The University of Hong Kong

Department of Physics

Experimental Physics Laboratory

PHYS2255 Introductory Electricity and Magnetism

2255-2 Supplementary notes for theories

Experiment 2: The A. C. Circuitry(Dated: March 26, 2014)

In the first part of this experiment, the properties of simple resistance-capacitance(RC) circuit

are demonstrated including the charging and discharging process of the capacitor. Then the differ-

entiating(CR)circuit is demonstrated and the phase difference between voltage drop on resistor and

capacitor is investigated, which is an remarkable feature of RC circuit. Finally, the demonstration

of Lissajou’s figures are carried out, which helps experimenter get an intuitive understanding of

complex harmonic oscillation.

1. Theoretical derivation of charging and discharging curve of a RC circuit

(a) (b)

FIG. 1: Charging and discharging processes of RC circuit in the case that a square wave is input-signal.

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The differential equation describing the a.c. circuit is

V = iR +qC

(1)

where V is electrostatic force of battery(signal generator), i is current, C is capacitance, and

q is charge on the capacitor.

Since we have relations i =dqdt and Vc =

qC which are the definition of current and basic

equation of capacitor, respectively, Eq. 1 can be written as

V = RCdVc

dt+ Vc (2)

The initial condition of charging process is defined as the square wave signal are of following

form, as shown in Fig. 1 (b)

Vi(t) =

V1 t < 0

V2 t > 0(3)

where V1 < V2.

Then Eq. 2 can be integrated along with the initial condition at t = 0 as∫ t

0τ−1dt′ =

∫ Vc

V1

V ′cV2 − V ′c

(4)

and the resultant charging curve is

Vc(t) = V2 + (V1 − V2)e−tτ , t > 0 (5)

The above equation is also applicable for discharging process and these two precessed have

been illustrated in Fig. 1 (b).

Note that the time constant τ = RC is the only characteristic parameter of an a.c. circuit and

the relation between period of square wave T and τ determines the behavior of charging and

discharging processes. By an approximately estimation, one may find that the half period of

square wave signal T2 dominate the charging(discharging) process for an given definite time

constant τ. The fully charged(discharged) situation is realized only in the case of T2 � τ as

shown in Fig. 2 (a).

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(a) (b) (c)

FIG. 2: Charging and discharging processes for (a) T2 � τ, (b) T

2 ≈ τ, and (c) T2 � τ, where Vi(t) is voltage

of square wave, Vc(t) is voltage drop of capacitor, and VR(t) is voltage drop on resistor.

We may also define a special time t 12

where the Vc(t) = 12 (V2−V1) + V1. Form Eq. 5 we have

t = τln[V1 − V2

Vc(t) − V2] (6)

Redefine Vs = V2 − V1 and Vd = Vc(t) − V1, we then arrive at

t = τln[Vs

Vs − Vd] (7)

when Vd = 12Vs,

t f rac12 = τln[Vs

Vs − 12Vs

] = τln2 (8)

One should notice that t 12

is only well-defined for T2 � τ.

2. Theory of differentiating circuit.

As shown in Fig. 3, the basic equation of the CR circuit is

Vin(t) =q(t)C

+ i(t)R (9)

when the time constant τ = RC is small enough, the second term in above equation can be

neglected

Vin(t) =q(t)C

(10)

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(a) (b)

FIG. 3: Differentiating circuit (a) and input-output curve (b). Tk here is half period of square wave [1].

whereas the voltage drop on resistor R still obeys Ohm’s law as

Vout(t) = iR =dqdt

R ⇒ dq =1R

Vout(t)dt (11)

Since the charging process is always continuous with time evolution, the we can take time

derivative with respect to Eq. 10 as

dVin(t) =1C

dq (12)

then we have, from Eq. 11 and 12,

Vout(t) = τdVin(t)

dt(13)

The above equation reveal the “differentiating feature’ of the circuit and its behavior has

been illustrated in Fig. 3 (b).

3. Theory of AC circuit

The phase retardation(advance) between the current i(t) and voltage drop u(t) on capacitor

is a remarkable feature due to the oscillation property of the input signal for an AC circuit,

which is different from the situation of a DC circuit. We now derive the effective reactance

of a capacitor and basic relation between voltage and current for a AC circuit. The symbols

used here are in consistence with that in lab manual on page 4.

