Chapter 35 Alternating Current Circuits

• ac-Circuits

• Phasor Diagrams

• Resistors, Capacitors and Inductors in ac-Circuits

• RLC ac-Circuits

• ac-Circuit power. Resonance

• Transformers

ac Circuits – Alternating currents and voltages

• An ac circuit consists of a combination of circuit elements and an ac-generator

with a time dependent sinusoidal output

where

v, i are the instantaneous voltage and current

V, I are the maximum (or peak) voltage and current

ƒ, ω are the frequency and angular frequency, in Hz

• The electric power is supplied to the public is ac form for reasons that will become

apparent soon

• The voltage and frequency vary by country:

• Elements of circuit (resistors, capacitors and inductors) behave somewheat

differently in ac-circuits than in dc-circuits

• We’ll look at the specific behavior of each one of them and combinations…

cos

cos

v V t

i I t

2 f

Ex: V / f are 110-127 V/60 Hz in the USA but 220-240V/50 Hz in Europe

Phasor Diagrams – Concept

• So, the ac-circuits are characterized by alternating

voltages and currents

• A common way to represent and analyze periodically

alternating quantities is by using phasors: vector-like

representations with the length corresponding to the

magnitude, and the angle ωt with respect to a selected

direction being the angle phase

• The alternating physical quantity – such as the emf

delivered by the generator – is the projection along the

respective direction

• The resulting diagram is called a phasor diagram

• Phasors allow the summation of quantities alternating

with different phases much like vector sums ωt

ω

ε0

0 cos t

a) Increasing

b) Decreasing

c) Constant

Quiz 1: Check out the adjacent current

phasor. The magnitude of the

instantaneous value of the current is

rms Current and Voltage • Note that, because the maximum current is reached alternately, the energy

dissipated by an ac-current across a resistor is less than that dissipated by a dc-

current equal to Imax across the same R

• The root-mean square or 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 values

• The average power dissipated in resistor in an ac-circuit:

• ac-Ammeters and voltmeters are designed to read rms values

maxrms max0.707

2

II I

maxrms max0.707

2

VV V

2

rmsP I R

Quiz 2: Since the 120 V specified when one speaks about the voltage of an outlet

represents the rms voltage, what is the maximum voltage of the respective source?

a) 120 V b) 170 V c) 110 V

Elements of ac-Circuits – Resistors

• Consider a circuit consisting of an ac-source delivering a

current i and a resistor R

• The current through and the voltage across the resistor

vary such that they reach their maximum values at the same

time: the current and the voltage are said to be in phase

cosR Rv V t

• We conclude that the variable direction of the current

has no effect on the behavior of the resistor

• The maximum current I occurs for a small amount of

time and the mean current is zero

• Ohm’s Law for a resistor R in an ac-circuit takes the

same form as in dc-circuits:

Rv iR

cosi I tin phase

cosi I t

rms rmsV RI RV RI

I

VR

i

vR

ωt or or

i

vR

i, v

I

VR

t T

vR and i-phasors

are aligned

C

R

vR

• Consider a capacitor C in circuit with an ac-source

• Notice that, when the current decreases (twice per

period) the capacitor voltage vC increases and vice-versa

• Therefore, the voltage reaches its maximum ¼ of a

period later than the current across the capacitor: we say

that the voltage lags behind the current by π/2

2 cosC Cv V t cosi I t

¼ cycle lag

Elements of ac-Circuits – Capacitors

1CX

C

• We see that we can formulate Ohm’s

Law for a capacitor in an ac-circuit:

• Here the resistance-like impeding effect

of the capacitor is called the capacitive

reactance and is given by

• However, Ohm’s law won’t work for

instantaneous currents and voltages

C CV IX

i vC

i, v

I

VC

t T

coscos sin

q idt

C Ci I t

q I Iv v tdt t

C C C

¼T

I

VC

i

vC

ωt

rms rms CV I XVC-phasor

lags

C

cosi I t

vC

vL

• Consider an inductor L in circuit with an ac-source

• The current in the circuit is impeded by the back emf

of the inductor

• As a result, the voltage across the inductor leads the

current by π/2:

