AC  Circuits  –  page  1  of  12  

RESISTORS, CAPACITORS AND INDUCTORS IN AC CIRCUITS

INTRODUCTION AND THEORY Many electric circuits use batteries and involve direct current (dc). However, there are considerably more circuits that operate with alternating current (ac), when the charge flow reverses direction periodically. In an “ac” circuit, the most common generators serve the same purpose as a battery serves in a dc circuit: they give energy to the moving electric charges but they change the direction of magnetic forces periodically.

Since electrical outlets in a house provide alternating current, we all use ac circuits routinely. If the voltage and the current alternate sinusoidally with time we can write:

€

v = v(t) =Vm ⋅ sin(ϖt +ϕ)

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i = i(t) = Im ⋅ sin(ϖt) Where

• v and i represent the instantaneous voltage and current when we are considering their variation with time explicitly.

• Vm and Im are the amplitude or peak value of the voltage and current • V = Vm/21/2 and I = Im/21/2 without subscripts refer to the RMS values. • f is the ordinary frequency and represents the number of complete oscillations per

second • ω = 2πf is the angular frequency. • φ is the phase difference between the voltage and current.

Resistors And Ohm's Law In AC Circuits

The schematic below is that of an ac circuit formed by plugging a toaster into a wall socket. The heating element of the toaster is essentially a thin wire of resistance R and becomes red hot when the electrical energy is dissipated in it    According to Ohm’s Law, the instantaneous voltage v across a resistor is proportional to the instantaneous current i flowing through it.

Figure 1: Resistor in an AC circuit

R   V  

i  

AC  Circuits  –  page  2  of  12  

The next two graphs show the voltage across the resistor and the current flowing through the resistor as a function of time.

Figure 2: V and I: Sinusoidal variation in phase Figure 3: Phasor Diagram

From the graph in Figure 2 the peak value of voltage across a pure resistor is reached at the same time with the peak value of the current. Therefore, the current and voltage are said to be in phase, and mathematically this is expressed as

In the Figure 3, the radial vectors also called phasors rotate with angular velocity ω representing the current and the voltage across the resistance. The lengths of these phasors represent the peak current Im and voltage Vm. The y components of these phasors are

and they are equal to the y – values from the graph in figure 2 at any time. The phasor diagram shows that the voltage and the current are in phase.

Resistance, Reactance And Impedance

The ratio of the voltage to the current in a resistor is called its resistance. As well, in a circuit where the current is proportional to the voltage, the circuit is called a linear circuit. This is happening when the circuit contains only resistors, capacitors and/or inductors. Resistance does not depend on frequency, and in a resistor the voltage and the current are in phase. However, circuits with only resistors, capacitors, and solenoids are not very useful in some

Instantaneous  voltage  V  –  (blue),    

and  current  i  –    (red)  

V(t),    

i(t)  

t  Im   Vm  

AC  Circuits  –  page  3  of  12  

cases. If the circuit contains also, diodes or transistors, the circuit is no longer linear. In most practical cases, the ratio of the voltage to the current depends on the frequency and in general there is a phase difference between the voltage and the current. In this general case, the ratio of the voltage to the current is called Impedance and it is denoted with the symbol Z. Resistance is a special case of impedance. A very important case is that in which the voltage and the current are out of phase by 90°: this is an important case because when this happens, no power is lost in the circuit and the ratio of the voltage to the current is called the reactance, denoted with the symbol X. Reactance can be caused by capacitors or by inductors.

Capacitors In AC Circuits

Capacitors store electric charge. They are used with resistors in timing circuits because it takes time for a capacitor to fill with charge. They are also used to smooth varying DC supplies by acting as a reservoir of charge. They are also used in filter circuits because capacitors easily pass AC signals but they block DC signals. The voltage on a capacitor depends on the amount of charge stored on its plates. If we denote the instantaneous value of the current with i(t):

The current flowing off the positive plate is equal to the current flowing into the negative plate and by definition is the rate at which the charge Q is being stored. From the definition of the capacitance as a function of the charge Q and the potential across the capacitor,

it follows that

But the charge Q on the capacitor equals the integral of the current with respect to time.

