The Phasor Transform

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Sinusoidal Sources

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Now we consider in detail circuits with sinusoidal sources A sinusoidal voltage source is one with the form v(t) = V0 sin( t + ) (722)

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Here V0 is the amplitude (in volts) that gives the largest value that (722) can attain We call the radial frequency with units rad/s and is the phase angle We can also have sinusoidally varying currents such as i(t) = I0 sin( t + ) (723)

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In this case the amplitude I0 is measured in amps The radial frequency is related to frequency by = 2 f (724)

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The units of f are hertz (Hz) The period T tells us the duration of a single cycle in the wave It is related to frequency using f = 1 T (725)

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A sine wave f (t) = 2 sin t is shown in Fig 7-1 Note that the amplitude gives the maximum value

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s in t 2

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1 1 2

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Fig 7-1 A plot of f (t) = 2 sin t

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Circuit Analysis Demysti ed

Radians can be related to degrees by using (radians) = (degrees) 180 180 (degrees) = (radians)

(726)

Leading and Lagging

In electrical engineering you often hear the terms leading and lagging Let v 1 (t) = V0 sin( t + ) and v 2 (t) = V0 sin( t) Both voltages have the same radial frequency, but we say that v 1 (t) leads v 2 (t) by radians or degrees (depending on the units used) This means that the features in the waveform v 1 (t) appear earlier in time than the features in v 2 (t) Otherwise they are the same waveforms Consider v 1 (t) = 170 sin(377t + 20 ), v 2 (t) = 170 sin(377t)

In this case v 1 (t) leads v 2 (t) by 20 Alternatively, we can say that v 2 (t) lags v 1 (t) by 20 In Fig 7-2, we show a plot of f (t) = 2 sin t together with g(t) = 2 sin(t + /6) The dashed line is f (t) = 2 sin t, which lags g(t) = 2 sin(t + /6) because the features of g(t) appear earlier Now suppose that g(t) = 2 sin(t /6) This wave lags f (t) = 2 sin t, meaning that its features appear later in time This is illustrated in Fig 7-3

sint 2

1 1 2

Fig 7-2 The wave g(t) = 2 sin(t + /6) leads f (t) = 2 sin t, its features appear earlier in time

The Phasor Transform

s in t 2

1 1 2

Fig 7-3 The wave g(t) = 2 sin(t /6) (dashed line) lags f (t) = 2 sin t (solid line), now f (t) = 2 sin t appears earlier in time

If two sinusoidal waveforms have a 0 phase difference, we say that they are in phase If the phase difference is 90 , we say that the waves are 90 out of phase

Effective or RMS Values

The effective or RMS value of a periodic signal is the positive dc voltage or current that results in the same power loss in a resistor over one period If the current or voltage is sinusoidal, we divide the amplitude by 2 to get the RMS value That is V0 Veff = , 2 I0 Ieff = 2

(727)

Dynamic Elements and Sinusoidal Sources

Suppose that a voltage v(t) = V sin( t + ) is across a capacitor C The current in the capacitor is i(t) = C dv = C V cos( t + ) dt

(728)

Circuit Analysis Demysti ed

The amplitude and hence the maximum value attained by the current is I = C V Rewriting this in terms of the voltage we have V = 1 I C

Notice that this resembles Ohm s law So we can denote a resistance by XC = 1 C (729)

We call this quantity the capacitive reactance The negative sign results from the phase shift that occurs relating voltage to current Now consider an inductor carrying a current i(t) = I sin( t + ) The voltage across the inductor is given by v(t) = L di = L I cos( t + ) dt

Following the same logic used when considering a capacitor, we note that the maximum voltage in the inductor is V = L I Once again this is an Ohm s law type relation with resistance R = L We call this the inductive reactance X L = L (730)

Notice that the inductive reactance and capacitive reactance have the same units as resistance Unlike a resistor, however, the resistance in an inductor or capacitor is frequency dependent As the frequency increases