Answer.1. I_t = I_R + Ij_C

Explanation:-

Phase Relationships of Currents and Voltages for the parallel RC circuit

Figure (a) shows all the currents in a basic parallel RL circuit. The total current I_t divides at the junction into the two branch currents, I_R and I_C. The applied voltage V_s appear across both the resistive and the inductive branches, so V_s, V_r, and V_C are all in phase and of the same magnitude.

The current through the resistor is in phase with the voltage. The current through the capacitor leads the voltage and the resistor current by 90°. By Kirchhoff’s current law, the total current is the phasor sum of the two branch currents, as shown by the phasor diagram in Figure (b). The total current is expressed as

I_t = I_R + Ij_C

This equation can be expressed in polar form as

${I_t} = \sqrt {I_R^2 + I_C^2} \angle {\tan ^{ – 1}}\left( {\frac{{{I_C}}}{{{I_R}}}} \right)$

Answer.3. 54.5 mA + j80 mA

Explanation:-

From the given figure the resistor current, the inductor current, and the total current are

I_R = V_s/R = 12∠0°/220∠0° = 54.5∠0° mA

I_L = V_s/X_C = 12∠0°/150∠90° = 80∠90° mA

I_t = I_R + Ij_C

I_t = 54.5 mA + j80 mA

Answer.4. Impedance, Phase angle

Explanation:-

For every parallel RC circuit, there is an equivalent series RC circuit for a given frequency. Two circuits are considered equivalent when they both present an equal impedance at their terminals; that is, the magnitude of impedance and the phase angle are identical.

To obtain the equivalent series circuit for a given parallel RC circuit, we need to find the impedance and phase angle of the parallel circuit. Then use the values of Z and θ to construct an impedance triangle, shown in Figure. The vertical and horizontal sides of the triangle represent the equivalent series resistance and capacitive reactance as indicated. These values can be found using the following trigonometric relationships:

R_eq = Zcosθ

X_Ceq = Zsinθ

Answer.2. 12.5 kΩ − j8.31 kΩ

Explanation:-

Resistance R = 330Ω

Conductance = 1/R = 1/18 = 55.6 µS.

Capacitive Suspectance

B_C = 1/X_C = 1/27 = 37.0 µS.

The total Admittance is

Y_t = G + JB_C

Y_t = 55.6 + j37.0

Converting to polar form

$\begin{array}{l} {Y_t} = \sqrt {{G^2} + {B^2}_C} \angle – {\tan ^{ – 1}}\left( {\frac{{{B_c}}}{G}} \right)\\ \\ = \sqrt {{{(55.6)}^2} + {{(37)}^2}} \angle – {\tan ^{ – 1}}\left( {\frac{{37}}{{55.6}}} \right) \end{array}$

Y_t = 66.8∠33.6° mS

Now total impedance

Z_t = 1/Y_t

Z_t = 1/(3.03 + j1.38)

Z_t = 15.0∠−33.6° kΩ

Converting to rectangular form yields

Z_tot = ZCosθ − jZsinθ = R_eq − jX_C(eq)

= 15.Cos(−33.6°) − j15.sin(−33.6°)

Z_tot = 12.5 kΩ − j8.31 kΩ

Answer.2. More energy is dissipated

Explanation:-

When the resistance in a series RC circuit is greater than the capacitive reactance, more of the total energy delivered by the source is converted to heat by the resistance than is stored by the capacitor. Likewise, when the reactance is greater than the resistance, more of the total energy is stored and returned than is converted to heat.

Answer.3. 19.2 V

Explanation:-

In series RC circuit the voltage is additive

$\begin{array}{*{20}{l}} {{E_{rms}} = \sqrt {{V^2}_R + {V^2}_C} }\\ {}\\ {{E_{rms}} = \sqrt {{{(12)}^2} + {{(15)}^2}} } \end{array}$

E_RMS = √369

E_RMS =19.2 V

Answer.1. R < XC

Explanation:-

The phase angle of an RC lead circuit is

φ = tan⁻¹(X_C/R)

To increase the phase angle we need to increase the value of capacitive reactance or decrease the value of resistance since capacitive reactance is directly proportional to the capacitive reactance.

X_C > R

Answer.3. 4.6 mA

Explanation:-

Given

Z = 4.33 kΩ

V = 20 V

The total current in the circuit I

I = V/Z = 20/1.33

I = 4.6 mA

Answer.1. Increase, Decrease

Explanation:-

The term Cosθ is called the power factor and is stated as

PF = Cosθ

As the phase angle between applied voltage and total current increases, the power factor decreases, indicating an increasingly reactive circuit. The smaller the power factor, the smaller the power dissipation.

Answer.2. Z = 47 Ω − j150 Ω

Explanation:-

Given

Resistance R = 47 Ω

Capacitive reactance X_C = 150 Ω

The total impedance in rectangular form for series RC circuit is

Z = R − jX_C

Z = 47Ω − j150 Ω