Chapter 33

Alternating Current (AC)

R, L, C in AC circuits

AC, the description A DC power source, like the one from a

battery, provides a potential difference (a voltage) that does not change its polarity with respect to a reference point (often the ground)

An AC power source is sinusoidal voltage source which is described as

Here

maxv V sin t

max

v

V

is the instantaneous voltage with respect to a reference (often not the ground).

is the maximum voltage or amplitude.

is the angular frequency, related to frequency f and period T as 2

2 fT

V

t

V

Symbol in a circuit diagram: or vv

The US AC system is 110V/60Hz. Many European and Asian countries use 220V/50Hz.

Resistors in an AC Circuit, Ohm’s Law

The voltage over the resistor:

R maxv v V sin t Apply Ohm’s Law, the current through the resistor:

maxRR max

Vvi sin t I sin t

R R

The current is also a sinusoidal function of time t. The current through and the voltage over the resistor are in phase: both reach their maximum and minimum values at the same time.

PLAYACTIVE FIGURE

The power consumed by the resistor is

22

2 2maxRR R R R

Vvp v i i R sin t

R R

We will come back to the power discussion later.

Phasor Diagram, a useful tool. The projection of a circular motion with

a constant angular velocity on the y-axis is a sinusoidal function.

To simplify the analysis of AC circuits, a graphical constructor called a phasor diagram is used. A phasor is a vector whose length is proportional to the maximum value of the variable it represents

The phasor diagram of a resistor in AC is shown here. The vectors representing current and voltage overlap each other, because they are in phase.

x

y

O

Rt

The projection on the y-axis is

yR R sin t

The power for a resistive AC circuit and the rms current and voltage

2R maxp p sin t

R maxv V sin t

R maxi I sin t

When the AC voltage source is applied on the resistor, the voltage over and current through the resistor are:

Both average to zero.

But the power over the resistor is

And it does not average to zero. The averaged power is: Pmax

Pav

Pmax

P

2

2

2

2 0

0

1

2 12

2 4 2

T

T

av maxT

max max

p p sin t dt

p ptsin t

T

2 12

2 4

xsin xdx sin x

The power for a resistive AC circuit and the rms current and voltage

22rms

av rms rms rms

Vp V I I R

R

So the averaged power the resistor consumes is

If the power were averaged to zero, like the current and voltage, could we use AC power source here?

The averaged power can also be written as:

PmaxPav

Pmax

P

1

2 2max

av max max

pp V I

221 1

2 2max

av max

Vp I R

R

Define a root mean square for the voltage and current:

2 2 2 21 1 and

2 2rms max rms maxV V , I I

or 2 and 2max rms max rmsV V , I I

One get back to the DC formula equivalent:

Resistors in an AC Circuit, summary

Ohm’s Law applies. Te current through and voltage over the resistor are in phase.

The average power consumed by the resistor is

From this we define the root mean square current and voltage. AC meters (V or I) read these values.

The US AC system of 110V/60Hz, here the 110 V is the rms voltage, and the 60 Hz is the frequency f, so

R maxv V sin t R maxi I sin t

22rms

av rms rms rms

Vp V I I R

R

and 2 2max max

rms rms

V IV , I

2 156 Vmax rmsV V 12 377 secf

Inductors in an AC circuit, voltage and current

The voltage over the inductor is

L maxv v V sin t To find the current i through the inductor, we start with Kirchhoff’s loop rule:

0Lv v

0max

diV sin t L

dt or

Solve the equation for i

maxVdi sin t dt

L

2

max maxmax

V Vi di sin t dt cos t I sin t

L L

or

with 2

maxmax max

Vi I sin t , I

L

Inductors in an AC circuit, voltage leads currentExamining the formulas for voltage over and current through the inductor:

L maxv v V sin t

Voltage leads the current by ¼ of a period (T/4 or 90° or π/2) . Or in a phasor diagram, the rotating current vector is 90° behind the voltage vector.

