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alternating currents & electromagnetic waves PHY232 – Spring 2007 Jon Pumplin http://www.pa.msu.edu/~pumplin/PHY232 (Ppt courtesy of Remco Zegers)

Alternating currents & electromagnetic waves PHY232 – Spring 2007 Jon Pumplin pumplin/PHY232 (Ppt courtesy of Remco Zegers)

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alternating currents & electromagnetic waves

PHY232 – Spring 2007Jon Pumplinhttp://www.pa.msu.edu/~pumplin/PHY232(Ppt courtesy of Remco Zegers)

PHY232 - Pumplin - alternating currents and electromagnetic waves 2

Question

At t=0, the switch is closed. After that: a) the current slowly increases from I = 0 to I =

V/R b) the current slowly decreases from I = V/R to I

= 0 c) the current is a constant I = V/R

LR

V

I

PHY232 - Pumplin - alternating currents and electromagnetic waves 3

Answer

At t=0, the switch is closed. After that: a) the current slowly increases from I=0 to I=V/R b) the current slowly decreases from I=V/R to I=0

c) the current is a constant I=V/R

The coil opposes the flow of current due to self-inductance, so thecurrent cannot immediately become the maximum I=V/R. It willslowly rise to this value (characteristic time Tau = L/R).

LR

V

I

PHY232 - Pumplin - alternating currents and electromagnetic waves 4

Alternating current circuits

Previously, we look at DC circuits: the voltage delivered by the source is constant, as on the left.

Now, we look at AC circuits, in which case the source is sinusoidal. A is used in circuits to denote this.

R

V V

R

I I

PHY232 - Pumplin - alternating currents and electromagnetic waves 5

A circuit with a resistor

The voltage over the resistor is the same as the voltage delivered by the source: VR(t) = V0 sint = V0 sin(2ft)

The current through the resistor is: IR(t)= (V0/R) sint Since V(t) and I(t) have the same behavior as a function of time, they

are said to be ‘in phase’. V0 is the maximum voltage V(t) is the instantaneous voltage is the angular frequency; =2f f: frequency (Hz)

SET YOUR CALCULATOR TO RADIANS WHERE NECESSARY

I

V(t)=V0sint

R

I R(A

)

V0=10 VR=2 Ohm=1 rad/s

PHY232 - Pumplin - alternating currents and electromagnetic waves 6

rms currents/voltages To understand energy

consumption by the circuit, it doesn’t matter what the sign of the current/voltage is. We need the absolute average currents and voltages (root-mean-square values) :

Vrms=Vmax/2

Irms=Imax/2

The following hold: Vrms=IrmsR

Vmax=ImaxR

I R(A

)|I

R|(A

) |V

R|(

V)

Vrms

Irms

PHY232 - Pumplin - alternating currents and electromagnetic waves 7

power consumption in an AC circuit

We already know for DC P = V I = V2/R = I2 R

For AC circuits with a single resistor:

P(t) = V(t) * I(t) = V0 I0

(sint)2

Average power consumption: Pave= Vrms* Irms = V2

rms/R = I2rms R

where Vrms = Vmax/2)

Irms = Imax/2

|IR

|(A

) |V

R|(

V)

Vrms

Irms

P(W

)

PHY232 - Pumplin - alternating currents and electromagnetic waves 8

vector representation

time (s)

V0

-V0

V

The voltage or current as a function of time can bedescribed by the projection of a vector rotating with constant angular velocity on one of the axes (x or y).

=t

PHY232 - Pumplin - alternating currents and electromagnetic waves 9

AC circuit with a single capacitor

I

V(t)=V0sint

C

Vc = V0sintQc = CVc= C V0 sintIc = Qc/t = C V0 costSo, the current peaks ahead of the voltage:There is a difference in phase of /2 (900).

I (A

)

Why? When there is not much charge on the capacitor it readily accepts more and current easily flows. However, the E-field and potential between the plates increase and consequently it becomes more difficult for current to flow andthe current decreases. If the potential over C is maximum, the current is zero.

PHY232 - Pumplin - alternating currents and electromagnetic waves 10

Capacitive circuit - continuedI (

A)

Note: Imax= C V0

For a resistor we have I = V0/R so ‘1/C’ is similar to ‘R’

And we write: I=V/Xc with Xc= 1/C the capacitive reactanceUnits of Xc are Ohms. The capacitive reactance acts like a resistancein this circuit.

