To study R-L-C series circuit and find the value of inductance. Also draw the phasor diagram.

Experiment No.: 05

Aim of The Experiment: –

Objective of the Experiment:

To be written by student.

Apparatus Required: –

Sl. No.	Name	Specification	Quantity
1	Single-phase auto transformer	5KVA, (0-270)V	1 Nos.
2	Voltmeter	(0-250)V, MI	2 Nos.
3	Voltmeter	(0-150)V, MI	2 Nos.
4	Ammeter	(0-5)A, MI	1 Nos.
5	Rheostat	(0-80)Ω, 5A	1 Nos.
6	Inductor		1 Nos.
7	Capacitor		1 Nos.
8	Connecting Wires	PVC Insulated Copper	As per required

Table 5.1

Circuit Diagram: –

Theory: –

In R-L-C series AC circuit a resistor of resistance R ohm, an Inductor of inductance L henry and a Capacitor of capacitance C farad are connected across single phase ac supply of rms V volts as shown in above fig.

As the circuit shown is in series the current across each element R,L and C will remain the same, but the voltage drops among these elements will differ depending upon their respective capacity.

Let V = rms value of applied voltage and I = rms value of supply current flows in the circuit.

As shown in phasor diagram:

\[\color\red{V_R = IR\; where\; V_R\; is\;in\;phase\;with\;I,}\] \[\color\red{V_L = IX_L\;where\;V_L\;leads\;I\;by\;90^0,}\] \[\color\red{V_c = IX_c\;where\;V_c\;lags\;I\;by\;90^0,}\]

It is better to say that the supply current I is in phase with V_R , but lags with V_L by 90⁰ and leads with V_C by 90⁰.

From phasor diagram of R-L-C series circuit an impedance triangle can be drawn by dividing each side of phasor diagram by the same factor I, we get a triangle whose sides represent R, ( X_L – X_C ), Z.

The impedance triangle is a right angle triangle.

When |V_L| > |V_C|, the resultant of phasor V_R and (V_L – V_C) will give the voltage across the impedance and should be equal to supply voltage.

\[V_Z=\sqrt{V_R^2+(V_L-V_C)^2}\] \[=IZ\] \[=\sqrt{(IR)^2+(IX_L-IX_C)^2}\] \[=I\sqrt{R^2+(X_L-X_C)^2}\]

Thus,

\[Z=\sqrt{R^2+(X_L-X_C)^2}\]

where Z is the impedance in ohm, X_L is the inductive reactance in ohm and X_C is capacitive reactance in ohm. The relation is shown in impedance triangle as shown in fig.: 5.4.

The phase angle of V_Z is given by

\[\color\red{\phi = tan^{-1}\left\{\dfrac{V_L-V_C}{V_R}\right\}}\] \[\color\red{= tan^{-1}\left\{\dfrac{IX_L-IX_C}{IR}\right\}}\] \[\color\red{= tan^{-1}\left\{\dfrac{X_L-X_C}{R}\right\}}\] \[ \color\red{= tan^{-1}\left\{\dfrac{X}{R}\right\}}\]

where X is the net reactance in ohm.

from impedance triangle

\[\color\red{Cos\phi=\dfrac{R}{Z}}\] \[\color\red{=\dfrac{R}{\sqrt{R^2+(X_L-X_C)^2}}}\]

Procedures: –

Connect all the instruments as per circuit diagram given above.
Before switch on the main power supply make sure that single-phase auto transformer knobe is at zero position.
Now slowly increase the supply voltage to the circuit after giving supply to the single-phase auto-transformer.
Take all the corresponding readings of the connected instruments in the circuit as per observation table.
Now calculate V_Z, power factor Cosɸ and % error as per formula given in observation table.

Precaution: –

Don’t switch on power supply without concerning teacher.
Single phase autotransformer must be kept at minimum potential point before switch on the experiment.

Observation Table: –

Sl. No.	V (volts.)	I (Amp.)	V_R (volts.)	V_L (volts.)	V_C (volts.)	V_Z (volts.)	Cosɸ= (V_R/V_Z)	% Error = [{(V-V_Z)/V}*100]
1
2
3
4
5
6
7

Table 5.2

Conclusion:

To be written by student.

For Your Reference: –

Studying an R-L-C series circuit involves analyzing the behavior of a circuit containing a resistor (R), an inductor (L), and a capacitor (C) connected in series with an AC voltage source. Here’s a step-by-step guide on how to study such a circuit:

Circuit Setup: Assemble the R-L-C series circuit on a breadboard or a circuit simulator. Ensure that the components are connected in series, with the AC voltage source providing the input.

