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AC CIRCUITS Every slide contains valuable and need-to-know information that has to be understood and retained before proceeding. Throughout this PowerPoint a color scheme has been employed. Current is depicted in purple, voltage across a resistor in orange, voltage across an inductor in green, voltage across a capacitor in blue, and generator voltage in red. IVLVRVCEIVLVRVCE

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GRAPHS OF SINUSOIDAL FUNCTIONS OF TIME. The graph is just a small piece of. A is the amplitude, is the angular frequency in radians/s. Related to is f, the frequency in Hz = s -1. Adding a phase angle, , shifts the graph. shifts the graph to the left if is positive.

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The additional graph shown in green is: T

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The average value of a function is the area under its graph divided by the width of the graph. Over one period the average value of is 0 since there is as much positive area above the horizontal axis as there is negative area below the horizontal axis. Examine the graphs of shown above. The two shaded areas in the first period lying above the midpoint can be cut off and fill in the troughs as shown. This completes a rectangular area equal to, further shaded in pink. Thus the average value of is over one period. We can now define what is called the root mean square of y to be. T A2A2 T

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EXAMPLE: Given:, identify the following: the amplitude, A the angular frequency, the frequency, f the period, T The basic sinusoidal equation is. Using this it can be seen that A = 5 (which may contain units) = 12 rad/s Then:

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Alternating Current AC Power Supply I t PHASOR i The time varying current, i, in this phasor diagram is represented by the projection of I on the y -axis. Graph of i Instantaneous current: Instantaneous emf:

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Resistance in an AC Circuit Maximum Voltage: Instantaneous Voltage: Peak-to-Peak Voltage: The Current and Voltage are in Phase. Maximum Current: RMS Current: This phasor diagram shows V R and I rotating at . v R and i are represented as projections on the y axis.

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I max t VRVR I, V Current through a Resistor Effective current, I rms Effective voltage, E rms Graphs of i and v R

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EXAMPLE: An AC generator has a maximum emf of 240 volts at a frequency of 90 Hz. Connected to the generator is a 500 resistor. What are the rms current through the resistor and the average power loss?

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Inductance in an AC Circuit Inductive Reactance: XLXL Reactance is an effective resistance to the flow of current due to the presence of electric or magnetic fields. V L leads I by 90 o This phasor diagram shows V L and I rotating at . v L and i are represented as projections on the y axis.

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I t I, V Current through an Inductor The current through an inductor lags the voltage across it by 90 o. The voltage peak across the inductor will be seen T /4 seconds before the current peak. ( T = period = 1/ f ) Graphs of i and v L

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EXAMPLE: A 300 peak voltage ac generator operating at 120 Hz is connected to a 15,000 turn solenoid of length 5.0 cm. and radius 1.0 cm. (A) what is the inductance of the solenoid. (B) What is the inductive reactance of the circuit? (C) What is the rms current in the circuit? (D) At T /3 what is the stored energy in the inductor?

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Capacitance in an AC Circuit Capacitive Reactance XCXC V C lags I by 90 o This phasor diagram shows V C and I rotating at . v C and i are represented as projections on the y axis.

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I, V Current through a Capacitor The current through a capacitor leads the voltage by 90 o. The current peak will be seen T /4 seconds before the voltage peak. I t Graphs of i and v C

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EXAMPLE: An AC generator operating at 140 V and 75 Hz is connected to a 12.7 F capacitor. (A) What is the capacitive reactance? (B) What is the rms current in the circuit? (C) What is the average power loss? Since there is no resistance, R, there is no power loss in the circuit.

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Series LRC Circuits The current i in the circuit is the same at all points at any instant of time and varies sinusoidally. The instantaneous voltage across the resistor is in phase with the current. The instantaneous voltage across the inductor leads the current. The instantaneous voltage across the capacitor lags the current.

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t Graphs of i, v I VCVC Phasors VRVR VLVL Phasors shown at t = 0. Click to see phasor rotation. View time-varying voltages and current as projections on the y -axis. I, V R, V L, V C, and E max are all maximum values. i, v R, v L, v C, E are all time-varying values. Color distinction has been used throughout for convenience. E max

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i – Purple v R - Orange i – Purple v L - Green i – Purple v C - Blue i – Purple E - Red I VRVR v R is in phase with i. I VLVL v L leads i by 90 o I VCVC v C lags i by 90 o I E max E may lead or lag i

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Z R X L -X C In an inductive circuit the generator voltage leads the current. In a capacitive circuit the generator voltage lags the current. Impedance, Z, is the effective resistance due to a combination of resistance and reactance. E max V L -V C VRVR IZ IR I(X L -X C )

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EXAMPLE: A series RLC circuit with R =300 , L =.75 H, C = 4.6 F, f = 60 Hz and E max = 120V. Since the phase angle is negative the circuit is capacitive and the generator emf peak lags the current peak by 44.4 o. VRVR VLVL VCVC E max X L -X C R Z

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POWER LOSS No power loss occurs at a pure inductor or a pure capacitor. Power loss occurs at a resistor where energy is lost as thermal energy. cos( is called the power factor.

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RESONANCE resonance angular frequency A maximum value of the current and power loss occurs when X L = X C, = o. At resonance: Generally:

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Resonance is related to the natural frequency of oscillation between a fully charged capacitor and an inductor connected in parallel. Initially the capacitor discharges through the inductor transferring all its stored energy from its electric field to an induced magnetic field inside the inductor. As the magnetic field collapses, its energy flows back to the capacitor by recharging the capacitor and reestablishing the electric field. This cycle repeats with each cycle reversing the polarity of the fields. the natural frequency is the same as the resonance frequency. image from Wikipedia

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RESONANCE A resonance: If there is no resistance, then Z = 0

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Power E max = 200 V R = 100 L = 4.0 H C = 12.0 F Current oo oo Current, I, and power, P, vary with and reach peak values at resonance when = o.

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IMPORTANT GRAPHS R At resonance X L = X C, Z = R, = 0, I and P peak.

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EXAMPLE: Given the RLC series circuit with R = 150 , L = 1.25 H, C = 8.4 F and E max = 150 V. (A) Calculate the resonance frequency, ; and the peak current and power loss at resonance. (B) Calculate the impedance, current, power loss, and phase angle at =.5 o.

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V IN V OUT A low-pass filter: A high-pass filter:

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V IN V OUT At resonance: Exercise: show that

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