In the indefinite integral above, the constant of integration was set to zero so that the average charge on the capacitor would be zero (we are starting with an uncharged capacitor).

AC  Circuits  –  page  4  of  12  

Therefore, the voltage across the capacitor:

The last equation shows that the current and the voltage are out of phase by 900. The capacitive reactance XC is equal to:

and it can be defined as the ratio of the magnitude of the voltage to magnitude of the current in a capacitor (and that is Ohm’s Law for the capacitor!) Looking at the difference of phase, the voltage across the capacitor is 90°, or one quarter of period, behind the current. The same phase difference φ = 90° is reflected in the phasor diagrams. Since the vertical component of any phasor arrow represents the instantaneous value of its quantity and the phasors are rotating counter clockwise the phasor representing VC is 90° behind the current.

Figure 4: V and I: Sinusoidal variation for Capacitor Figure 5: Capacitor Phasor Diagram

As we have seen before, when the voltage and the current differ in phase by 900, the resistance is called reactance. Another important difference between reactance and resistance is that the reactance is frequency dependent and for a capacitor, it decreases with frequency.

Instantaneous  voltage  V  –  (blue),  and  current  i  –    (red)  

AC  Circuits  –  page  5  of  12  

Inductors In AC Circuits

An inductor is usually a coil of wire. The resistance of an ideal inductor is negligible, as is its capacitance. However, the voltage across an inductor is influenced by changes of its own magnetic field. Faraday's law of electromagnetic induction states that the current i(t) in the coil sets up a magnetic field, whose magnetic flux ΦB is proportional to the field strength B, which in turn is proportional to the current. Therefore, the self inductance of the coil, denoted L is defined as: However, Faraday's law gives the emf induced in a coil due to a change in the magnetic flux

According to Kirchhoff’s First Law the emf is a voltage rise; therefore, the voltage drop vL across the inductor should be:

vL (t) = −eL =dΦB

dt

=ddt

L ⋅ i(t)[ ]

= L ⋅ ddt

Im ⋅sin(ϖ t)[ ]

=ϖL ⋅ Im ⋅cos(ϖ t)

=Vm ⋅sin(ϖ t +π2)

As in the case of the capacitor, we define the inductive reactance XL as the ratio of the magnitudes of the voltage and the current, and from the equation above we see that

XL = ωL.

It is worth noting the analogy to Ohm's law: the voltage is proportional to the current, and the peak voltage and currents are related by

Vm = XL.Im

AC  Circuits  –  page  6  of  12  

Figure 6: V and I: Sinusoidal variation for Solenoid Figure 7: Solenoid Phasor Diagram

From figures 6 and 7 it follows that the voltage and the current through the solenoid are in phase: the voltage across the inductor has its maximum when the current is changing most rapidly, which is when the current is passing through zero. Therefore, the voltage across the ideal inductor is 90° (or ) ahead of the current, (i.e. it reaches its maximum one quarter of the cycle before the current does). The same conclusion is drawn from the phasor diagram.

We should also note that for a coil the reactance is frequency dependent in the sense that it increases with frequency.

Summary:

Resistance, Reactance and Impedance

The following is a summary of the relationship between voltage and current in linear circuits: • The impedance is the general term for the ratio of the voltage to the current. • Resistance is the special case of impedance when φ = 0, • Reactance the special case when φ = ± 90°.