2. Measurement of Parameters:

a. Resistance (R): Measure the resistance of the resistor using a multimeter.

b. Inductance (L): If the inductance is not specified, you can measure it using an LCR meter.

c. Capacitance (C): Similarly, if the capacitance is not known, you can measure it using a capacitance meter.

3. AC Voltage Source: Set the frequency and amplitude of the AC voltage source. Note down the values for further analysis.

4. Voltage and Current Measurement:

a. Use an oscilloscope or a voltmeter to measure the voltage across the entire series circuit. This voltage will be the same across all components.

b. Measure the current flowing through the circuit using an ammeter. Ensure that both voltage and current measurements are taken simultaneously and accurately.

5. Impedance Calculation:

Calculate the impedance (Z) of the circuit using the formula:

\[\color\red{𝑍=\sqrt{𝑅^2+(𝑋_𝐿−𝑋_𝐶)^2}}\] \[\color\red{where\; 𝑋_𝐿 \;is\; the\; inductive\; reactance}\] \[ \color\red{and \;𝑋_𝐶\; is\; the\; capacitive\; reactance.}\] \[\color\red{𝑋_𝐿=2𝜋𝑓𝐿}\] \[\color\red{and\; 𝑋_𝐶=\frac{1}{2𝜋𝑓𝐶}, \;where\; 𝑓\; is}\] \[\color\red{the\; frequency\; of\; the\; AC\; source.}\]

6. Phase Angle Calculation:

Determine the phase angle (ϕ) between the voltage and current using trigonometric relationships:

\[\color\red{\tan⁡(𝜙)=\frac{𝑋_𝐿−𝑋_𝐶}{𝑅}}\]

The phase angle indicates the phase difference between the voltage and current in the circuit.

7. Analysis:

Analyze the impedance, phase angle, and voltage-current relationship to understand the behavior of the R-L-C series circuit.

Observe how the circuit behaves at different frequencies, especially near resonance (where X_L=X_C).

8. Phasor Diagram:

Draw a phasor diagram to represent the voltage, current, resistance drop, inductive reactance, and capacitive reactance as vectors. This helps visualize the phase relationship between voltage and current.

9. Interpretation:

Interpret the results obtained from the measurements and calculations to understand the circuit’s behavior, including impedance, resonance, phase angle, and power factor.

By following these steps, you can effectively study an R-L-C series circuit and gain insights into its behavior under AC excitation.

Viva-voce Questions: –

What is an R-L-C series circuit, and why is it studied ?

Answer: An R-L-C series circuit is a circuit configuration consisting of a resistor (R), an inductor (L), and a capacitor (C) connected in series. It is studied to understand the behavior of circuits containing reactive components under alternating current (AC) excitation, and it finds applications in various electrical and electronic systems.

2. How would you find the value of inductance in an R-L-C series circuit experimentally ?

Answer: To find the value of inductance ((L)) experimentally in an R-L-C series circuit, we can vary the frequency of the AC source and measure the impedance ((Z)) of the circuit using an oscilloscope or a multimeter. By using the formula (L = \frac{Z^2 – R^2}{4\pi^2 f^2}), where (R) is the resistance and (f) is the frequency, we can calculate the inductance.

3. What is a phasor diagram, and how is it relevant to the study of an R-L-C series circuit ?

Answer: A phasor diagram is a graphical representation of the magnitude and phase relationship between different voltage and current components in an AC circuit. In an R-L-C series circuit, the phasor diagram helps visualize the phase relationship between voltage ((V)) and current ((I)), as well as the voltage drop across the resistor, inductor, and capacitor.

4. How do you draw a phasor diagram for an R-L-C series circuit ?

Answer: To draw a phasor diagram for an R-L-C series circuit, we represent the voltage ((V)), current ((I)), and the voltage drops across the resistor ((VR)), inductor ((VL)), and capacitor ((VC)) as vectors on a complex plane. The angle between the voltage and current phasors represents the phase angle ((\phi)) between them, and the lengths of the phasors represent their magnitudes.

5. What are the key components of a phasor diagram for an R-L-C series circuit ?

Answer: The key components of a phasor diagram for an R-L-C series circuit include the voltage ((V)) phasor, the current ((I)) phasor, and the voltage drops across the resistor ((VR)), inductor ((VL)), and capacitor ((VC)). These phasors are drawn such that the vector sum of (VR), (VL), and (VC) equals the voltage phasor (V), illustrating the phase relationships between them.

6. How do you interpret the results obtained from the phasor diagram of an R-L-C series circuit ?

Answer: The interpretation of the phasor diagram results involves understanding the phase relationships between voltage and current components in the circuit. For example, the angle between the voltage and current phasors ((\phi)) indicates the phase difference between them, while the lengths of the phasors represent their magnitudes. This information helps analyze the circuit’s behavior, including impedance, phase angle, and power factor.

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