LX L

rms rms LV I X

2 cosL Lv V t cosi I t

¼ cycle lead

Elements of ac-Circuits – Inductors

i vL

i, v

I

VL

t T ¼T

I VL

i

vL

ωt

sinL

div L LI t

dt

• Again, we can formulate Ohm’s Law

for an inductor in an ac-circuit:

• Here the resistance-like impeding effect

of the inductor is called the inductive

reactance and is given by

• However, Ohm’s law won’t work for

instantaneous currents and voltages

L LV IXVL-phasor

leads

L

cosi I t

Problems:

1. Resistor in ac circuit: An ac voltage source is connected to a resistor R = 40 Ω. The source

has an output:

a) Find the rms voltage and current, and the angular frequency of the source of the source.

c) How does the current vary with time? Represent its phasor.

2. Inductor in ac circuit: An inductor is connected to a 20.0 Hz power supply that produces a

52.0 V rms voltage.

a) What is the angular frequency of the source?

b) How does the current vary with time? Represent its phasor.

3. Capacitor in ac circuit: When a 4.0 µF capacitor is connected to a generator whose rms

output is 27 V, the rms current in the circuit is observed to be 0.25 A.

a) What is the angular frequency of the source?

b) How does the current vary with time? Represent its phasor.

24.0 10 V sin 314 Hzv t

Quiz 3: What happens with the maximum current when the frequency is decreased in a

capacitive ac-circuit?

a) Increases

b) Decreases

c) Stays constant

The RLC Series Circuit – Functionality

• The resistor, inductor, and capacitor can be combined in circuits where the three

elements will compete in order to impose their respective behavior, resulting into a

phase difference φ between voltage and current different from 0, π/2 or –π/2

• For instance, consider the RLC series circuit:

Comments:

• The instantaneous voltage vR across the resistor is

in phase with the current

• The instantaneous voltage vL across the inductor

leads the current by π/2

• The instantaneous voltage vC across the capacitor

lags the current by π/2

cosi I t

cosv V t

φ = 0

φ = π/2

φ = –π/2

φtotal

L

vL

R

vR vC

C

Let’s build the phasor diagram for an RLC series circuit

contains the voltage phasors on the same diagram with a

reference current

• The voltage across the resistor is along the current

phasor since it is in phase with the current

• The voltage across the inductor is perpendicular

anticlockwise on the current phasor since it leads the

current by π/2

• The voltage across the capacitor is perpendicular

clockwise on the current phasor since it lags behind the

current by π/2

• Since the voltages are not in phase, to get the voltage

across the combination of elements we can add the

phasors like vectors:

where φ is the phase angle between the net voltage and

current, since VR has the same phase as the current

I

VC = IXC

If XL > XC such that VL > VC, the

voltage leads the current by φ

The RLC Series Circuit – Phasor diagram

cosi I t

L R C

VL – VC

φ

VL = IXL

VR = IR

V = IZ

I VL = IXL

VC – VL

φ

VC = IXC

VR = IR

V = IZ

If XL < XC such that VL < VC, the

voltage lags the current by φ

22

R L CV V V V 1tan L C

R

V V

V

• The resistance of a circuit determining the ac current is given by its impedance Z

• To find the impedance and phase, we can use the phasor diagram and Ohm’s law:

• So, Ohm’s Law can also be applied to the whole RLC circuit:

where the impedance for RLC series can be rewritten

Impedance of series ac-Circuits

rms rmsV IZ V I Z

22 L CV I R X X IZ

2

2 1 RZ L

C

1tan L CX X

R

• This form for Ohm’s law can

be regarded as a generalized

form applied to any ac-circuit

even though the impedance may

have a different form than the

one of a series ac-combination

•The results can be extrapolated

to variants of the RLC series

circuit:

0

–π/2

π/2

–π/2

π/2 0

Power in ac-Circuits

• Single capacitors and inductors in ac-circuits are associated with no power losses:

• In a capacitor, energy is stored during one-half of a cycle and returned and

returned to the circuit during the other half

• In an inductor, the ac-source does work against the back emf of the inductor

and energy is stored in the inductor, but when the current begins to decrease in

the circuit, the energy is returned to the circuit

• Therefore, the net average power delivered by the ac-generator is converted to

internal energy in the resistor

• Consequently, the average power dissipated in a generic ac-circuit is

where cos φ is called the power factor of the circuit

• So, we see that phase shifts can be used to maximize power outputs, by making the

power factor 1

1av rms rms2

cos cosP IV P I V

2

av

1 2

cos cos cos cosP vi VI t t VI t

I

φ Vcosφ

V

ωt

Resonance in an ac-Circuit

• The resonance of an ac-circuit occurs at a certain

frequency ω0 where the current takes an extreme

value for the respective arrangement of elements

• For instance, in a series RLC, the resonance is

achieved when the current reaches a maximum which

occurs when the impedance has a minimum value

• Based on the expression for impedance, we get

22

series min RLC L CZ R X X Z R

0 0

0

1 1 L

C LC

Ex: a) Tuning a radio: a varying capacitor changes the resonance frequency of the tuning

circuit in your radio to match the station to be received

b) Metal Detector: The portal is an inductor, and the frequency is set to a condition with

no metal present. When a metal is present, it changes the effective inductance, which

changes the current. The change in current is detected and an alarm sounds

• This occurs when XL = XC, such that the resonance

frequency ω0 for the RLC series circuit is given by:

• Theoretically, if R = 0, the current would be infinite at

resonance, but real circuits always have some resistance

v lags i v leads i

v in phase with i

cosv V t

cosv V t

L

R

C Problem:

4. Parallel RLC circuit: A resistor R, a capacitor C and an inductor L

are connected in parallel across an ac source that provides a voltage

a) What are the phases of the currents through each element with respect

to v?

b) Use the respective current phasors to find out the current i through the

source in terms of the currents through the elements and the phase angle

with respect to v.

c) Find the impedance of the circuit

d) Calculate the respective resonance angular frequency ω0 and the angle

phase φ0 for this frequency. Then calculate the minimum current I0.

Transformers – Properties

• An ac-transformer consists of two coils of

wire wound around a core of soft iron

• The side connected to the input AC voltage

source is called the primary and has N1 turns

• The other side, called the secondary, is

connected to a resistor and has N2 turns

• The core is used to increase the magnetic flux

and to provide a medium for the flux to pass

from one coil to the other

• When N2 > N1 ↔ V2 > V1 the transformer is called a step up transformer

• When N2 < N1 ↔ V2 < V1 the transformer is called a step down transformer

22 1

1

NV

NV 1

2 1

2

NI

NI2 2 1 1I V I V

Symbol:

Properties:

1. The rate of change of the magnetic flux

is the same through both coils, such that

the potential differences across the

primary and secondary are related by

1 11 2

2 2

B B

t t

NN N

N

2. On the other hand, the energy must be

conserved, so the power input and output

must be the same, such that the primary

and secondary currents are related by

33.7

Application:

• When transmitting electric power over long distances, it is most economical to use

high voltage and low current since this minimizes the I2R power losses

• In practice, voltage is stepped up to about 230 000 V at the generating station and

stepped down to 20 000 V at the distribution station and finally to 120 V at the

customer’s utility pole

Transformers – Comments and an important application

• While ideally the energy is conserved across a transformer and the only energy loss

is via resistive dissipation, in reality a transformer also loses energy due to eddy

currents in the iron core

Comments:

• Besides voltages and currents, the

transformer also “transforms” impedance: 1 1 2 2

2

1 2 1 2 2 1

V N N V Z

I N N I N N

• A modern solution to this loss of energy is to use superconductive cables that would

reduce the resistance rather than current

Problem:

5. Transformer in an ac-circuit: A transformer is to be used to provide power for a computer

disk drive that needs 5.8 V (rms) instead of the 120 V (rms) from the wall outlet. The number

of turns in the primary is 400, and it delivers 500 mA (the secondary current) at an output

voltage of 5.8 V (rms).

a) Should the transformer have more turns in the secondary compared to the primary, or fewer

turns?

b) Find the current in the primary.

c) Find the number of turns in the secondary.