄ ߱angular resonance frequency Radian ሺݐ݅݉݁ሻ⁄ ߩ resistivity Ohm · ሺ݈݁݊݃ݐ݄ሻ ߶ current phase angle ሺ݈ܽ݊݃݁ሻ
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݅ሺݐሻ ൌ ݀ݍሺݐሻ݀ݐ For constant current: ݅ሺݐሻ ൌ ܫ
Voltage ݒሺݐሻ ൌ ݅ሺݐሻ · ܴ For constant voltage: ݒሺݐሻ ൌ ܸ
Kirchhoff’s Current Law
Kirchhoff’s Voltage Law
Sum of Voltages equals zero for any closed loop I.E: Loop 1: െ ܸ ܸ ܸ ൌ 0 Loop 2: െ ܸ െ ௗܸ ܸ ൌ 0 Loop 3: ܸ െ ܸ ௗܸ െ ܸ ൌ 0
Node 1 ܫ ܫௗ
B D
C
Sum of Currents entering a node is zero I.E: Node 1: െ ܫ െ ܫ ܫௗ ൌ 0
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Resistors in Parallel
Resistance ܴ ൌ ߩ · ݈ܣ
ܴ ൌ ܴ · ൫1 ߙ௧ · ሺ ܶ െ ܶሻ൯
Voltage ݒሺݐሻ ൌ ݅ሺݐሻ · ܴ
Power ሺݐሻ ൌ ݒሺݐሻ · ݅ሺݐሻ ൌ ሺݒሺݐሻሻ
ܴ ൌ ሺ݅ሺݐሻሻ ଶ · ܴ
1 ܴଵ
1 ܴଶ
1 ܴଷ
ܴଷ ܴ ൌ ܴଵ ܴଶ ܴଷ
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Capacitance ܥ ൌ ݍሺݐሻݒሺݐሻ
Charge ݍሺݐሻ ൌ න ݅ሺݐሻ ݀ݐ ݍሺݐଵ
Voltage ݒሺݐሻ ൌ 1ܥ · න ݅ሺݐሻ ݀ݐ ݒሺݐଵ
Current ݅ሺݐሻ ൌ ܥ · ݀ݒሺݐሻ݀ݐ
Power ሺݐሻ ൌ ݒሺݐሻ · ݅ሺݐሻ ൌ ܥ · ݒሺݐሻ · ݀ݒሺݐሻ݀ݐ
Energy ݓሺݐሻ ൌ 12 · ܥ · ሺݒሺݐሻሻ
ଶ ൌ 12 · ݒሺݐሻ · ݍሺݐሻ ൌ 1 2 ·
ሺݍሺݐሻሻଶ ܥ
ܥ ൌ 11 ܥଵ
1 ܥଶ
1 ܥଷ
ܥ ൌ ܥଵ ܥଶ ܥଷ
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Inductors in Parallel
Voltage ݒሺݐሻ ൌ ݀݅ሺݐሻ݀ݐ · ܮ
Current ݅ሺݐሻ ൌ 1ܮ · න ݒሺݐሻ ݀ݐ ݅ሺݐଵሻ
Power ሺݐሻ ൌ ݒሺݐሻ · ݅ሺݐሻ ൌ ܮ · ݅ሺݐሻ · ݀݅ሺݐሻ݀ݐ
Energy ݓሺݐሻ ൌ 12 · ܮ · ሺ݅ሺݐሻሻ
1 ܮଵ
1 ܮଶ
1 ܮଷ
ܮଷ ܮ ൌ ܮଵ ܮଶ ܮଷ
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݂ݎ ݐ 0
݅ሺݐሻ ൌ ሺ ௦ܸ െ ݒሺ0ሻሻܴ · ݁ ି௧ ோ·⁄
ݒோሺݐሻ ൌ ݅ሺݐሻ · ܴ ൌ ሺ ௦ܸ െ ݒሺ0ሻሻ · ݁ି௧ ோ·⁄ ݒሺݐሻ ൌ ݒሺ0ሻ · ݁ି௧ ோ·⁄ ௦ܸ · ൫1 െ ݁ି௧ ோ·⁄ ൯
݂ݎ ݐ 0
݅ሺݐሻ ൌ ݅ሺ0ሻ · ݁ିோ·௧ ⁄ ௦ܸܴ · ൫1 െ ݁ ିோ·௧ ⁄ ൯
