Derive impedance and power in an RC circuit with the help of a phasor diagram

In actual practice, AC circuits contain two or more than two components connected in series. In a series circuit, each component carries the same current. An AC series circuit may be classified as under:

• RL series circuit
• RC series circuit
• RLC series circuit

 
In an RL series circuit, a pure resistance (R) is connected in series with a coil having the pure inductance (L). To draw the phasor diagram of RL series circuit, the current I (RMS value) is taken as reference vector because it is common to both elements.  

Voltage drop VR is in phase with current vector, whereas, the voltage drop in inductive reactance VL leads the current vector by 90o since current lags behind the voltage by 90o in the purely inductive circuit. The vector sum of these two voltage drops is equal to the applied voltage V (RMS value).

The power waveform for RL series circuit is shown in the figure. In this figure, voltage wave is considered as a reference. The points for the power waveform are obtained from the product of the corresponding instantaneous values of voltage and current.  

It is clear from the power waveform that power is negative between 0 and φ and between 180o and (180o + φ). The power is positive during rest of the cycle.

Since the area under the positive loops is greater than that under the negative loops, the net power over a complete cycle is positive. Hence a definite quantity of power is consumed by the RL series circuit. But power is consumed in resistance only; inductance does not consume any power.

RC Series Circuit

 
In an RC series circuit, a pure resistance (R) is connected in series with a pure capacitor (C). To draw the phasor diagram of RC series circuit, the current I (RMS value) is taken as reference vector. Voltage drop VR is in phase with current vector, whereas, the voltage drop in capacitive reactance VC lags behind the current vector by 90o, since current leads the voltage by 90o in the pure capacitive circuit. The vector sum of these two voltage drops is equal to the applied voltage V (RMS value).

The power waveform for RC series circuit is shown in the figure. In this figure, voltage wave is considered as a reference. The points for the power waveform are obtained from the product of the corresponding instantaneous values of voltage and current. It is clear from the power waveform that power is negative between (180o – φ) and 180o and between (360o – φ) and 360o. The power is positive during rest of the cycle.  

Since the area under the positive loops is greater than that under the negative loops, the net power over a complete cycle is positive. Hence a definite quantity of power is consumed by the RC series circuit. But power is consumed in resistance only; capacitor does not consume any power.

RLC Series Circuit

In an RLC series circuit a pure resistance (R), pure inductance (L) and a pure capacitor (C) are connected in series. To draw the phasor diagram of RLC series circuit, the current I (RMS value) is taken as the reference vector. The voltages across three components are represented in the phasor diagram by three phasors VR, VL and VC respectively.  

The voltage drop VL is in phase opposition to VC. It shows that the circuit can either be effectively inductive or capacitive. In the figure, phasor diagram is drawn for the inductive circuit. There can be three cases of RLC series circuit.

• When XL > XC, the phase angle φ is positive. In this case, RLC series circuit behaves as an RL series circuit. The circuit current lags behind the applied voltage and power factor is lagging. In this case,   if the applied voltage is represented by the equation;

  v = Vm sin ωt

  then, the circuit current will be represented by the equation;

  i = Im sin (ωt – φ).

• When XL < XC, the phase angle φ is negative. In this case, the RLC series circuit behaves as an RC series circuit. The circuit current leads the applied voltage and power factor is leading. In this case, the circuit current will be represented by the equation:  

  i = Im sin (ωt + φ).

• When XL = XC, the phase angle φ is zero. In this case, the RLC series circuit behaves like a purely resistive circuit. The circuit current is in phase with the applied voltage and power factor is unity. In this case, the circuit current will be represented by the equation:  

  i = Im sin (ωt).

Thanks for reading about “rl series circuit”.

A circuit that contains pure resistance R ohms connected in series with a pure capacitor of capacitance C farads is known as RC Series Circuit. A sinusoidal voltage is applied and current I flows through the resistance (R) and the capacitance (C) of the circuit.

The RC Series circuit is shown in the figure below:

Where,

• VR – voltage across the resistance R
• VC – voltage across capacitor C
• V – total voltage across the RC Series circuit

Contents:

Phasor Diagram of RC Series Circuit

The phasor diagram of the RC series circuit is shown below:

Steps to draw a Phasor Diagram

The following steps are used to draw the phasor diagram of RC Series circuit

• Take the current I (r.m.s value) as a reference vector
• Voltage drop in resistance VR = IR is taken in phase with the current vector
• Voltage drop in capacitive reactance VC = IXC is drawn 90 degrees behind the current vector, as current leads voltage by 90 degrees (in the pure capacitive circuit)
• The vector sum of the two voltage drops is equal to the applied voltage V (r.m.s value).

Now,

VR = IR and VC = IXC

Where XC = I/2πfC

In right triangle OAB,

Where,

Z is the total opposition offered to the flow of alternating current by an RC series circuit and is called impedance of the circuit. It is measured in ohms (Ω).

Phase angle

From the phasor diagram shown above, it is clear that the current in the circuit leads the applied voltage by an angle ϕ and this angle is called the phase angle.

Power in RC Series Circuit

If the alternating voltage applied across the circuit is given by the equation

Then,

Therefore, the instantaneous power is given by p = vi

Putting the value of v and i from the equation (1) and (2) in p = vi

The average power consumed in the circuit over a complete cycle is given by:

Where cosϕ is called the power factor of the circuit.

Putting the value of V and cosϕ from the equation (3) the value of power will be

From the equation (4) it is clear that the power is actually consumed by the resistance only and the capacitor does not consume any power in the circuit.

Waveform and Power Curve of the RC Series Circuit

The waveform and power curve of the RC circuit is shown below:

The various points on the power curve are obtained from the product of the instantaneous value of voltage and current.

The power is negative between the angle (180° – ϕ) and 180° and between (360° -ϕ) and 360° and in the rest of the cycle, the power is positive. Since the area under the positive loops is greater than that under the negative loops, therefore the net power over a complete cycle is positive.