2maxi I sin t

PLAYACTIVE FIGURE

Inductive Reactance, the “resistance” the inductor offers in the circuit.Examining the formulas for voltage over and current through the inductor again:

L maxv V sin t

with 2

maxmax max

Vi I sin t , I

L

This time pay attention to the relationship between the maximum values of the current and the voltage:

maxmax

VI

L

This could be Ohm’s Law if we define a “resistance” for the inductor to be:

LX L

And this is called the inductive reactance. Remember, it is the product of the inductance, and the angular frequency of the AC source. I guess that this is the reason for it to be called a “reactance” instead of a passive “resistance”. The following formulas may be useful:

and max L max rms L rmsV X I , V X I L max Lv I X sin t

Capacitors in an AC circuit, voltage and currentThe voltage over the capacitor is

C maxv v V sin t To find the current i through the capacitor, we start with Kirchhoff’s loop rule:

0Cv v

0 with max

q dqV sin t , i

C dt or

Solve the first equation for q, and take the derivative for i

maxq C V sin t

2max max

dqi C V cos t I sin t

dt

or

1 with 2

maxmax max

Vi I sin t , I

C

Here I still like to keep the Ohm’s Law type of formula for voltage, current and a type of resistance.

Capacitors in an AC circuit, current leads the voltage

Examining the formulas for voltage over and current through the capacitor:

C maxv V sin t 2maxi I sin t

Current leads the voltage by ¼ of a period (T/4 or 90° or π/2) . Or in a phasor diagram, the rotating voltage vector is 90° behind the current vector.

PLAYACTIVE FIGURE

Capacitive Reactance, the “resistance” the capacitor offers in the circuit.Examining the formulas for voltage over and current through the capacitor again:

This time pay attention to the relationship between the maximum values of the current and the voltage:

1max

max

VI

C

This could be Ohm’s Law if we define a “resistance” for the capacitor to be:

1CX

C

And this is called the capacitive reactance. It is the inverse of the product of the capacitance, and the angular frequency of the AC source. The following formulas may be useful:

and max C max rms C rmsV X I , V X I L max Cv I X sin t

C maxv V sin t

1 with 2

maxmax max

Vi I sin t , I

C

The RLC series circuit, current and voltage

R L Cv v v v

maxv V sin t

The voltage over the RLC is

Now let’s find the current. From the this equation, write out each component:

max

di qV sin t iR L

dt C

Apply to both sides, and remember

That we have

d

dtdq

idt

2

2max

d i iV cos t R L

dt C

“Simply” solve for the current i :

maxmax

Vi sin t I sin t

Z

Where: 22L CZ R X X L CX X

tanR

Phase angle between current and voltage

Overall resistance

Phase angle

The RLC series circuit, current and voltage, solved with Phasor Diagrams

maxv V sin t

The RLC are in serial connection, the current i is common and must be in phase:

R L Ci i i i

So use this as the base (the x-axis) for the phasor diagrams:

i

The RLC series circuit, current and voltage, solved with Phasor Diagrams

Now overlap the three phasor diagrams, we have:

The RLC series circuit, current and voltage, solved with Phasor Diagrams

Now from final phasor diagram, we get the voltage components in x- and y-axes:

22max R L CV V V V

22L CZ R X X

2 2

max max max L max CI Z I R I X I X

From:

or:

We have:

Here Z is the overall “resistance”, called the impedance.

From the diagram, the phase angle is

L CX Xtan

R

L C max L max C

R max

V V I X I Xtan

V I R

We have: PLAYACTIVE FIGURE

Determining the Nature of the Circuit

If is positive XL> XC (which occurs at high frequencies) The current lags the applied voltage The circuit is more inductive than capacitive

If is negative XL< XC (which occurs at low frequencies) The current leads the applied voltage The circuit is more capacitive than inductive

If is zero XL= XC (which occurs at )

The circuit is purely resistive and the impedance is minimum, and current reaches maximum, the circuit resonates.

Often this resonant frequency is called

21 1 or L ,

C LC

maxmax

VI

Z

22L CZ R X X

L CX Xtan

R

0

1

LC