ݒோሺݐሻ ൌ ݅ሺݐሻ · ܴ ൌ ݅ሺ0ሻ · ܴ · ݁ିோ·௧ ⁄ ௦ܸ · ൫1 െ ݁ିோ·௧ ⁄ ൯
ݒሺݐሻ ൌ ܮ · ݀݅ሺݐሻ݀ݐ ൌ െ݅ሺ0ሻ · ܴ · ݁ ିோ·௧ ⁄ ௦ܸ · ݁ିோ·௧ ⁄
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damping coefficient: ߙ ൌ ܴ2 · ܮ
angular resonance frequency: ߱ ൌ 1√ܮ · ܥ
damping factor: ߞ ൌ ߙ߱
natural frequency: ߱ ൌ ඥ߱ଶ െ ߙଶ
Underdamped ࣀ ൏ 1 ݂ݎ ݐ 0 ݅ሺݐሻ ൌ ܣଵ · ݁ିఈ·௧ · ܿݏሺ߱ · ݐሻ ܣଶ · ݁ିఈ·௧ · ݏ݅݊ሺ߱ · ݐሻ
Overdamped ࣀ 1 ݂ݎ ݐ 0 ݅ሺݐሻ ൌ ܤଵ · ݁ቀିఈାඥఈమିሺఠబሻమቁ·௧ ܤଶ · ݁ቀିఈିඥఈమିሺఠబሻమቁ·௧
Critically damped ࣀ ൌ ݂ݎ ݐ 0 ݅ሺݐሻ ൌ ܥଵ · ݐ · ݁ିఈ·௧ ܥଶ · ݁ିఈ·௧
Note: A1, A2, B1, B2, C1, and C2 are determined by boundary conditions
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Frequency Angular Frequency ݂ ൌ 1ܶ ߱ ൌ
2 · ߨ ܶ ൌ 2 · ߨ · ݂
RMS Values for general wave form:
ோܸெௌ ൌ ඨ1ܶ · න ൫ݒሺݐሻ൯ ଶ்
݀ݐ ܫோெௌ ൌ ඨ1ܶ · න ൫݅ሺݐሻ൯
for sinusoidal wave form: ோܸெௌ ൌ ெܸ√2 ܫோெௌ ൌ
ܫெ √2
ݒሺݐሻ ெܸ
െ ெܸ
െ ோܸெௌ
ݐ
ܶ
݅ሺݐሻ ܫெ
െܫெ
ܫோெௌ
െܫோெௌ
ݐ
ܶ
ߠ ߚ
߶
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ߚ ൌ ߠ െ ߶ Average Power
ܲ௩ ൌ ோܸெௌ · ܫோெௌ · cos ሺߚሻ
Reactive Power ܲ௧௩ ൌ ோܸெௌ · ܫோெௌ · sinሺߚሻ
Average Power for ࢼ ൌ (Pure Resistance)
ܴ ൌ ሺܫோெௌሻ ଶ · ܴ ൌ ோܸெௌ · ܫோெௌ
Phasors for sinusoidal voltage and current in the form: ݒሺݐሻ ൌ ெܸ · ܿݏሺ߱ · ݐ ߠሻ ݄ܽݏݎ ݁ݍݑ݈ܽݏ: ܄ ൌ ெܸסߠ ݅ሺݐሻ ൌ ܫெ · ܿݏሺ߱ · ݐ ߶ሻ ۷ ൌ ܫெס߶
Impedance ܼ ൌ ܄۷
Resistor: ܼோ ൌ ܴ Capacitor: ܼ ൌ 1݆ · ߱ · ܥ ൌ െ݆ ·
1 ߱ · ܥ ൌ
1 ߱ · ܥ ס െ 90
Inductor: ܼ ൌ ݆ · ߱ · ܮ ൌ ߱ · ܮ ס90
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Resistor, Inductor in Series
Inductor, Capacitor in Series
Resistor, Inductor, Capacitor, in Series
