single particle described by newton’s equation with...

Physics 227 Lecture 5 1 Autumn 2008

Lecture 5 Complex Variables II (Applications in Physics) (See Chapter 2 in Boas.)

To see why complex variables are so useful consider first the (linear) mechanics of a

single particle described by Newton’s equation with viscous damping (as appeared

first in Lecture 1), a linear restoring force and a driving force,

.mx bx kx F t (5.1)

Along the same line recall Kirchoff’s equation describing a (series) RLC circuit with

a voltage source. With the current I as the free variable (coordinate) we have

,dt

IR I LI V tC

(5.2)

where the first term is the voltage across the resistor, the second term is the voltage

across the capacitor (CV Q C ), the third term is the voltage across the inductor, and

the right-hand-side is the applied voltage. Written in terms of the charge on the

capacitor Kirchoff’s equation is identical in form to Newton,

,Q

LQ RQ V tC

(5.3)

i.e., in both situations we have a linear, second order, inhomogeneous differential

equation (due to the driving term on the RHS). Such a situation with multiple similar

equations is simplified by application of the “rule of Feynman” (one of many rules

with the same name), which in this case states that the “same equations have the same

solutions”. Only the names of the variables and constants have changed in going

from Eq. (5.1) to Eq. (5.3), m L , b R , 1k C and F V . Here we will

study Eq. (5.3), but the discussion applies also to Eq. (5.1). With only a small loss of

generality (as we will see later in the course) we can assume that the driving voltage

is a periodic function of time, V t V t , with period and we expand the time

dependence in a Fourier Series (more about this later), 00cosn nn

V t V n t

where 0 2 (the fundamental frequency). This set of functions (with both a

magnitude and phase to be specified for each term) constitutes a complete set of

functions with the required periodicity – any function with this periodicity can be

represented as such a sum! We will prove this essential point later in the course.


Note that the n = 0 terms allows for the possibility of a constant term, while the

phases are equivalent to including both sines and cosines.

We focus first on the fundamental frequency term, which we can rewrite as

0 01

1 0 1 1 1cos Re Re .i t i tiV t V t V e e V e

(5.4)

We have defined the complex constant, 1

1 1

iV V e , which carries the information on

both the magnitude, 1V , and phase,

1 , of the fundamental component of the driving

voltage. By the rule of Feynman we can apply exactly the same decomposition to the

driving force in the oscillator problem (assuming that it is also periodic),

00cosn nn

F t F n t

.

The corresponding Ansatz (i.e., educated guess) for the charge on the capacitor is

00cosn nn

Q t Q n t

, where each term requires us to solve for a

magnitude, nQ , and a phase,

n , i.e., for the complex constant ni

n nQ Q e

,

0

0Re

in t

nnQ t Q e

. The corresponding current is then given by

0

00Re

in t

nnI t Q t in Q e

. (Note the 2 symbols for the 2 phases, n n , of

the applied voltage and of the charge on the capacitor.)

Magic Point #1: The essential feature here is that both Eq. (5.1) and Eq. (5.3) are

linear equations in the sense that the free dynamic variable (x and Q, respectively)

appears linearly (to the power unity) in each term on the left-hand-side. As a result

we can use linear superposition to solve the general equation. We break up the

right-hand-side of the equation into “bite-sized” pieces (individual frequencies), as

we are discussing here, solve the equation corresponding to each piece of the right-

hand-side, then sum up all of these individual solutions to find the particular

solution to the original equation. The general solution to the original equation plus

initial conditions, 0 0Q t x t and 0 0Q t x t , can be found by

summing the particular solution and the solution to the homogeneous problem with

zero right-hand-side. (Understand this result and you will have mastered much of

this course!)

The next big step is to rewrite the real differential equation as a complex equation.

