9. EXPERIMENT E9: THE RLC CIRCUIT

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9. EXPERIMENT E9: THE RLC CIRCUIT

OBJECTIVES

In this experiment, you will measure the electric current, voltage, reactance,

impedance, and understand the resonance phenomenon in an alternating-current circuit,

which consists of an inductor, a resistor, and a capacitor in series.

THEORY

Alternating-current (AC) circuits have a power source, which supply a time-varying

current , such that the voltage and thus the current are periodic functions of time with

frequency and angular frequency :

(9.1)

(9.2)

When there is a single resistor with resistance in an AC circuit (Fig. 9.1), the

instantaneous voltage is

( ) . (9.3)

So that the voltage and the current are in phase, which means that the phase angle

between the voltage and current is zero.

Figure 9.1 Circuit with AC source and resistor.

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Now suppose there is a single inductor with self-inductance and zero resistance

connected to an AC power supply (Fig. 9.2). The alternating current will then induce an

emf

, which is along the direction of the current. The voltage between the

terminals of the inductor can be obtained:

( ) (9.4)

Figure 9.2 Circuit with AC source and inductor.

In the case of a pure inductor, the phase angle is equal to . In general, the phase

of the voltage is defined with respect to the phase of the current. For an AC circuit

supplying a current given in Eq. (9.2), the voltage is given by

( ) (9.5)

where is the phase angle of the voltage relative to the current (note that: (

) ).

The amplitude (maximum value) of voltage for a pure inductor is

(9.6)

and the inductive reactance of an inductor is defined as

(9.7)

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The SI unit of inductive reactance is ohm, as for resistance. With this definition, we

can rewrite Eq. (9.6) the voltage amplitude over the inductor in the following compact

form:

. (9.8)

If we connect a single capacitor with capacitance to the AC power source (Fig. 9.3),

the voltage across the capacitor is given by

(9.9)

Figure 9.3 Circuit with AC source and capacitance.

This means that the phase angle for the capacitor voltage (relative to the current), is

. The voltage amplitude becomes

(9.10)

and we define the capacitive reactance associated with the capacitor as

(9.11)

This can be recast into Eq. (9.10) to obtain

(9.12)

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If we connect a resistor, an inductor, and a capacitor in series to the AC power source

(Fig. 9.4), the circuit is called as RLC-circuit.

Figure 9.4 RLC circuit.

The impedance of an AC circuit is defined as the ratio of the voltage amplitude to

the current amplitude across the circuit:

√ ( )

(9.13)

Using Eqs. (9.7) and (9.11), we can write the current amplitude as

√ [ ( )] (9.14)

To calculate the phase angle of voltage ( ) relative to the current

, we can make use of a phase diagram, as shown in Fig. 9.5.

Figure 9.5 Phase diagrams.

XL

XC

R

Şekil 1

XLXC

R

Şekil 2

-

Z

XLXC

R

Şekil 3

-

Z

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The phase angle is given by

( )

(9.15)

If , the voltage leads the current, whereas for , the voltage lags the

current.

When the inductive reactance exactly equals the capacitive reactance, and

, so the phase angle becomes zero. In this case, we have the phenomenon called

resonance. The voltage and the current reach their maxima and minima simultaneously. If

the resonant frequency of the RLC circuit is , means , so

√ . (9.16)

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EXPERIMENTAL PROCEDURE

Part I

1. Set up the circuit with an inductor, as shown in Fig. 9.2.

2. Measure and for a range of frequencies, calculate ⁄ and record them on

the table 9.1.

3. Plot a graph showing as a function of the angular frequency (Graph 9.1).

4. Using the slope of the line, estimate the self-induction coefficient, , in Henry.

5. Calculate the percentage error (see appendix). Record it on the table 9.1.

Part II

6. Set up the circuit with a capacitor, as shown in Fig. 9.3.

7. Measure and for a range of frequencies, calculate ⁄ and record the

measurements on the table 9.2.

8. Plot a graph showing as a function of the angular frequency (Graph 9.2).

9. Using the slope of the line, estimate the capacitance, , in Farads.


Part III

11. Set up the RLC circuit with an inductor and a capacitor, as shown in Fig. 9.4. The

“resistor” consists of all the conducting material within the circuit elements. There is no

need to introduce to an extra resistor in circuit.

12. Measure and for a range of frequencies and record your measurements on the table

9.3.

13. Calculate the corresponding impedance values, ⁄ , and record on the table 9.3.

14. Plot the graph of versus the angular frequency (Graph 9.3).

15. Estimate the value of resonance frequency graphically, and compare

with the theoretical value, .


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LABORATORY REPORT: CALCULATIONS AND GRAPHS

Name : _______________________

Faculty number : _______________________

Date : _______________________

Instructor name : _______________________ Instructor’s signature: ____________

These blanks must be written in pen (not pencil).

Table 9.1 Calculation of the inductance.

( ) ( ) ( ) ( ) ( )

Show your work here:

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Graph 9.1 XL versus .


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Table 9.2 Calculation of the capacitance.

( ) ( ) ( ) ( ) ( )⁄


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Graph 9.2 1/XC versus .


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Table 9.3 Calculation of the resonant frequency.

( ) ( ) ( ) ( ) ( )


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Graph 9.3 Z versus .


BIBLIOGRAPHY

[1] P Young, H.D., Freedman, R. 2012, University Physics with Modern Physics, 13th ed.,

Addison-Wesley, Chapter 31.

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9. EXPERIMENT E9: THE RLC CIRCUIT