Voltage and Current Source
Series Parallel Battery Cells
Mesh Network and Analysis
Kirchhoff Current and Voltage Law
Maximum Power Transfer Theorem
Star - Delta Transformation
RL Series Circuit
RL Parallel Circuit
Series RLC Circuit
Parallel RLC Circuit
Resonance in Series RLC Circuit
Consider a simple RL circuit in which resistor, R and Inductor, L are connected in series with a voltage supply of V volts. Let the current flowing in the circuit is I (amp) and current through resistor and inductor is IR and IL respectively. Since both resistance and inductor are connected in series so the current in both the elements and the circuit remains the same. i.e IR = IL = I. Let VR and Vl be the voltage drop across resistor and inductor. Apply Kirchhoff voltage law ( i.e sum of voltage drop must be equal to apply voltage) to this circuit we get
V2 = VR2 + VL2
Phasor diagram for RL circuit
Before drawing the phasor diagram of series RL circuit, one should know the relationship between voltage and current in case of resistor and inductor.
1. Resistor - In case of resistor the voltage and the current are in same phase or we can say that the phase angle difference between voltage and current is zero.
2. Inductor - In inductor the voltage and the current are not in phase . The voltage leads that of current by 90° or in other words voltage attains its maximum and zero value 90° before the current attains it.
3. RL circuit - For drawing the phasor diagram of series RL circuit follow the following steps:
Step- I. In case of series RL circuit resistor and inductor are connected in series so current flowing in both the elements are same i.e IR = IL = I. So take current phasor as reference and draw it on horizontal axis as shown in diagram.
Step- II. In case of resistor both voltage and current are in same phase. So draw the voltage phasor, VR along same axis or direction as that of current phasor. i.e VR is in phase with I
Step- III. We know that in inductor voltage leads current by 90° so draw VL (voltage drop across inductor) perpendicular to current phasor.
Step- IV Now we have two voltages VR and VL. Draw the resultant vector(VG) of these two voltages. Such that
VR2 + VL2 = VG2 and
From right angle triangle : phase angle θ = tan - 1(VL2/VR2)
CONCLUSION : In case of pure resistive circuit, the phase angle between voltage and current is zero and in case of pure inductive circuit phase angle is 90° but when we combine both Resistance and inductor, the phase angle of a series RL circuit is between 0º to 90º
Impedance of Series RL Circuit
The impedance of series RL circuit is defined as the total opposition to flow of AC current caused by resistance (R) and inductive reactance (XL) of the circuit . The impedance Z in ohms is given by
Z = (R2 + XL2)0.5 and from right angle triangle phase angle θ = tan - 1(XL/R)
Series RL Circuit Analysis
In series RL circuit the values of frequency f, voltage V, resistance R and Inductance L are known and there is no instrument for directly measuring the value of inductive reactance and impedance so, for complete analysis of series RL circuit follow these simple steps:
Step 1.Since the value of frequency and inductance are known so firstly calculate the value of inductive reactance XL: XL = 2πfL ohms
Step 2. From the value of XL and R, calculate the total impedance of the circuit which is given by
Z = (R2 + XL2)0.5 ohms
Step 3. Calculate the total phase angle for the circuit θ = tan - 1(XL/ R)
Step 4. Use Ohm's Law and find the value of the total current: I = V / Z amp
Step 5 . Calculate the voltages across resistor R and inductor L by using Ohm's Law. Since the resistor and the inductor are connected in series so current in them remains the same
VR = RI volts and VL = XLI volts
Power in RL Circuit
In series RL circuit some energy is dissipated by the resistor and some energy is alternately stored and returned by inductor
• The instantaneous power deliver by voltage source V is P = VI (watts)
• power dissipated by the resistor in the form of heat, P = I2R (watts)
• The rate at which energy is stored in inductor,
So total power in series RL circuit is given by adding the power dissipated by the resistor and the power absorbed by the inductor
Power triangle for series RL circuit is as shown below
The electrical power factor cosθ is defined as ratio of true power to apparent power.
Variation of Impedance and Phase Angle with Frequency
The above diagram shows the impedance triangle. The base of this impedance triangle represents resistance. The resistance is independent of frequency so if frequency either increases or decreases resistance remains constant. The formulae for inductive reactance is XL = 2πfL so if frequency increases inductive reactance XL also increases and if inductive reactance increases total impedance of circuit also increases and this leads to variation in Phase angle θ with frequency. So in series RL circuit if frequency increases,
• inductive reactance also increases as it is directly proportional to frequency
• total impedance Z increases
• phase angle θ increases
• resistance remains constant
Expression for current flowing in series RL circuit
Consider a circuit in which resistance is connected in series with inductance and voltage source of V volts is applied across it .Initially the switch is open . Let us say at time 't' we close the switch and the current 'i' starts flowing in the circuit but it does not attains its maximum value rapidly due to presence of inductor in the circuit as we know inductor has a property to oppose the change in the current flowing through it.
Apply Kirchhoff's voltage law in the above series RL circuit
Rearranging the above equation
Integrating both sides, we get
Now integrate right hand side by using substitution method
Substituting the values we get
We know that integration of
So we get
By applying limits we get
Taking antilog on both sides
We know that e ln x = x, so we get,
Moving the term containing 'i' on one side we get,
The term L/R in the equation is called the Time Constant, ( τ ) of the RL series circuit and it is defined as time taken by the current to reach its maximum steady state value and the term V/R represents the final steady state value of current in the circuit.