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.

V IR VI

R

max sin

Vt

R

Imax

max sinI t

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.

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0

2

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VoltageCurrent

Maxima line up

t

Imax

DVmax

T

The voltage amplitude and the current amplitudes are different.

The phase constant for the voltage and the current are the same. f = 0 for the plot shown.

The period of oscillation describes the time it takes to complete one full cycle. Is used to determine f or w.

IR, DVR

wtImax

DVmax

Another method for showing the phase difference between two sinusoidally varying quantities is through the use of a phasor diagram.

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.

If you sweep the vector over 2p radians you will sweep out a circle with a radius equal to the max amplitude.

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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VoltageCurrent

t

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.

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.

2dUP I R

dt 2dU I Rdt 2

0

T

U I Rdt 2 2max

0

sinT

R I t dt

2 2max

0

sinT

U RI t dt 2

max

0

sin 2

2 4

Ttt

RI

2max

2sin 2

2 4

TTT

RI

0

2max2

TU RI

2max

2

IU

TR

Average power for one period

2avgavg

PI

R

2max

2

I max

2rms

II

rms is the type of average that we have obtained

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.

max

2rms

II The expression for the rms of the current we have derived is only valid for

sinusoidally varying functions.

For a triangle wave: max

3rms

II

For any sinusoidally varying function the rms value is determined by the amplitude divided by the square root of 2.

max

2rms

VV

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.

Inductor:

LV(t)

Let us consider the following circuit consisting of an ac voltage source and an inductor.

max sinV t V t

0

dI tV t L

dt

max sindI t

V t Ldt

max sinV

dI t t dtL

max

0 0

sinI t t VdI t t dt

L

maxcos tV

I tL

max cos

Vt

L

cos sin2

max sin2

VI t t

L

We use this form so we can compare this with the current through the resistor.

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VoltageCurrent

t

Imax

DVmax

IR, DVR

Imax

DVmax

wt

f

The presence of the inductor introduces a phase shift of –p/2 for the current relative to the voltage.

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.

f The phasor diagram shows the phase shift. The current and voltage vectors no longer line up. They are offset by –p/2 radians.

The lag in the current means that the current maximum occurs at a time after the voltage maximum has occurred.

Capacitor:

C~V(t)

Let us consider the following circuit consisting of an ac voltage source and a capacitor.

max sinV t V t

0

q tV t

C q t C V t

dq t dI t C V t

dt dt

max sind

I t C V tdt

max cosC V t

cos sin2

max sin 2I t C V t

The current has a phase shift of +p/2 relative to the voltage.

The current leads the voltage.

IR, DVR

Imax

DVmax

wtf

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VoltageCurrent

t

Imax

DVmax

f

The presence of the capacitor introduces a phase shift of +p/2 for the current relative to the voltage.

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.

The phasor diagram shows the phase shift. The current and voltage vectors no longer line up. They are offset by +p/2 radians.

The leading of the current means that the current maximum occurs at a time before the voltage maximum has occurred.