I

V(t) = V0 sint

C

PHY232 - Pumplin - alternating currents and electromagnetic waves 11

Power consumption in a capacitive circuit

There is no power consumption in a purely capacitive circuit:Energy (1/2 C V2) gets stored when the (absolute) voltage over thecapacitor is increasing, and released when it is decreasing.

Pave = 0 for a purely capacitive circuit

PHY232 - Pumplin - alternating currents and electromagnetic waves 12

AC circuit with a single inductor

I

V(t) = V0 sint

L

VL= V0 sint = L I/tI = -(V0/(L)) cost(no proof here: you need calculus…)the current peaks later in time than the voltage:there is a difference in phase of /2 (900)

I (A

)

Why? As the potential over the inductor rises, the magnetic flux produces a current that opposes the original current. The voltage across the inductor peaks when the current is just beginning to rise.

PHY232 - Pumplin - alternating currents and electromagnetic waves 13

Inductive circuit - continued

Note: Imax= V0/(L)

For a resistor we have I = V0/R so ‘L’ is similar to ‘R’

And we write: I = V/XL with XL = L the inductive reactanceUnits of XL are Ohms. The inductive reactance acts as a resistancein this circuit.

I L(A

)

I

V(t) = V0 sint

L

I(A

)

PHY232 - Pumplin - alternating currents and electromagnetic waves 14

Power consumption in an inductive circuit

There is no power consumption in a purely inductive circuit:Energy (1/2 L I2) gets stored when the (absolute) current through theinductor is increasing, and released when it is decreasing.

Pave = 0 for a purely inductive circuit

PHY232 - Pumplin - alternating currents and electromagnetic waves 15

Reactance

The inductive reactance (and capacitive reactance) are like the resistance of a normal resistor, in that you can calculate the current, given the voltage, using I = V/XL (or I = V/XC ).

This works for the Maximum values, or for the RMS average values.

But I and V are “out of phase”, so the maxima occur at different times.

PHY232 - Pumplin - alternating currents and electromagnetic waves 16

Combining the three: the LRC circuit

Things to keep in mind when analyzing this system:

1) The current in the system has the same value everywhere I = I0 sin(t-)

2) The voltage over all three components is equal to the source voltage at any point in time: V(t) = V0 sin(t)

I

V(t)=V0sint

L C R

PHY232 - Pumplin - alternating currents and electromagnetic waves 17

An LRC circuit

For the resistor: VR = I R and VR and I are in phase

For the capacitor: Vc = I Xc (“Vc lags I by 900”)

For the inductor: VL= I XL (“VL leads I by 900”)

at any instant: VL+Vc+VR=V0 sin(t). But the maximum values of VL+Vc+VR do NOT add up to V0

because they have their maxima at different times.

VR

I

VC

VL

I

V(t)=V0sint

L C R

PHY232 - Pumplin - alternating currents and electromagnetic waves 18

impedance

Define X = XL-Xc = reactance of RLC circuit

Define Z = [R2+(XL-Xc)2]= [R2+X2] = impedance of RLC cir

Then Vtot = I Z looks like Ohms law!

I

V(t)=V0sint

L C R

PHY232 - Pumplin - alternating currents and electromagnetic waves 19

Resonance

If the maximum voltage over the capacitor equals the maximum voltage over the inductor, the difference in phase between the voltage over the whole circuit and the voltage over the resistor is:a) 00

b)450

c)900

d)1800

In this case, XL

PHY232 - Pumplin - alternating currents and electromagnetic waves 20

Power consumption by an LRC circuit

Even though the capacitor and inductor do not consume energy on the average, they affect the power consumption since the phase between current and voltage is modified.

P = I2rms R

PHY232 - Pumplin - alternating currents and electromagnetic waves 21

Example

questions: what is the angular frequency of the system?what are the

inductive and capacitive reactances? what is the impedance, what is the phase angle what is the maximum current and peak voltages over each

element compare the algebraic sum of peak voltages with V0. Does this

make sense? what are the instantaneous voltages and rms voltages over

each element? what is power consumed by each element and total power

consumption

I

V(t)=V0sint

L C R

Given:R=250 OhmL=0.6 HC=3.5 Ff=60 HzV0=150 V

PHY232 - Pumplin - alternating currents and electromagnetic waves 22

answers a) angular frequency of the system?

