Single Phase Ac Circuits

SINGLE PHASE AC CIRCUITS

Introduction:

The resistance, inductance and capacitance are three basic elements of any electrical network. In order to analyze any electric circuit, it is necessary to understand the following three cases.

1) AC Through pure resistive circuit.

2) AC Through pure inductive circuit.

3) AC Through pure capacitive circuit.

 

AC through Pure Resistance

Consider a circuit consisting of a pure resistance 'R' ohms connected across a voltage v=V_m sin ot. The circuit is shown in Figure 1.29.

According to ohm's law the current equation is written as:

 

AC through Pure Inductance

Pure inductance has zero ohmic resistance. Its internal resistance is zero. The coil has pure inductance of L henries (H).

According to Faraday law of electromagnetic induction.

Thus induced emf will oppose the supply voltage according to Lenz's law

The above equations clearly shows that the current is purely sinusoidal and having phase angle of - π/2 radians. i.e., - 90°. That means that the current lags voltage applied by 90o.

The inductive reactance is defined as the opposite offered by the inductance of a circuit to the flow of an alternating sinusoidal current.

Power:

The power equation can be obtained by taking the product of instantaneous voltage and current.

The average power over a one complete cycle is zero.

 

AC through Pure Capacitor

Consider a simple circuit consisting of a pure capacitor of C-farads, connected across a voltage given by the equation:

v = V_m sin ωt

The circuit is shown in the Figure 1.33.

The current i charges the capacitor C. The instantaneous charge 'q' on the plates of the capacitor is given by:

q=CV

The current through the capacitor is equal to rate of change of charge. S

The current through the pure capacitor leads the voltage by the angle of π/2 radians.

The capacitive reactance is defined as the resistance offered by the capacitor.

The pure capacitor never consumes the power. The power factor for pure capacitive circuit is zero (cos 90° = 0).

 

AC through R.L Circuit

Consider a circuit with pure resistance and inductance connected to series as shown in Figure 1.28. The series combination is connected across a supply given by:

V = V_m sin ot

The circuit draws current with two voltage drops.

(a) Drop across pure resistance, V_R =I x R

(b) Drop across pure inductance, V_L =I x X_l

where X_L=2πf L.

By applying KVL in the circuit:

Lets draw the phasor diagram for the above case.

 

Steps to Draw the Phasor Diagram:

1) Take current as a reference phasor.

2) The case of resistance, voltage and current are in phase, so VR will be along current phasor.

3) In case of inductance, current lags voltage by 90°. But as current as reference, V_L must be shown leading with respect to current by 90°.

4) The supply voltage being vector sum of these two vectors V1 and VR by law of parallelogram.

Impedance is defined as the opposition of circuit to flow of alternating current. It is denoted by Z and its unit is ohms.

 

Power and Power Triangles

The expression for the current in the series R-L circuit is:

If we multiply the voltage equation by current I, we get the power equation:

Apparent Power (S):

S = VI (VA)

Real (or) Active Power (P):

P=VI co Φ (watts)

Reactive Power (Q):

Q = VI sin Φ (VAR)

Power Factor:

 

AC through Series R-C Circuit

The series connection of resistance and capacitance are connected with the source ac voltage as shown in the Figure 1.39.

A voltage V = Vm sin ωt is applied.

Impedance:

In R-C series circuit, current leads voltage by angle o or supply voltage V lags current by angle Φ as shown in Figure 1.41.

If all the sides of the voltage triangle are divided the current, we get a triangle called impedance triangle.

Power and Power Triangle:

 

AC through R-L-C Circuit

Consider a circuit consisting of resistance, inductance, capacitance connected in series with each other across ac supply as shown in figure 1.43.

1) Drop across resistance R is V_R = IR

2) Drop across inductance L is V_L = IX_L

3) Drop across capacitance C is V_C = IX_C

The characteristics of three drops are:

1) V_R is in phase with current I

2) V_L leads current I by 90°

3) V_C lags current I by 90o.

According to Kirchoff's Laws.

Lets draw the phasor diagram.

The phasor diagram depends on the conditions of the magnitudes of VL which ultimately depends on the values of X and XC. Let us consider different cases.