Component Resistor Capacitor Inductor

Difference of Phase between Voltage and Current

Voltage and Current are in phase

Voltage lags behind Current by π/2

Current lags behind Voltage by π/2

Ohm’s Law R = VRI

Xc =VCI=1ϖC

XL =VLI=ϖL

V(t),    

i(t)  t  

AC  Circuits  –  page  7  of  12  

Resistor and Capacitor connected in Series

When we connect components together, Kirchhoff's laws apply at any instant. So the voltage vs(t) across a resistor and capacitor in series is just

vs(t) = vR(t) + vC(t) However, the addition is more complicated because the two are not in phase: they add to give a new sinusoidal voltage, but the amplitude VS is less than VR + VC as it can be seen from figure 11.

Figure 10: Resistor and Capacitor in Series Figure 11: R-C Series Phasor Diagram

However, using Pythagoras' theorem in figure 11 we have: VS

2 =VR2 +VC

2 Using Ohm’s Law and expressing the three voltages and substituting in the equation above, we obtain the impedance ZRCS and the phase difference ϕ between voltage VRCS and the current I:

I ⋅ZRCS( )2 = I ⋅R( )2 + I ⋅XC( )2

ZRCS = R2 + 1ϖC!

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$

%&2

and tanϕ = −VCVR

= −XC

R= −

1ϖRC

R   VS  

i  

C vs

er

ie

s = vR + vC but Vs

er

ie

s > VR + VC.

VC vseri

es = vR + vC but Vseri

es > VR + VC. The amplitudes and the RMS voltage

VR vseri

es = vR + vC but Vseri

es > VR + VC. The amplitudes

I  VR  

VC   VRCS  

ϕ  

AC  Circuits  –  page  8  of  12  

Resistor and Inductor connected in Series

Similarly, when we connect a solenoid in series with a resistor, the instantaneous voltage vs(t) across the resistor and inductor in series will be:

vs(t) = vR(t) + vL(t) And again, the amplitude VRLS will be always less than VR + VL as it can be seen from figure 13.

Figure 12: Resistor and Inductor in Series Figure 13: R-L Series Phasor Diagram

Applying again Pythagoras' theorem, from figure 13 we have: VRLS

2 =VR2 +VL

2 Using Ohm’s Law to express the three voltages in the equation above, we obtain the impedance ZRLS and the phase difference ϕ between voltage VRLS and the current I:

I ⋅ZRCS( )2 = I ⋅R( )2 + I ⋅XL( )2

ZRCS = R2 + ϖL( )2

and tanϕ = VLVR

=XL

R=ϖLR

R   VS  

i  

L vs

eri

es = vR + vC but Vs

eri

es > VR + VC. The ampli

VL vseri

es = vR + vC but Vseri

es > VR + VC. The amplitudes and the RMS voltages V do not

VR vseri

es = vR + vC but Vseri

es > VR + VC. The amplitudes and the RM

I  

VR  

VL  VRLS  

ϕ  

AC  Circuits  –  page  9  of  12  

Resistor, Capacitor and Inductor connected in Series

Now we connect a capacitor, and solenoid in series with a resistor, the instantaneous voltage vs(t) across the resistor, capacitor and inductor in series will be:

VRCLS(t) = vR(t) + vC(t) + vL(t)

But, the amplitude VRCLS will be always less than VR + VC + VL because of the same reason explained before.

Figure 14: Resistor, Capacitor and Inductor in Series Figure 15: R-C-L Series Phasor Diagram

Applying again Pythagoras' theorem, from figure 15 we have: VRCLS

2 =VR2 + VL −VC( )2

Using Ohm’s Law to express the four voltages in the equation above, we obtain the impedance ZRCLS and the phase difference ϕ between voltage VRCLS and the current I:

I ⋅ZRCLS( )2 = I ⋅R( )2 + I ⋅XL − I ⋅XC( )2

ZRCS = R2 + ϖL − 1ϖC

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#$

%

&'2

and tanϕ = VL −VCVR

=XL − XC

R=ϖL − 1

ϖCR

Since the inductive and capacitive phasors are 180° out of phase, their reactances tend to cancel each other. This happens at resonance when XL = XC. At resonance ϕ = 0, the impedance Z = R has a minimum and the current through the circuit can reach very large

I  

VR  

VL  

VRLS  

ϕ  R   VS  

i  

VC vseri

es = vR + vC but Vseri

es > VR + VC. The amplitudes and the RMS voltages V do

VL vseri

es = vR + vC but Vseri

es > VR + VC. The amplit

VC vseri

es = vR + vC but Vseri

es > VR + VC. The amplitude

AC  Circuits  –  page  10  of  12  

values that could be damaging to the circuit.

