EEE Power Factor Project Report

eee-1109 EEE-1109 Basic Electrical Engineering

Objectives

To understand the effect of inductive loads on the power factor in AC circuits.
To analyze the relationship between resistance, inductance, and capacitance in a series RLC circuit and their impact on phase angle and impedance.
To implement and evaluate power factor improvement using a series capacitor and determine the optimal capacitance for maximum efficiency.

Introduction

In alternating current (AC) circuits, loads are rarely composed of pure resistance. Most practical electrical devices motors, transformers, inductors introduce inductance into the circuit. Inductance causes the current to lag behind the voltage, reducing the power factor (PF), which is the ratio of real power to apparent power in the system. The power factor indicates how effectively the supplied electrical power is converted into useful work. A power factor less than unity means a portion of the input power oscillates in the circuit as stored energy rather than delivering useful output.

In a series combination of a resistor and inductor (R–L circuit), the inductive reactance $(X_{L})$ leads to a phase angle $(ϕ)$ between voltage and current. The relationship between the circuit components and this phase shift is given by:

ϕ = tan^{- 1} (\frac{X _{L}}{R})

The cosine of this angle determines the power factor:

cos ϕ = \frac{R}{Z}

where ( Z ) is the total impedance of the circuit. Improving the power factor means reducing the phase angle, which in turn requires reducing the net reactive component of the circuit.

To achieve this, a capacitor can be added in series or parallel to provide a leading reactive effect. In a series RLC circuit, the capacitor’s reactance $(X_{C})$ counteracts the inductive reactance:

X = X_{L} - X_{C}

As ( $X_{C}$ ) increases (which depends on the value of $(C)$ and supply frequency), the net reactance decreases, reducing the phase shift and increasing the power factor. The total impedance becomes:

Z = R^{2} + (X_{L} - X_{C})^{2}

By choosing an appropriate capacitor value, the reactive power balance can be improved, enhancing the efficiency of power usage while reducing energy losses and current burden on the supply.

Apparatus

Table 01: Apparatus needed to perform the experiment

Sl. No.	Name of apparatus	Rating	Quantity (piece)
1	Bulb	230V ; 100W	1
2	Ballast	40W ; 0.43A	1
3	Capacitor	3.5 $μ$ F ; 230V	1
4	LED	2V	1
5	Resistor	136K $Ω$	1
6	Switch	8A ; 250V	1
7	Connecting wires	-----	As required
8	PBC board	-----	As required
9	Thermocol	-----	As required
10	Screw	-----	As required

### Cost Estimation

Table 02: Cost estimation of apparatus used while performing the experiment

Sl. No.	Name of Apparatus	Rating/Specification	Quantity	Cost (BDT)
1	Bulb	230V ; 100W	1	35
2	Inductor Ballast	40W ; 0.43A	1	160
3	Capacitor	3.5 $μ$ F ; 230V AC	1	90
4	Switch	8A ; 250V	1	20
5	Connecting Wires	Copper, insulated	As required	60
6	Holder	-	1	70
7	2 pin Plug	-	1	20
8	LED	Red, 2V	1	2
Total				457

Experimental setup:

Fig 01: Circuit diagram for RL circuit

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Fig 02: Circuit diagram for RLC circuit

Fig 03: Experimental Setup for RLC circuit

Experimental Table

Table 03: Data table for RL circuit

Supply voltage $V_{s}$ (v)	Current, I (A)	Voltage of Bulb, $V_{R}$	Voltage of Inductor, $V_{L}$	Power of bulb, $P_{R}$	Power of Inductor, $P_{L}$	Resistance of Bulb, $R = P_{R} / I^{2}$
230	0.33	133.7	165	44	10	404.04

Resistance of Inductor, $r_{L} = P_{L} / I^{2}$	$Z_{L} = \frac{V _{L}}{I}$	$X_{L} = Z_{L}^{2} - r_{L}^{2}$	$θ = t a n^{- 1} (\frac{X _{L}}{R + r _{L}})$	Power Factor, $PF = cos (θ)$
91.83	500	491.83	44.74	0.71 (Lagging)

Table 04: Data for RLC circuit

Supply voltage $V_{s}$ (v)	Current, I (A)	Voltage of Bulb, $V_{R}$	Voltage of Inductor, $V_{L}$	Voltage of Capacitor, $V_{C}$
230	0.36	151.8	173	313.3

Power of bulb, $P_{R}$	Power of RL, $P_{R L}$	Power of RLC, $P_{R L C}$	Power of Inductor, $P_{L}$	Power of bulb, $P_{R}$	Power Factor, PF
54	62	66	8	4	0.8

