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Each of the circuit elements will have a different ac current response to an applied ac voltage. We need to look at each of these elements. Resistor: Voltage and current are related through Ohm’s Law. Imax The time variation of the voltage and the current are the same. The voltage and current are in phase. In Phase means that f is the same for both the voltage and the current. This means that the amplitudes for the voltage and the current are a maximum at the same time. The voltage amplitude and the current amplitudes are different. DVmax Voltage Current T The phase constant for the voltage and the current are the same. f = 0 for the plot shown. Imax t The period of oscillation describes the time it takes to complete one full cycle. Is used to determine f or w. Maxima line up

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Another method for showing the phase difference between two sinusoidally varying quantities is through the use of a phasor diagram. IR, DVR Plot the function as a vector where the amplitude corresponds to the magnitude of the vector and wt is the angle the vector is swept out. DVmax If you sweep the vector over 2p radians you will sweep out a circle with a radius equal to the max amplitude. Imax wt The projection of the vector on to the y-axis (we are using sinq) will give the magnitude of the function at any time. The phase angle would be the angle between the two vectors being compared. In this case the two vectors line up and is therefore zero. Phasor diagrams are very useful representations of sinusoidally varying functions, especially when a circuit contains capacitors and inductors. This type of analysis is normally done in a complex plane, so one axis would correspond to the real axis and the other to the complex axis. Discussion of complex numbers in beyond the scope of this class.

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We are not always interested in the magnitude of a function at a specific time. It is sometimes useful to look at an average value. What is the average value of the sinusoidally varying current and voltage functions? Zero for both – there are equal positive and negative values in each case, so when summed the result is zero. Voltage Current t Does this mean that the average power delivered to a resistor is zero? No, the resistor does not care about the direction of current flow. Let us determine the average power dissipated by a resistor. Average power for one period rms is the type of average that we have obtained

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**The average we have determined is called a root mean square (rms)**

The average we have determined is called a root mean square (rms). This is a method used to determine the average for a variety functions (usually periodic) where a normal average calculation will result in zero. The expression for the rms of the current we have derived is only valid for sinusoidally varying functions. For a triangle wave: For any sinusoidally varying function the rms value is determined by the amplitude divided by the square root of 2. A similar expression can be determined for the voltage. Multimeters and other similar devices for measuring ac voltage and ac current always measure rms values. If you want to know the peak amplitude you must calculate it from the rms value.

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Inductor: Let us consider the following circuit consisting of an ac voltage source and an inductor. We use this form so we can compare this with the current through the resistor.

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The presence of the inductor introduces a phase shift of –p/2 for the current relative to the voltage. IR, DVR The inductor delays the current. The voltage is changed instantaneously, but the current increases to its max value at a rate consistent with the exponential decay of the induced current. DVmax wt f The phasor diagram shows the phase shift. The current and voltage vectors no longer line up. They are offset by –p/2 radians. Imax Voltage Current t Imax DVmax The lag in the current means that the current maximum occurs at a time after the voltage maximum has occurred. f

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Capacitor: Let us consider the following circuit consisting of an ac voltage source and a capacitor. The current has a phase shift of +p/2 relative to the voltage. The current leads the voltage.

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The presence of the capacitor introduces a phase shift of +p/2 for the current relative to the voltage. IR, DVR The capacitor supplies current to the circuit in the same direction as the voltage source. The current will therefore reach its maximum value quicker than the voltage can change. DVmax f Imax wt The phasor diagram shows the phase shift. The current and voltage vectors no longer line up. They are offset by +p/2 radians. Voltage Current t Imax DVmax The leading of the current means that the current maximum occurs at a time before the voltage maximum has occurred. f

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