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Ch.9 Sinusoids and Phasors. 1. Introduction. AC is more efficient and economical to transmit over long distance Sinusoid is a signal that has the form of the sine or cosine function Sinusoidal current = alternating current (ac) Nature is sinusoidal Easy to generate and transmit

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Ch.9 Sinusoids and Phasors

1. Presentation AC is more proficient and temperate to transmit over long separation Sinusoid is a flag that has the type of the sine or cosine work Sinusoidal current = exchanging current (air conditioning) Nature is sinusoidal Easy to produce and transmit Any reasonable occasional flag can be spoken to by a total of sinusoids Easy to handle numerically Electric Circuit, 2007

2. Sinusoids Consider the sinusoidal voltage T: time of the sinusoid Electric Circuit, 2007

Sinusoids (2) Periodic capacity Satisfies f(t) = f(t+nT), for all t and for all whole numbers n Hence Cyclic recurrence f of the sinusoid Electric Circuit, 2007

Sinusoids (3) Let us analyze the two sinusoids Trigonometric characters Electric Circuit, 2007

Sinusoids (4) Graphical approach Used to include two sinusoids of similar recurrence where Electric Circuit, 2007

Example 9.1 Find the adequacy, stage, period, and recurrence of the sinusoid Electric Circuit, 2007

Example 9.2 Sol) Electric Circuit, 2007

3. Phasors Phasor is an unpredictable number that speaks to the plentifulness and period of a sinusoid Provides a basic method for breaking down direct circuits energized by sinusoidal sources Complex number with Electric Circuit, 2007

Phasors (2) Operations of complex number Addition: Subtraction: Multiplication: Division: Reciprocal: Square Root: Complex Conjugate: Electric Circuit, 2007

Phasors (3) Euler\'s character with Given a sinusoid Thus, where Plot of the Electric Circuit, 2007

Phasors (4) Phasor representation of the sinusoid v(t) Electric Circuit, 2007

Phasors (5) Derivative & indispensable of v(t) Derivative of v(t) Phasor area representation of subsidiary v(t) Phasor space rep. of Integral of v(t) Electric Circuit, 2007

Phasors (6) Summing sinusoids of similar recurrence Differences amongst v(t) and V v(t) is time area representation, while V is phasor space rep. v(t) is time subordinate, while V is not v(t) is constantly genuine with no unpredictable term, while V is for the most part complex Phasor investigation Applies just when recurrence is consistent Applies in controlling at least two sinusoidal flags just on the off chance that they are of similar recurrence Electric Circuit, 2007

Example 9.3 Evaluate these mind boggling numbers Sol) a) then Taking the square root Electric Circuit, 2007

Example 9.4 Transform these sinusoids to phasors Example 9.5 Find the sinusoids spoke to by these phasors Electric Circuit, 2007

Example 9.6 Example 9.7 Using the phasor approach, decide the current i(t) Electric Circuit, 2007

4. Phasor Relationships for Circuit Elements Voltage-current relationship Resistor: ohm\'s law Phasor frame Inductor Phasor shape Electric Circuit, 2007

Phasor Relationships for Circuit Elements(2) Inductor The present slacks the voltage by 90 o . Capacitor: Phasor frame The present leads the voltage by 90 o . Electric Circuit, 2007

Example 5.6 The voltage v=12cos(60t+45 o ) is connected to a 0.1H inductor. Locate the consistent state current through the inductor Sol) Converting this to the time area, Electric Circuit, 2007

5. Impedance and Admittance Voltage-current relations for three detached components Ohm\'s law in phasor frame Imdedance Z of a circuit is the proportion of the phasor voltage to the phasor current I, gauged in ohms When , When , Electric Circuit, 2007

Impedance and Admittance (2) Impedance = Resistance + j Reactance where Adimttance Y is the complementary of impedance, measured in siemens (S) Admittance = Conductance + j Susceptance Electric Circuit, 2007

Example 9.9 Find v(t) and i(t) in the circuit Sol) From the voltage source The impedance Hence the current The voltage over the capacitor Electric Circuit, 2007

6. Kirchhoff\'s law in the recurrence space For KVL, Then, KVL holds for phasors KCL holds for phasors Time area Phasor area KVL & KCL holds in recurrence area Electric Circuit, 2007

7. Impedance Combinations Consider the N arrangement associated impedances Voltage-division relationship Electric Circuit, 2007

Impedance Combinations (2) Consider the N parallel-associated impedances Current-division relationship Electric Circuit, 2007

Example 9.10 Find the information impedance of the circuit with w=50 rad/s Sol) The info impedance is Electric Circuit, 2007

Example 9.11 Determine v o (t) in the circuit Sol) Time area recurrence space Voltage-division guideline Electric Circuit, 2007