alternating currents & electromagnetic waves

alternating currents & electromagnetic waves

PHY232Remco [email protected] W109 – cyclotron buildinghttp://www.nscl.msu.edu/~zegers/phy232.html

PHY232 - Remco Zegers - alternating currents and electromagnetic waves

2

Alternating current circuits

previously, we look at DC circuits (the voltage delivered by the source was constant).

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

I

R

V

I

V

R


3

A circuit with a resistor

The voltage over the resistor is the same as the voltage delivered by the source: VR(t)=V0sint=V0sin(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


4

lon-capa

you should now do problem 1 from set 7.


5

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


6

power consumption by an AC circuit

We already saw (DC): P=VI=V2/R=I2R

For AC circuits with a single resistor:

P(t)=V(t)*I(t)=V0I0sin2t

The average power consumption:

Pave=Vrms*Irms=V2rms/R=I2

rmsR

Pave=(Vmax/2)( Imax/2)= ImaxVmax/2

|IR

|(A

) |V

R|(

V)

Vrms

Irms

P(W

)


7

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


8

phasorsI R

(A)

I(t) V(t) t

=t

The instantaneous current and voltage over R are the projections on the t-axis (horizontal axis) of vectors rotating with ang. frequency .The length of the vectors indicate the maximum current or voltage.

I(t)=V(t)=0

t

I(t)=5A V(t)=10 Vt V(t)=-10V I(t)=-5A t

I(t), V(t) are in phase, sopoint in the same direction


9

question

V(t) I(t) t

Given a phasor diagram for a single resistor in circuit.If the voltage scale is V and current scale Ampere,then the resistor has a resistancea) < 1 Ohmb) > 1 Ohmc) 1 Ohm


10

A circuit with a single capacitor

I

V(t)=V0sint

C

Vc= V0sintQc=CVc=CV0sintIc=Qc/t= CV0cost= CV0sin(t+/2)So, the current peaks ahead in time (earlier) of the voltageThere is a difference in phase of /2 (900)

I C(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.


11

phasor diagram for capacitive circuit

I(t) V(t) t

=tI C(A

)

Note: Imax= CV0

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 as a resistancein this circuit.


12

power consumption in a capacitive circuit

There is no power consumption in a purely capacitive circuit:Energy (1/2CV2) gets stored when the (absolute) voltage over thecapacitor is increasing, and released when it is decreasing.

Pave = 0 for a purely capacitive circuit


13

question

I(t) V(t) t

=t

The angle between the current vector and voltagevector in a phasor diagram for a capacitive circuit isa) 00

b) 450

c) 900

d) 1800


14

A circuit with a single inductor

I

V(t)=V0sint

L

VL= V0sint=LI/tIL=-V0/(L)cost= V0 /(L )sin(t-/2)(no proof here: you need calculus…)So, the current peaks later in time than the voltageThere is a difference in phase of /2 (900)

I L(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, due to this tug of war.


15

phasor diagram for inductive circuit

I(t) V(t) t

=tI L(A

)

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.


16

power consumption in an inductive circuit

There is no power consumption in a purely inductive circuit:Energy (1/2LI2) gets stored when the (absolute) current through theinductor is increasing, and released when it is decreasing.

Pave = 0 for a purely inductive circuit


17

question

The inductive reactance (and capacitive reactance) are just like the resistance of a normal resistor, I.e. if I know the inductive reactance, I can calculate the current at any time given the voltage using I=V/XL.

a) True b) False


18

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=I0sin(t-)

2) The voltage over all three components is equal to the source voltage at any point in time: V(t)=V0sin(t)

I

V(t)=V0sint

L C R


19

An LRC circuit

For the resistor: VR=IRR and VR and IR=I are in phase

For the capacitor: Vc=IXc and Vc lags Ic=I by 900

For the inductor: VL=IXL and VL leads IL=I by 900

at any instant: VL+Vc+VR=V0sin(t), that is the total voltage Vtot is the vector addition of the three individual components

VR

I

VC

VLt

VC

I

VR

VL

Vtot

=t


20

impedance

Vtot= VL+Vc+VR (vectors) Vtot=[VR

2+(|VL|-|VC|)2]=

[ (IR)2+(IXL-IXC)2]=I[R2+(XL-Xc)2] define X=XL-Xc : reactance of an RLC circuit define Z=[R2+(XL-Xc)2]= [R2+X2] : impedance of RLC

circ. Vtot=IZ & I=Vtot/Z looks like Ohms law

t

VC

VR

VL

t

VC

VR

VL

Vtot

vector sum VL+VC


21

phase angle

The current I and the voltage Vtot are out of phase by an angle . This angle can be calculated with:

tan=opposite/adjacent=(|VL| -|Vc| )/VR=X/R

t

VC

I

VR

VL

Vtot

=t

t

VC

VR

VL

Vtot

vector sum VL+VC


22

question

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


23

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=I2rmsR=IrmsVR

VR=Vrmscos (since cos=VR/Vtot)

So: P=VrmsIrmscos

cos: power factor of a circuitt

VC

VR

VL

Vtot

VL+VC


24

lon-capa

you should now do problem 4 from LON-CAPA 7


25

example

questions: a) what is the angular frequency of the system? b) what are the inductive and capacitive reactances? c) what is the impedance, what is the phase angle d) what is the maximum current and peak voltages over each element

Compare the algebraic sum of peak voltages with V0. Does this make sense?

e) make the phasor diagram. Include I,VL,VC,VR,Vtot, . Assume VR is in the first quadrant.

f) what are the instantaneous voltages and rms voltages over each element. Consider Vtot to have zero phase.

g) 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


26

lon-capa

you should now try problem 6 of lon-capa set 7, except for the last part


27

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=IZ

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)

t

VC

I

VR

VL

Vtot

=t

vector sum VL+VC


28

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 (see question on slide 23)Z

I

0


29

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?


30

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


31

loncapa

You should now try question 6, part 7 and question 5 of lon-capa set 7.


32

transformers

transformers are used to convertvoltages to lower/higher levels


33

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


34

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?


35

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


36

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


37

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)


38

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


39

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


40

producing the electric field wave

antenna

++

++

++

----

----

--+

++

++

+--

----

----


41

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


42

question

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

a) yesb) no


43

c=f


44

lon-capa

now try questions 2 and 7 from set 7.

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alternating currents & electromagnetic waves