=2f=260=377 rad/s b) Reactances?

XC=1/C=1/(377 x 3.5x10-6)=758 Ohm

XL= L=377x0.6=226 Ohm c) Impedance and phase angle

Z=[R2+(XL-Xc)2]=[2502+(226-758)2]=588 Ohm

=tan-1[(XL-XC)/R)=tan-1[(226-758)/250]=-64.80 (or –1.13 rad) d) Maximum current and maximum component voltages:

Imax=Vmax/Z=150/588=0.255 A

VR=ImaxR=0.255x250=63.8 V

VC=ImaxXC=0.255x758=193 V

VL=ImaxXL=0.255x266=57.6 V

Sum: VR+VC+VL=314 V. This is larger than the maximum voltage delivered by the source (150 V). This makes sense because the relevant sum is not algebraic: each of the voltages are vectors with different phases.

Given:R=250 OhmL=0.6 HC=3.5 Ff=60 HzV0=150 V

PHY232 - Pumplin - alternating currents and electromagnetic waves 23

answers

f) instantaneous voltages over each element (Vtot has 0 phase)? start with the driving voltage V=V0sint=Vtot

VR(t)=63.8sin(t+1.13) (note the phase relative to Vtot)

VC(t)=193sin(t-0.44) phase angle : 1.13-/2=-0.44

VL(t)=57.6sin(t+2.7) phase angle : 1.13+/2=2.7

rms voltages over each element? VR,rms=63.8/2=45.1 V

VC,rms=193/2=136 V

VL,rms=57.6/2=40.7 V

Imax=Vmax/Z=0.255 A

VR=ImaxR=63.8 V

VC=ImaxXC=193 V

VL=ImaxXL=57.6 V=-64.80 (or –1.13 rad)Vtot=150 V

PHY232 - Pumplin - alternating currents and electromagnetic waves 24

answers

g) power consumed by each element and total power consumed? PC=PL=0 no energy is consumed by the capacitor or

inductor PR=Irms

2R=(Imax/2)2R=0.2552R/2=0.2552*250/2)=8.13 W

or: PR=Vrms2/R=(45.1)2/250=8.13 W (don’t use

Vrms=V0/2!!)

or: PR=VrmsIrmscos=(150/2)(0.255/2)cos(-64.80)=8.13 W

total power consumed=power consumed by resistor!

PHY232 - Pumplin - alternating currents and electromagnetic waves 25

LRC circuits: an overview

Reactance of capacitor: Xc= 1/C

Reactance of inductor: XL= L

Current through circuit: same for all components ‘Ohms’ law for LRC circuit: Vtot=I Z

Impedance: Z=[R2+(XL-Xc)2] phase angle between current and source voltage:

tan=(|VL| -|Vc| )/VR=(XL-Xc)/R

Power consumed (by resistor only): P=I2rmsR=IrmsVR

P=VrmsIrmscos VR=ImaxR in phase with current I, out of phase by with Vtot

VC=ImaxXC behind by 900 relative to I (and VR)

VL=ImaxXL ahead of 900 relative to I (and VR)

PHY232 - Pumplin - alternating currents and electromagnetic waves 26

Question

The sum of maximum voltages over the resistor, capacitor and inductor in an LRC circuit cannot be higher than the maximum voltage delivered by the source since it violates Kirchhoff’s 2nd rule (sum of voltage gains equals the sum of voltage drops).

a) true b) false

answer: false The maximum voltages in each component arenot achieved at the same time!

PHY232 - Pumplin - alternating currents and electromagnetic waves 27

Resonances in an RLC circuit If we chance the (angular) frequency the reactances will

change since: Reactance of capacitor: Xc= 1/C

Reactance of inductor: XL= L

Consequently, the impedance Z=[R2+(XL-Xc)2] changes

Since I=Vtot/Z, the current through the circuit changes

If XL=XC (I.e. 1/C= L or 2=1/LC), Z is minimal, I is maximum)

= (1/LC) is the resonance angular frequency At the resonance frequency =0

PHY232 - Pumplin - alternating currents and electromagnetic waves 28

example

Given:R=250 OhmL=0.6 HC=3.5 Ff=60 HzV0=150 V

Using the same given parameters as the earlier problem,what is the resonance frequency?