ܥ ܮ ܼ ൌ ඨܴଶ ൬߱ · ܮ െ 1߱ · ܥ൰
ס െ 90 ൏ ߚ ൏ 0
ܫ ݈݁ܽ݀ݏ ܸ ݂ݎ ߱ · ܮ ൏ 1߱ · ܥ
ס0 ൏ ߚ ൏ 90 ܫ ݈ܽ݃ݏ ܸ
݂ݎ ߱ · ܮ 1߱ · ܥ
ܼ ൌ ሺ߱ · ܮሻ െ ൬ 1߱ · ܥ൰
ܼ ൌ ඥܴଶ ሺ߱ · ܮሻଶ
ס0 ൏ ߚ ൏ 90
ܫ ݈ܽ݃ݏ ܸ
ܼ ൌ ඨܴଶ ൬ 1߱ · ܥ൰ ଶ
ס െ 90 ൏ ߚ ൏ 0 ܫ ݈݁ܽ݀ݏ ܸ
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Resistor, Inductor in Parallel
Inductor, Capacitor in Parallel
Resistor, Inductor, Capacitor, in Parallel
ܼ ൌ 1 ටቀ1ܴቁ
ଶ ቀ 1߱ · ܮ െ ߱ · ܥቁ
ס െ 90 ൏ ߚ ൏ 90
ܼ ൌ ሺ߱ · ܮሻ
1 െ ߱ଶ · ܮ · ܥ
ܼ ൌ 1 ටቀ1ܴቁ
ଶ ቀ 1߱ · ܮቁ
ס0 ൏ ߚ ൏ 90
ܫ ݈ܽ݃ݏ ܸ
ܼ ൌ 1 ටቀ1ܴቁ
ଶ ሺ߱ · ܥሻଶ
ס െ 90 ൏ ߚ ൏ 0 ܫ ݈݁ܽ݀ݏ ܸ
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Resistor Color Codes (3 and 4 Band)
A B C D Color Significant
Figure Significant Figure
Multiplier Tolerance
Black 0 0 1 Brown 1 1 10 േ 1 % Red 2 2 100 േ 2 % Orange 3 3 1,000 Yellow 4 4 10,000 Green 5 5 100,000 േ 0.5 % Blue 6 6 1,000,000 േ 0.25 % Violet 7 7 10,000,000 േ 0.1 % Gray 8 8 100,000,000 White 9 9 1,000,000,000 Gold 0.1 േ 5 % Silver 0.01 േ 10 % No Color േ 20 %
ܣ ܤ ܥ ܦ
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Resistor Color Codes (5 and 6 Band)
A,B,C D E F Color Significant
Figure Multiplier Tolerance Temperature
Coefficient Black 0 1 250 ppm/K Brown 1 10 േ 1 % 100 ppm/K Red 2 100 േ 2 % 50 ppm/K Orange 3 1,000 15 ppm/K Yellow 4 10,000 25 ppm/K Green 5 100,000 േ 0.5 % 20 ppm/K Blue 6 1,000,000 േ 0.25 % 10 ppm/K Violet 7 10,000,000 േ 0.1 % 5 ppm/K Gray 8 100,000,000 1 ppm/K White 9 1,000,000,000 Gold 0.1 േ 5 % Silver 0.01 േ 10 % No Color േ 20 %
ܣ ܤ ܥ ܦ ܧ ܨ
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