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Chapter: Electrical and electronics - Circuit Theory - Transient Response For DC Circuits

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Phasor diagram for a series RLC circuit

Drawing of the phasor diagram for a series RLC circuit energized by a sinusoidal voltage showing the relative position of current, component voltage and applied voltage for the following case a) When XL > Xc b) When XL < Xc c) When XL = Xc.


Drawing of the phasor diagram for a series RLC circuit energized by a sinusoidal voltage showing the relative position of current, component voltage and applied voltage for the following case

 

a)     When XL > Xc

 b)    When XL < Xc

c)      When XL = Xc.

 

RLC Circuit:





Consider a circuit in which R, L, and C are connected in series with each other across ac supply as shown in fig.

 

The ac supply is given by,

V = Vm sin wt

The circuit draws a current I. Due to that different voltage drops are,

1)     Voltage drop across Resistance R is VR = IR

2)     Voltage drop across Inductance L is VL = IXL

3) Voltage drop across Capacitance C is Vc = IXc The characteristics of three drops are,

1.     VR is in phase with current I

2.     VL leads I by 900

3.     Vc lags I by 900

According to krichoff’s laws

 

Steps to draw phasor diagram:

1.     Take current I as reference

2.     VR is in phase with current I

 

3.     VL leads current by 900

4.     Vc lags current by 900

5.     obtain resultant of VL and Vc. Both VL and Vc are in phase opposition (1800 out of phase)

 

6.     Add that with VRby law of parallelogram to getsupply voltage.

 

The phasor diagram depends on the condition of magnitude of VL and Vc which ultimately depends on values of XL and Xc.

Let us consider different cases:

Case(i): XL > Xc

When X L > Xc

Also VL > Vc (or) IXL > IXc

So, resultant of VL and Vc will directed towards VL i.e. leading current I. Hence I lags V i.e. current I will lags the resultant of VL and Vc i.e. (VL - Vc). The circuit is said to be inductive in nature.

From voltage triangle,

V = √ (VR2 + (VL – Vc) 2) = √ ((IR) 2 + (IXL – IXc) 2)

V = I √ (R2 + (XL – Xc) 2)

V = IZ

Z = √ (R2 + (XL - Xc) 2 )

If , V = Vm Sin wt    ;  i = Im Sin (wt - ф)

 

i.e I lags V by angle ф

Case(ii): XL < Xc

When XL < Xc

Also VL < Vc (or) IXL < IXc

 

Hence the resultant of VL and Vc will directed towards Vc i.e current is said to be capacitive in nature Form voltage triangle




i.e I lags V by angle ф

Case(iii): XL = Xc

When XL = Xc

Also VL = Vc (or) IXL = IXc

 

So VL and Vc cancel each other and the resultant is zero. So V = VR in such a case, the circuit is purely resistive in nature.

 

Impedance:

In general for RLC series circuit impedance is given by,

Z = R + j X

X = XL – Xc = Total reactance of the circuit

 

If   XL > Xc ;         X is positive & circuit is Inductive

If   XL < Xc ;         X is negative & circuit is Capacitive

If  XL = Xc ; X =0 & circuit is purely Resistive

Tan ф = [(XL - Xc )∕R]

Cos ф = [R∕Z]

Z = √ (R2 + (XL - Xc ) 2)

Impedance triangle:

In both cases        R = Z Cos ф

 

X = Z Sin ф

 

Power and power triangle:

The average power consumed by circuit is,

 

Pavg = (Average power consumed by R) + (Average power consumed by L) + (Average power consumed by C)

 

Pavg = Power taken by R = I2R = I(IR) = VI

 

V = V Cos ф

P = VI Cos ф

Thus, for any condition, XL > Xc or XL < Xc General power can be expressed as

 

P = Voltage x Component in phase with voltage

Power triangle:

S = Apparent power = I2Z = VI

P = Real or True power = VI Cos ф = Active power

 

Q = Reactive power = VI Sin ф

 

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