FIGURE 1620 in Software

Generator PDF-417 2d barcode in Software FIGURE 1620

2 L

(1680)

The vector drop across an inductance may therefore be de ned, in magnitude and phase position, as follows:

(1681a) (1681b)

BL is called the inductance susceptance of the circuit and is de ned as follows: BL 1 XL 1 2 L (1682)

C of Fig 1621 be e Em sin ( t ) (1683)

The charge accumulating on the capacitor (condenser) C at any instant is, by Eq (1617a), q Ce CEm sin ( t ) (1684)

Hence the current owing through the circuit is i dq dt Im sin where Im C Em cos ( t t BcEm ) BcEm sin ) t 2 (1685a) (1685b) (1685c)

Observe that in this case the current i leads the voltage by / 2 radians or is in leading quadrature with the voltage The vectors representing the current and voltage, respectively, are given in Fig 1621 and may be expressed as follows:

/ 2)

j( / 2)

(1686a) (1686b)

) / 2, between the voltage and current waves, as well as The angle ( between the voltage and current vectors (see Fig 1621), is the power-factor angle C 2 C is called the capacitive susceptance of the circuit, of the circuit Bc and jbc j C is called the complex capacitive susceptance of the circuit The potential drop across a capacitance may be de ned as follows:

(1687)

where Xc and Bc are known, respectively, as the capacitive reactance and capacitive susceptance of the circuit and are de ned as Xc Bc 1 2 C 2 C (1688a) (1688b)

(Fig 1622) Let e Em sin ( t ) (1689)

(1690)

It is required to ascertain the relation between the effective values E and I and to determine the phase angle between the voltage and current waves or vectors

(1687),

(1691a) (1691b) (1691c)

By Kirchhoff s emf law,

j( )

(1692b) (1693a) (1693b) (1693c)

Z is known as the impedance of the series circuit (Fig 1622) Equation (1692b) states that the effective value of the current and its phase angle in the series circuit in Fig 1622 are, respectively, I E Z ( The power-factor angle is, by Eq (1692b), (1695)

(1694a) ) (1694b)

circuit (Fig 1623) Let e Em sin ( t ) (1696)

Since all the characteristics are across the same emf, from Eqs (1676), (1681), and (1687),

(1697a) (1697b) (1697c)

Applying Kirchhoff s current law to the junction a (Fig 1623),

BL)]

(1698) (1699a) (1699b) (1699c)

Y is called the admittance of the parallel circuit in Fig 1623 Equation (1698) states that the effective value of the current and its phase angle in the parallel circuit (Fig 1623) are, respectively, I EY (16100a) (16100b) The power-factor angle is, by Eq (1699c) and by Fig 1623, (16101)

(16102)

When the sum of the voltage drops across the inductance and capacitance is nil, the series circuit in Fig 1622 is said to be in resonance Obviously, this occurs when XL Xc 0 or XL Xc (16103)

In order to determine the resonant frequency r , substitute in Eq (16103) for XL and Xc their values as given by Eqs (1680) and (1688), and obtain

(16104a) (16104b)

(16105)

When the sum of the Currents in the inductance and capacitance of a parallel circuit is nil, that circuit is said to be in resonance This occurs when Bc BL 0 or BL Bc (16106)

In order to determine the resonant frequency r , substitute in Eq (16106), for BL and Bc , their values as given in Eqs (1682) and (1688), and obtain 2 rC or r 1 2 rL 1 2 LC (16107a) (16107b)

Comparing Eqs (16104b) and (16107b), it may be concluded that the frequency at which resonance occurs in a series or parallel circuit is the same