APPARATUS

• RLC circuit board with resistors, capacitors and inductor o Resistors: 100 Ω, 1 W; 33 Ω, 5 W; 10 Ω, 10 W o Capacitors: 100 µF, 16 V and 330 µF, 16 V (capacitance

values may vary by ±20 %) o Inductor: 8.2 mH @ 1 kHz, 6.5 Ω maximum DC resistance,

0.8 A current rating RMS, 3/4” I.D. x 1-3/4” O.D.

• Dual channel oscilloscope with sinusoidal voltage signal generator incorporated.

• BNC cables.

• Banana plug patch cords.

PROCEDURE

In this experiment you will be applying a sinusoidal signal to different circuits and will analyze the effect of the resistors, capacitors and inductors on the current and the relative phase between the voltage applied and the current.

Familiarize yourself with the apparatus to be used. The function generator and the oscilloscope are a single unit. There should be a “TEE” connected to the output of the function generator portion of the unit, with one end of the “TEE” connected directly to CH1 of the oscilloscope (to give you the input signal) and the other end going to the points of the circuit where the source is going to be connected. The voltage collected across different components of the circuit will be going to CH2 of the oscilloscope.

1. Select the mode of the function generator to “sinusoidal” and then select a signal in the range of 100KHz.

AC  Circuits  –  page  11  of  12  

2. Using the “T” splitter to apply this signal to “Chanel 1” of the oscilloscope and to the “source” in the circuit in figure 10.

3. Adjust the amplitude of the sinusoidal signal, and make sure that the ‘OFFSET’ knob is pushed in. Setup the time/div so that only one cycle appears on the oscilloscope.

4. Collect the voltage across the resistor and the capacitor and apply it on “Chanel 2” of the oscilloscope. Compare with what you see on CH1. Are CH1 and CH2 in phase for both the resistor and the capacitor? Explain.

5. Read the relative phase for each component from the oscilloscope by comparing the position of the signal on CH1 relative to CH2.

6. Measure the resistance of the resistor R with an ohm-meter and calculate the capacitive reactance XC and the capacitance of the capacitor C.

Table # 1: Data for Resistor and Capacitor in Series

7. Repeat steps 2 through 6 for the circuits in figure 12 and determine the relative phase, the inductive reactance and the induction of the solenoid.

Table # 2: Data for Resistor and Inductor in Series

Component

ϕ (µs)

ϕ (rad)

tan ϕ

R ( Ω )

f (Khz)

ϖ = 2πf (S-1)

C (F)

CapacitiveReactance

XC

( Ω ) Resistor

Capacitor

Component

ϕ (µs)

ϕ (rad)

tan ϕ

R ( Ω )

Inductive

Reactance XL

( Ω )

f (Khz)

ϖ = 2πf (S-1)

L (H)

Resistor

Inductor

AC  Circuits  –  page  12  of  12  

8. Repeat steps 2 through 6 for the circuit in figure 14 and determine the relative phase, the overall reactance XL - XC and the impedance Z of the circuit.

Table # 3: Data for Resistor, Capacitor and Inductor in Series

Component

ϕ (µs)

ϕ (rad)

tan ϕ

R ( Ω )

Overall Reactance XL - XC

( Ω )

f (Khz)

ϖ = 2πf (S-1)

Z ( Ω )

Resistor

Capacitor & Inductor