= (1/LC)=690 rad/sf= /2=110 Hz

PHY232 - Pumplin - alternating currents and electromagnetic waves 29

question

An LRC circuit has R=50 Ohm, L=0.5 H and C=5x10-

3 F. An AC source with Vmax=50V is used. If the resistance is replaced with one that has R=100 Ohm and the Vmax of the source is increased to 100V, the resonance frequency will:

a) increase b)decrease c) remain the same

answer c) the resonance frequency only dependson L and C

PHY232 - Pumplin - alternating currents and electromagnetic waves 30

transformers

transformers are used to convertvoltages to lower/higher levels

PHY232 - Pumplin - alternating currents and electromagnetic waves 31

transformers

Vp Vs

primary circuitwith Np loops incoil

secondary circuit with Ns loops in coil

iron core

If an AC current is applied to the primary circuit: Vp=-NpB/tThe magnetic flux is contained in the iron and the changing flux actsin the secondary coil also: Vs=-NsB/t

Therefore: Vs=(Ns/Np)Vp if Ns<Np then Vs<Vp

A perfect transformer is a pure inductor (no resistance), so no power loss: Pp=PS and VpIp=VsIs ; if Ns<Np then Vs<Vp and IS>Ip

PHY232 - Pumplin - alternating currents and electromagnetic waves 32

question

a transformer is used to bring down the high-voltage deliveredby a powerline (10 kV) to 120 V. If the primary coil has 10000 windings, a) how many are there in the secondary coil? b) If the current in the powerline is 0.1 A, what is the maximum current at 120 V?

a) Vs=(Ns/Np)Vp or Ns=(Vs/Vp)Np = 120 windingsb) VpIp=VsIs so Is=VpIp/Vs=8.33 A

PHY232 - Pumplin - alternating currents and electromagnetic waves 33

question

Is it more economical to transmit power from the power station to homes at high voltage or low voltage?

a) high voltage b) low voltage

answer: high voltageIf the voltage is high, the current is lowIf the current is low, the voltage drop over the powerline (with resistance R) is low, and thus the power dissipated in the line ([V]2/R=I2R) also low

PHY232 - Pumplin - alternating currents and electromagnetic waves 34

electromagnetic waves

James Maxwell formalized the basic equations governing electricity and magnetism ~1870:Coulomb’s lawMagnetic forceAmpere’s Law (electric currents make magnetic

fields)Faraday’s law (magnetic fields make electric

currents) Since changing fields electric fields produce magnetic

fields and vice versa, he concluded: electricity and magnetism are two aspects of the

same phenomenon. They are unified under one set of laws: the laws of electromagnetism

PHY232 - Pumplin - alternating currents and electromagnetic waves 35

electromagnetic waves

Maxwell found that electric and magnetic waves traveltogether through space with a velocity of 1/(00)v=1/(00)=1/(4x10-7 x 8.85x10-12)=2.998x108 m/s which is just the speed of light (c)

PHY232 - Pumplin - alternating currents and electromagnetic waves 36

electromagnetic waves can be used to broadcast…

Consider the experiment performed by Herz (1888)

I

Herz made an RLC circuit with L=2.5 nH, C=1.0nFThe resonance frequency is = (1/LC)=6.32x108 rad/sf= /2=100 MHz. Recall that the wavelength of waves =v/f=c/f=3x108/100x106=3.0 m

wavelength: =v/f

PHY232 - Pumplin - alternating currents and electromagnetic waves 37

He then constructed an antenna

charges and currents vary sinusoidally in the primary and secondary circuits. The charges in the two branches also oscillate at the same frequency f

I

dipole antenna

PHY232 - Pumplin - alternating currents and electromagnetic waves 38

producing the electric field wave

antenna

++

++

++

----

----

--+

++

++

+--

----

----

PHY232 - Pumplin - alternating currents and electromagnetic waves 39

producing the magnetic field wave

antenna

++

++

++

----

----

--

I

I

++

++

++

----

----

--

I

I

E and B are in phaseand E=cB withc: speed of lightThe power/m2=0.5EmaxBmax/0

The energy in the wave isshared between the E-field and the B-field

PHY232 - Pumplin - alternating currents and electromagnetic waves 40

question

Can a single wire connected to the + and – poles of a DC battery act as a transmitter of electromagnetic waves?

a) yesb) no

answer: no: there is no varying current and hence nowave can be made.

PHY232 - Pumplin - alternating currents and electromagnetic waves 41

c=f