We can always take the real part in the end to find the desired physical solution. For


now we focus on a single form of the time dependence, i.e., a single frequency, for

the driving voltage. We find after switching to complex notation, taking derivatives

and canceling common factors that (recall our Ansatz is 0

1Rei t

Q t Q e

)

0

0 0 0

0

0 0 0

0

0 0 0

21

1 1 12

2

11 1 12

2 10 1 0 1 1

2 10 1 0 1

ReRe Re Re

i t

i t i t i t

i ti t i t i t

i ti t i t i t

QLQ RQ V t

C

Q ed dL Q e R Q e V e

dt dt C

d d Q eL Q e R Q e V e

dt dt C

Q eL Q e iR Q e V e

C

QL Q iR Q

C

0

1 canceling the factor .i t

V e

(5.5)

Magic Point #2: By using complex notation, cos Re i t i tt e e , and the

special feature of the exponential function, t tde dt e , we have succeeded in

converting the original differential equation into an algebraic equation, which is

solvable by elementary means (i.e., arithmetic). This is a major step in simplifying

our task. (Recall that we are lazy and smart!)

It is now a simple matter to solve for the complex form of the charge and the current,

11 2

0 0

1 1 1 0 1

0 1 11 2

0 0 00

,1

,

.1 1

VQ

L iRC

I t Q t I i Q

i V VI

L iR R iLC i C

(5.6)

This last expression plus our previous experience with DC circuits, I V R ,

suggests that we define a complex, frequency dependent impedance via


1 1

,Z R i L R i Li C C

(5.7)

where

2

2

1

1,

1

phase tan .Z

Z R LC

LC

ZR

(5.8)

These expressions suggest that the frequency 1LC LC , where Z R and

0Z , must play a special role. This is just the natural frequency of the LC

circuit, when there is no real resistance ( 0R ), in the same way that the natural

frequency of the undamped oscillator in Eq. (5.1) ( 0b ) is HO k m . In terms of

this parameter we can write

22

2 2 2

2 2

1

,

phase tan .

LC

LC

Z

LZ R

LZ

R

(5.9)

Then starting with the complex form of the applied voltage of frequency , the

corresponding complex charge and current are

, .V V

I QZ i Z

(5.10)

In particular, for the case of a single fundamental frequency we have


1 01 0

1 0

211 0 1 0

0 0

11 0 1

0

,

Z

Z

ii

i

VQ Q e e

Z

VI i Q e

Z

(5.11)

where

22

2 2 2

0 0

0

2 2

01

0

0

,

tan .

LC

LC

Z

LZ R

L

R

(5.12)

Thus for the single driving voltage 1 0 1cosV t V t , the use of complex

variables (and the 2 bits of magic noted above) allows us to quickly solve for the

corresponding current in a general RLC circuit. We find

0 1 11 1 0 0 1 0

0

Re cos ,i t i

Z

VI t I e t

Z

(5.13)

i.e., the resulting current differs from the applied voltage

in magnitude by 1 over the modulus of the complex impedance (evaluated at the

driving frequency),

in phase from the applied voltage by the negative of the phase of the complex

impedance (evaluated at the driving frequency).

Note that the corresponding charge (on the capacitor) differs from the current by a

factor of 01 in magnitude, and 2 in phase (“the charge lags behind the current

by 90 ”, i.e., it takes time for the current to build up the charge),


0 1 11 1 0 0 1 0

0 0

10 1 0

0 0

Re cos2

sin .

i t i

Z

Z

VQ t Q e t

Z

Vt

Z

(5.14)

Also note that, if the driving voltage is given instead by a sine function, i.e.,

1 2 , the resulting current in Eq. (5.13) is also given by a sine function and the

charge in Eq. (5.14) will be given by minus a cosine function (with all the functions

having the same argument 0 0Zt ).

Looking at these results we observe first that, as expected, the response of the circuit

has its maximum amplitude when the driving frequency equals the natural frequency

( 0 LC , this is how radio tuners work), where the impedance has its minimum

magnitude and vanishing phase (i.e., the driving voltage and the current are in-phase,

while the charge on the capacitor lags the driving voltage by 90 ),

1 11 0 1 0

, ,

0,

, .

LC

LC LC

LC LC

Z R Z R

V V LCI Q

R R

(5.15)

Next we observe that in the limit where the driving frequency is well below the

natural frequency (and then goes to zero in the limit with 1 1 10,V t V ),

0 0 0

00

1

0

1 0 1 0 1

1 1 0 1 1

1 1,

0 ,

tan2

cos 0,2

sin ,2

LC

Z

Z ZC C

I CV t

Q CV t CV

(5.16)


which is the expected result for a DC circuit. The charge just builds up on the

capacitor to match the applied DC voltage.