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Without loss of generality, we may set up the entire theory with a choice that the initial

phase of charging on a capacitor is zero

q(t) = Qcos(ωt) (14)

where Q is maximum value of charge. In order to find out the relation among phases of

different physical quantities, we define the current and voltage as following

i(t) = Icos(ωt + ϕi) (15)

u(t) = Ucos(ωt + ϕu) (16)

where ϕi and ϕu are phases of current and voltage, respectively. Then we have

i(t) =dq(t)

dt= −ωQsin(ωt) (17)

u(t) =q(t)C

=QC

cos(ωt) (18)

Compare Eq. 15 and 17, Eq. 16 and 18, we obtain

I = ωQ, ϕi =π

2(19)

QC

= U, ϕu = 0 (20)

From the above two equations, we arrive at

X =UI

=1ωC

, ∆ϕ = ϕi − ϕu =π

2(21)

This is the definition of reactance of a capacitor X = 1ωC and phase difference of the i(t) and

u(t). The amplitude of total voltage can be calculated via parallelogram law, as shown in

Fig. 2 in the manual(due to the phase difference of π2 ) as

V0 =

√V2

co + V2R =

√I2R2 + I2 1

ω2C2 = I

√R2 +

1ω2C2 (22)

Xtotal =V0

I=

√R2 +

1ω2C2 (23)

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and the voltage drop on capacitor and angle φ are

Vco = IX =V0√

R2 + 1ω2C2

1ωC

=V0√

1 + (ωRC)2(24)

φ = tan−1[VR

Vc] = tan−1[

IRIX

] = tan−1[RωC] (25)

4. The Principle of triggering

(a) (b)

FIG. 4: (color online)Improper triggering case where the Y-plate and X-plate signals are not synchronized

(a) and Proper triggering case where X-plate and Y-plate signals are synchronized by setting the starting

scanning-time point of each frame always at the identical location within one period of Y-plate signal (b).

Different colors have been used to differentiate different frames.

CRO is an apparatus aimed at visualizing the signal coming into it, which is transformed

into a voltage applied to Y-plate. However, since the sweeping process along horizontal

direction is always carried out with time evolution, we have to determine which part of

the signal being measured we are going to visualize. The sweeping time base in CRO is

a sawtooth wave which can be used to simulate the time evolution because it is a linear

function of time. In order to get a stable displaying of the waveform coming in Y-plate;

stable here means that in each displaying time interval, the signal being displayed on the

screen are identical; we have to synchronized the time evolution of X- and Y-plate signals.

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That is why people design a triggering unit in CRO. As shown in Fig. 4 (a), an improper

triggering time points T0, T1, T2, and T3 result in a mixed display of signals of each time

interval(as shown in dashed box). Alternatively, if we reset the triggering time points exactly

at the same point in each period, as the case shown in Fig. 4 (b), The three displayed results

are identical since they correspond to the same part of Y-plate signal. This is analogous to

the case of making a movie or animation, the only difference is that we always want same

content be displayed in each frames.

In part (E) of our experiment, we use internal and external triggering to differentiate phase

difference of V and Vco, which means that when we measure the phase of Vco, the triggering

time points are controlled by the Vco itself, whereas external triggering means that the trig-

gering time points is dominated by V . Put it in another word, it is V that triggered Vco, such

that the phase difference are revealed.

5. Introduction of theoretical aspect of Lissajou’s pattern

We can summarized the physics of Lissajou’s pattern in just one sentence: a superposition of

two harmonic oscillations along x and y directions with different frequencies and different

initial phases. Let us first look at a more simple case of superposition where two harmonic

oscillation along x and y directions with same frequency but different initial phases. We

define

x = A1cos(ωt + α1) (26)

y = A2cos(ωt + α2)

where the variable t can be eliminated using sin2(ωt) + cos2(ωt) = 1, and the equation of

motion is a elliptic type equation as

x2

A21

+y2

A22

− 2xyA1A2

cos(α2 − α1) = sin2(α2 − α1) (27)

We take α2 − α1 = ±π2 as an example and then the equation of motion is

x2

A21

+y2

A22

= 1 (28)

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FIG. 5: Superposition of phase difference is π2 for counterclockwise and clockwise polarizations [2].

Note that the sign of phase difference corresponds a retardation or an advance of motion

along x and y directions, as shown in Fig. 7. Fig. 5 gives some results of motions with

different values of α1 − α2.

FIG. 6: Superposition of oscillation along x and y directions for various phase differences [2].

The situation of superposition of oscillation of different frequencies and phases are more

complicated than that have been discussed hitherto. Let first define

x = Acos(mωt) (29)

y = Acos(nωt + α)

α =Kπ4m

(30)

we do not solve the equation of motion explicitly and instead the motions can be simulated

by computer programme. We just give some patterns in Fig. 5, where m, n, and K are

integers. The physics behind Lissajou’s pattern is nonlinear behaviors of a dynamic system,

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which in some sense makes our world beautiful.

FIG. 7: Lissajou’ patterns for different proportions of frequencies and different phases [2]. Note that the

superposition can be only stable when m, n, and K are integers.

[1] K. H. Zhao and X. M. Chen, Electromagnetics, Higher Education Press, 1985(Writen in Chinese)

[2] A. S. Qi and C. Y. Du, Mechanics, Higher Education Press, 1997(Writen in Chinese)