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Chapter 4 Alternating Current Circuits

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AC Circuit An AC circuit consists of a combination of circuit elements and an AC generator or source An AC circuit consists of a combination of circuit elements and an AC generator or source The output of an AC generator is sinusoidal and varies with time according to the following equation The output of an AC generator is sinusoidal and varies with time according to the following equation –ΔV = ΔV max sin 2 ƒt Δv is the instantaneous voltage Δv is the instantaneous voltage ΔV max is the maximum voltage of the generator ΔV max is the maximum voltage of the generator ƒ is the frequency at which the voltage changes, in Hz ƒ is the frequency at which the voltage changes, in Hz –Same thing about the current (if only a resistor) –I = I max sin 2 ƒt

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Resistor in an AC Circuit Consider a circuit consisting of an AC source and a resistor Consider a circuit consisting of an AC source and a resistor The graph shows the current through and the voltage across the resistor The graph shows the current through and the voltage across the resistor The current and the voltage reach their maximum values at the same time The current and the voltage reach their maximum values at the same time The current and the voltage are said to be in phase The current and the voltage are said to be in phase Voltage varies as ΔV = ΔV max sin 2 ƒt Same thing about the current I = I max sin 2 ƒt

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More About Resistors in an AC Circuit The direction of the current has no effect on the behavior of the resistor The direction of the current has no effect on the behavior of the resistor The rate at which electrical energy is dissipated in the circuit is given by The rate at which electrical energy is dissipated in the circuit is given by –P = i 2 R = (I max sin 2 ƒt) 2 R where i is the instantaneous current where i is the instantaneous current the heating effect produced by an AC current with a maximum value of I max is not the same as that of a DC current of the same value the heating effect produced by an AC current with a maximum value of I max is not the same as that of a DC current of the same value The maximum current occurs for a small amount of time The maximum current occurs for a small amount of time Averaging the above formula over one cycle we get Averaging the above formula over one cycle we get

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rms Current and Voltage The rms current is the direct current that would dissipate the same amount of energy in a resistor as is actually dissipated by the AC current The rms current is the direct current that would dissipate the same amount of energy in a resistor as is actually dissipated by the AC current Alternating voltages can also be discussed in terms of rms valuesAlternating voltages can also be discussed in terms of rms values

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Ohms Law in an AC Circuit rms values will be used when discussing AC currents and voltages rms values will be used when discussing AC currents and voltages –AC ammeters and voltmeters are designed to read rms values –Many of the equations will be in the same form as in DC circuits Ohms Law for a resistor, R, in an AC circuit Ohms Law for a resistor, R, in an AC circuit –ΔV rms = I rms R –Also applies to the maximum values of v and i

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Example: an AC circuit An ac voltage source has an output of DV = 150 sin (377 t). Find (a) the rms voltage output, (b) the frequency of the source, and (c) the voltage at t = (1/120)s. (d) Find the maximum current in the circuit when the generator is connected to a 50.0W resistor.

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Capacitors in an AC Circuit Consider a circuit containing a capacitor and an AC source Consider a circuit containing a capacitor and an AC source The current starts out at a large value and charges the plates of the capacitor The current starts out at a large value and charges the plates of the capacitor –There is initially no resistance to hinder the flow of the current while the plates are not charged As the charge on the plates increases, the voltage across the plates increases and the current flowing in the circuit decreases As the charge on the plates increases, the voltage across the plates increases and the current flowing in the circuit decreases

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More About Capacitors in an AC Circuit The current reverses direction The current reverses direction The voltage across the plates decreases as the plates lose the charge they had accumulated The voltage across the plates decreases as the plates lose the charge they had accumulated The voltage across the capacitor lags behind the current by 90° The voltage across the capacitor lags behind the current by 90°

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Capacitive Reactance and Ohms Law The impeding effect of a capacitor on the current in an AC circuit is called the capacitive reactance and is given by The impeding effect of a capacitor on the current in an AC circuit is called the capacitive reactance and is given by –When ƒ is in Hz and C is in F, X C will be in ohms Ohms Law for a capacitor in an AC circuit Ohms Law for a capacitor in an AC circuit –ΔV rms = I rms X C

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Inductors in an AC Circuit Consider an AC circuit with a source and an inductor Consider an AC circuit with a source and an inductor The current in the circuit is impeded by the back emf of the inductor The current in the circuit is impeded by the back emf of the inductor The voltage across the inductor always leads the current by 90° The voltage across the inductor always leads the current by 90°

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Inductive Reactance and Ohms Law The effective resistance of a coil in an AC circuit is called its inductive reactance and is given by The effective resistance of a coil in an AC circuit is called its inductive reactance and is given by –X L = 2 ƒL When ƒ is in Hz and L is in H, X L will be in ohms When ƒ is in Hz and L is in H, X L will be in ohms Ohms Law for the inductor Ohms Law for the inductor –ΔV rms = I rms X L

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Example: AC circuit with capacitors and inductors A 2.40mF capacitor is connected across an alternating voltage with an rms value of 9.00V. The rms current in the capacitor is 25.0mA. (a) What is the source frequency? (b) If the capacitor is replaced by an ideal coil with an inductance of 0.160H, what is the rms current in the coil?

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