In the opposite limit of a driving frequency well above the natural frequency we have

2

0 0 0 0 0

0 1 00

1 11 0 1 0 1

0 0

1 11 0 1 0 12 2

0 0

,

,tan

2

cos sin 0,2

sin cos 0.2

LC

Z

Z L Z L

L

R

V VI t t t

L L

V VQ t t t

L L

(5.17)

The amplitudes of both the charge and the current vanish in the limit of very high

frequency, 0 . The circuit simply cannot respond to a driving voltage that

oscillates at a frequency much larger than its natural frequency.

To see what sets the scale for how rapidly the impedance varies with frequency, we

consider the impedance in the form

22 220

0 2 2

0

.LCR

Z LL

(5.18)

Then we ask how far 0 must vary from LC before 0Z varies by a factor 2 ,

2 2 2LCZ Z R . This is given by the equation

2 22 2 2 22 22 2

2 2 222 2 2

2 2

2

24

.2

LC LC

LC

LC

R R

L L

R

L

(5.19)


The width of the peak, typically called a resonance, in the quantity 01 Z is

determined by the damping in the system. This characteristic resonance shape will

appear often in physics and is illustrated in the next figure corresponding to 1L ,

0.01R L and the x-axis is the scaled frequency LC

ASIDE 1: Another way to think about the quantity 1 Z is to consider where the

(complex) impedance vanishes,

2

2

20 .

2 4LC

R RZ i

L L (5.20)

Thus the quantity 1 Z has (2) simple poles at the complex frequencies in Eq.

(5.20) ,

1

.i

Z L

(5.21)

Of course, in physical applications with 0 and real we can get close only to the

pole at . We can easily check that the expression in Eq. (5.21) vanishes in the

limits 0 and ( Z , see Eqs. (5.16) and (5.17)) and goes to its

maximum value for real frequencies, 1 R , at LC (see Eq. (5.15)). The reader is

encouraged to perform this check. (Note that the maximum value for real

0.9 0.95 1 1.05 1.1

LC

20

40

60

80

100

120

1Z


frequencies is not at Re due to the factor of in the numerator.) The distance

of the pole(s) from the real axis, 2R L , is just what sets the width of the peak in Eq.

(5.19). It is these poles that characterize the solution to the homogeneous

(undriven) equation as we discuss below.

ASIDE 2: The (time) average power consumed in the circuit corresponding to the

frequency 0 can also be easily calculated in terms of the complex current and

voltage,

0

0

2

10 1 0 0 1

0 0

2

10 1 1

0

1

1cos cos

1cos Re ,

22

Z

Z

P dt I t V t

Vdt t t

Z

VI V

Z

(5.22)

which is very much like the DC result except for the factor of ½ (from the time

average of 2

0cos t ) and the cosine of the phase difference between the I and V. If

the current and driving voltage are 2 out of phase (as when the R vanishes),

energy is only stored in the circuit and no power (energy) is dissipated.

Now let’s apply the rule of Feynman to simply write down the form of the motion of

a damped harmonic oscillator driven with the frequency 0 , 1 0 1cosF t F t .

Changing the names in Eq. (5.6) we have

01

1 11 0 2

2 20 00 0

11 00 2 22

2 02 2 2

0 0

tan .HO

HO

ii

HO

HO

HO

F Fx

bm ib km i

m

Fe e b

mbm

m

(5.23)


Again the maximum amplitude of motion arises for a driving frequency equal to the

natural frequency, HO k m ,

11 0 1 0

1

0

,

tan .2

HO HO

HO HO

F mx x

b k

(5.24)

As we did in Eq. (5.20) we can think of the amplitude in Eq. (5.23) as having 2

simple poles in the complex frequency plane, i.e., we can write

1 11 0

2 2 0 , 0 ,0 0

2 22

, 2 2

,

.2 4 2 4

HO HOHO

HO HO

F Fx

b mm i

m

b b b k bi i

m m m m m

(5.25)