Calculations:

Calculator for RL circuit:

$P_{R} = 44 W$

$I = 0.33 A$

$R = \frac{P _{R}}{I ^{2}} = \frac{44}{( 0.33 ) ^{2}} = 404.04 Ω$

$P_{L} = 10 w$

$R_{L} = \frac{P _{L}}{I ^{2}} = \frac{10}{0.3 3 ^{2}} = 91.83 \ohm$

$Z_{L} = \frac{V _{L}}{I} = \frac{165}{0.33} = 500 Ω$

X_{L} = (500)^{2} - (91.83) = 491.83 Ω

θ = tan^{- 1} (\frac{X _{L}}{R + r _{L}}) = tan^{- 1} (\frac{491.49}{404.04 + 91.83}) = 44.7 4^{\circ}

$cos θ = cos (44.7 4^{\circ}) \approx 0.71$

$∴ Power factor (RL) = 0.71 (lagging)$

Calculation for RLC circuit:

$Desired Power factor = 0.8$

$cos θ = 0.8$ $∴ θ = cos^{- 1} (0.8)$

Neglecting the resistance of the capacitor,

t a n^{- 1} (\frac{X}{R _{P} + r _{L}}) = cos^{- 1} (0.8)

So,

t a n^{- 1} (\frac{X}{404.04 + 91.83}) = cos^{- 1} (0.8)

∴ X = 371.9025 Ω

Since in RLC series circuit,

X = X_{L} - X_{C}

Heew $(X_{L} > X_{C})$ , therefore

X_{L} - X_{C} = 371.9025

Again ( $X_{L} = 491.49 Ω$ ), so

491.49 - X_{C} = 371.9025

X_{C} = 863.3925 Ω

Now,

X_{C} = \frac{1}{2 π f C}

So,

863.3925 = \frac{1}{2 π \cdot 50 \cdot C}

C = \frac{1}{2 π \cdot 50 \cdot 863.3925}

C = 3.68 μ F

Vector Diagram

Fig 04: Vector diagram for RL circuit

Fig 05: Vector Diagram for RLc Circuit

Discussion:

In the initial RL configuration (Table 03), the circuit showed a lagging power factor due to the inductive reactance of the ballast coil . The supply voltage was measured at approximately 230 V with a current of about 0.33 A. The voltage across the inductor was higher than that across the resistor, confirming that the circuit was inductive dominant. The calculated power factor from impedance and phase angle analysis was approximately 0.71, while the power factor meter indicated a slightly higher measured value of about 0.76, which is expected due to practical factors such as internal resistance of windings, instrument precision, and core losses.

To correct the power factor, a capacitor of 3.68 μF was needed in series with the RL combination. But we could only find capacitors of value 3.5 μF. Despite of that small deviation, our measured power factor with power factor meter was 0.8. Here, the presence of the capacitor introduced a capacitive reactance $(X_{C})$ , which opposed the inductive reactance $(X_{L})$ of the inductor. As a result, the net reactance of the circuit decreased, reducing the impedance and increasing the circuit current slightly. This decreased the phase angle between voltage and current, leading to an improvement in the power factor. The power factor observed after adding the capacitor increased to approximately 0.80, indicating successful compensation.

The change in the brightness of the load (lamp) also reflected this improvement. With a higher power factor, more of the supplied power was converted to useful real power rather than oscillating reactive power. In other words, less energy was being wasted in the exchange between the electric and magnetic fields of the reactive components.

Theoretically, the required capacitor value for achieving the targeted power factor can be determined from the condition of balancing part of the inductive reactance. The approximate calculated capacitance required to improve the power factor to around 0.8 aligns closely with the 3.5 μF value used experimentally, which validates the choice. Minor deviation between calculated and observed behavior can be attributed to:

Tolerance of capacitor value
Winding resistance and iron losses in the inductor
Non-linear behavior of the lamp filament (resistance increases with temperature)
Measurement accuracy of analog meters

Overall, the experiment clearly demonstrated that the addition of a capacitor effectively reduces phase shift, increases real power transfer, and improves power factor in a series RL circuit. The circuit behavior after compensation shifted from inductive dominant to slightly capacitive dominant, which matched the observed leading nature of the current waveform after correction.

Conclusions:

The RL circuit initially showed a lagging power factor due to the inductive ballast. By adding a 3.5 μF capacitor in series, the reactive effects were partially balanced and the phase angle was reduced. As a result, the power factor improved from about 0.71–0.74 to approximately 0.8. This confirms that series capacitive compensation is an effective method for improving power factor in inductive loads.

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