Resistance-Inductance (R-L) series circuit is the case most generally met with in practice; nearly all circuits contain both resistance and inductance.
Consider an ac circuit consisting of resistance of R ohms and inductance of L henrys connected in series, as shown in Fig. 4.5 (a).
Let the supply frequency be f and current flowing through the circuit be of I amperes (rms value).
Now voltage drop across resistance, VR = I R in phase with the current.
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Voltage drop across inductance, VL = I XL = I ω L leading I by π/2 radians, as shown in Fig 4.5 (c).
The applied voltage, being equal to the phasor sum of VR and VL, will be given by the diagonal of the parallelogram.
Quantity √R2 + XL2 is known as impedance, denoted by Z and is expressed in ohms.
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From phasor diagram it is also evident that the current lags behind the applied voltage V by angle which is given by:
Since XL and R are known, the value of phase angle ɸ can be computed.
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If the applied voltage v = Vmax sin ω t, then expression for the circuit current will be:
Impedance Triangle:
If a triangle ABC is drawn so that AB = VR/I = R, BC = VL/I = XL and AC = V/I = Z, it is a triangle similar to that produced by the voltage triangle. Such a triangle is called an impedance triangle, which is most useful in letting one see at a glance how R, X, and Z are related to each other. The angle between Z and R sides of the impedance triangle is known as phase angle of the circuit and cos of this angle is known as power factor of the circuit.
Power factor = Cos ɸ = R/Z
Power in Resistance—Inductance (R-L) Circuit:
Where V and I are the rms values of voltage and current and ɸ is the phase angle between applied voltage V and circuit current I.
Alternatively power consumed by the circuit:
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p = Power consumed by resistance
... Power consumed by inductance is zero
= I2 R = I (I R) = V/Z. I R = V I R/Z = V I cos ɸ
Since from impedance triangle cos ɸ = R/Z
So the power in an ac circuit is given by the product of rms values of current and voltage and cosine of the phase angle between voltage and current. Cosine of the phase angle between the voltage and current, cos ɸ is known as the power factor of the circuit, and is equal to R/Z which is obvious from impedance triangle [Fig. 4.6 (b)]
