LCR series circuit|Alternating Current
LCR series circuit
- Figure below shows a circuit containing a capacitor ,resistor and inductor connected in series through an alternating voltage source
- Same amount of current will flow in all the three circuit components and vector sum of potential drop across each component would be equal to the applied voltage
- If i be the amount of current in the circuit at any time and V_{L},V_{C} and V_{R} the potential drop across L,C and R respectively then
V_{R}=iR ⇒ Voltage is in phase with i
V_{L}=iωL ⇒ Voltage is leading i by 90^{0}
V_{C}=i/ωC ⇒ Voltage is lagging behind i by 90^{0}
- Since V_{L} is ahead of i by 90 and V_{C} is behind by 90 so that phase difference between V_{L} and V_{C} is 180 and they are in direct opposition to each other as shown in the figure 12(b)
- In figure 12(b) we have assumed that V_{L} is greater than V_{C} which makes i lags behind V.If V_{C} > V_{L} then i lead V
- In this phasors diagram OA represent V_{R},AD represent V_{C} and AC represent V_{L}.So in this case as we have assumed that V_{L} > V_{C} ,there resultant will be (V_{L} -V_{C}) represented by vector AD
- Vector OB represent resultant of vectors V_{R} and (V_{L} -V_{C}) and this vector OB is the resultant of all the three ,which is equal to applied voltage V,thus
is called impedance of the circuit
- From phasors diagram 12(b),current i lag behind resultant voltage V by an phase angle given by,
- From equation (20) three cases arises
(i) When ωL > 1/ωC then tanφ is positive i.e. φ is positive and voltage leads the current i
(ii) When ωL < 1/ωC,then tanφ is negative i.e. φ is negative and voltage lags behind the current i
(iii) When ωL = 1/ωC ,then tanφ is zero i.e. φ is zero and voltage and current are in phase
- Again considering case (iii) where ωL = 1/ωC,we have
which is the minimum value Z can have.
- This is the case where X_{L}=X_{C},the circuit is said to be in electric resonance where the impedance is purely resistive and minimum and currents has its maximum value
- Hence at resonance
ωL = 1/ωC
or ω=1/√LC ---(21)
But ω=2πf where f is the frequency of applied voltage .Therefore
f_{0}=1/2π√LC ---(22)
This frequency is called resonant frequency of the circuit and peak current in this case is
i_{0}=V_{0}/R
and reactance is zero
- We will now define resonance curves which shows the variation in circuit current (peak current i_{0}) with change in frequency of the applied voltage
- Figure below shows the shape of resonance curve for various values of resistance R
- for small value of R,the resonance is sharp which means that if applied frequency is lesser to resonant frequency f_{0},the current is high otherwise
- For large values of R,the curve is broad sided which means that those is limited change in current for resonance and non -resonance conditions
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