Chapter 35 Alternating Current Circuits

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



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

 



Quiz 1: Check out the adjacent current

phasor. The magnitude of the

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 I_max 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

max rms

0.707

2

I

I





I

0.707

2

V

V





V

P



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

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

cos

R R

v



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:

R

v



iR

cos

i



I



t



V



RI

V



RI

• Consider a capacitor C in circuit with an ac-source • Notice that, when the current decreases (twice per

period) the capacitor voltage v_C 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





cos

C C

v



V



t



_i



_I

_cos



_t

¼ cycle lag

Elements of ac-Circuits –

Capacitors

1 C X 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 C

V



IX

cos cos sin

q idt C i I t C q I I v v tdt t C  C









V



I

X



v_L • 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: L

X





L

V



I

X





cos

v



V



t



_i

_

_I

_cos



_t

Elements of ac-Circuits –

Inductors

i v_L i, v I V_L t T ¼T I V_L i v_L ωt

sin

di

v

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 L

V



IX



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.









4.0 10 V sin 314 Hz

v   _ 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 v_R across the resistor is

in phase with the current

• The instantaneous voltage v_L across the inductor

leads the current by π/2

• The instantaneous voltage v_C across the capacitor

lags the current by π/2

cos

i



I



t





cos

v



V



t





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 V_R has the same phase as the current

I

V_C = IX_C

If X_L > X_C such that V_L > V_C, the

voltage leads the current by φ

The RLC Series Circuit –

Phasor diagram

cos i  I







V



V



V



V

tan

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 rms

V



IZ



V



I

Z





V



I

R



X



X



IZ

1

R

Z

L

C













_



_







• 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 1 av 2

cos

cos

P



IV





P



I V







cos

cos

cos

cos

P

vi

VI

 

t



t

VI



t











Resonance in an ac-Circuit

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





Z



R



X



X



Z



R

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 X_L = X_C, 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

cos

v



V



t

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

• 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 > V₁the transformer is called a step up transformer • When N₂ < N₁ ↔ V₂ < V₁ the transformer is called a step down transformer

2 2 1 1

N

V

N

V



N

I

N

I



I V



I V



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 1 1 2 2 2 B B t t N N 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

Application:

• When transmitting electric power over long distances, it is most economical to use high voltage and low current since this minimizes the I2_{R 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.