The corresponding time dependence for general 0 is given by (see Eq. (5.14))

1

1 0 1 022

2 2 2

0 0

cos .HO

HO

Fx t t

bm

m

(5.26)

For a general driving force, described by a sum over Fourier components,

0

0 0

Re cosn oi in t

n n n

n n

F t F e e F n t

, linear superposition yields the

complete form of the response to the given driving force, called the particular

solution

0 02

0 22 2 2 2 2

0 0

cos .np n HO

n

HO

Fx t n t n

bm n n

m

(5.27)


Similar expressions, with the names changed, arise for the general driving voltage in

the RLC circuit problem,

0 0

0 0

0 020 2

2 2 2 2

0

0

cos

cos ,

np n Z

n

nn Z

n

LC

VI t n t n

Z n

Vn t n

LR n

n

0 022 2 2 2 2 2 20

0 0

sin .np n Z

nLC

VQ t n t n

n R L n

(5.28)

Although Eqs. (5.27) and (5.28) appear very similar, the astute student will recognize

that there is a subtle underlying issue arising from our (conventional) choice to define

the complex impedance to be real at the natural frequency while the denominator in

Eq. (5.23) is pure imaginary at the natural frequency of the oscillator (this is why the

expression for Q t , which is the closest analog to x t , is a sine function instead of

a cosine function).

We close this discussion of the use of complex variables to solve differential

equations by considering the homogeneous version of Eq. (5.1) (a similar analysis

works for Eq. (5.3) with 0V t ). This will allow us to write down the complete

solution to such linear mechanics problems including both the particular and

homogenous (also called the complementary) solutions (and to fit any initial

conditions). In particular, consider the equation

0mx bx kx (5.29)

and assume an Ansatz of the familiar exponential form t

Hx xe with x a constant

(note that in general we are allowing both x and to be complex). As above this

leads to an algebraic equation,

222 4

0 .2 2 2

b b km b k bm b k i

m m m m

(5.30)


Note that when we translate to frequencies and RLC circuits these are just the

positions of the poles of 1 Z in Eq. (5.20)

2

, , , 1

1.

2 2

i m L b R k C

R Ri i

L LC L

(5.31)

For the usual case of small damping where 2b km , the second term is truly

imaginary and the motion corresponds to damped, oscillatory motion with frequency

2

2HO HOk m b m . We can express this behavior in a variety of forms,

2 21 2 3

24 5

cos

cos sin .

HO HObt bt

i t i tm mH HO

btm

HO HO

x t e x e x e x e t

e x t x t

(5.32)

We can use the 2 constants in each of the expressions (1 2,x x or 3,x or

4 5,x x ) to fit

the initial conditions for 0Hx and 0Hx . Note that, unlike the (inhomogeneous)

driven case, we are not explicitly taking any real parts here. On the other hand, when

we fit to the (presumably real) initial conditions, we will find a real Hx t , i.e., 3,x

and 4 5,x x will be real, while

1 2,x x will be appropriately complex (1 2 4x x x ,

1 2 5i x x x ). In the opposite limit with 2b km , called the over-damped case,

we have instead only damped behavior, i.e., the are pure real and both negative

( 2

2 2b m k m b m ),

.t t

Hx t x e x e

(5.33)

In the special case of (so-called) critical damping, 2b km with

2b m , we find (the reader should confirm that the following

expression satisfies Eq. (5.29))

6 7 .t

Hx t e x x t (5.34)


In all three cases (with 0b ) the homogenous solution damps to zero asymptotically,

0Hx t . Similar expressions arise for the undriven RLC circuit as you may

recall from Physics 122.

Finally, for the general problem with a driving force and initial conditions, we sum

(linear superposition again) the homogeneous solutions and the particular solution

,H px t x t x t (5.35)

to obtain a solution that (still) satisfies the inhomogeneous equation, Eq. (5.1), that

can be matched to the initial conditions using the 2 constants in Hx t and that is just

the particular solution due to the driving force at asymptotic times (after the

homogenous solution has damped out). This is an extremely powerful result and

should be added to your knowledge base as quickly as possible!

single particle described by newton’s equation with...

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