# AC Circuits RLC Circuits

## Purely Resistive Circuit

There is no capacitance as well as inductance available in a purely resistive circuit. If a current starts flowing through a purely resistive circuit there will be no back EMF generated in the circuit. Like DC, there will be only resistive drop occurs in the AC circuit. In a resistive circuit, the current is directly proportional to the voltage. That means when the voltage is zero, the current will also become zero. That means the voltage and current alternate simultaneously. At any instant, the power of the circuit is the simple multiplication of the value of voltage and current at that instant. When the voltage is negative the current is also negative, therefore the product of the voltage and current is positive. Even when the voltage is positive the current is also positive therefore the product of voltage and current is positive. That means the value of power in a purely resistive circuit varies from 0 to its positive value only. In other words, the power will not have any negative cycle.

## Purely Inductive Circuit

#### What is an Inductor?

When an alternating current flows through a coil there will be an alternatingly changing magnetic field induced surround the coil. The magnetic flux of the field opposes the supply voltage. This is because of Lenz’s law. This property is described by the inductance of the coil. The inductance is denoted by the capital letter L. Henry is the unit of inductance. Henry is denoted with capital H. The expression inductance can be written as

φ is the magnetic flux in Weber, N is the number of turns of the coil and ‘I’ is the current flowing through the coil. The opposition due to inductance is called inductive reactance. The inductive reactance is denoted with XL. The unit of inductive reactance is ohm just similar to the unit of resistance. The expression of an inductive reactance can be written as

There is no resistance and capacitance in a purely inductive circuit. When an alternating current flows through an inductor a self-induced EMF appears across the inductor. This self-induced EMF directly opposite the source voltage. Since there is no resistance in a purely inductive circuit or an ideal inductive circuit, no resistive voltage drop occurs in the circuit. Obviously the induced EMF across the inductor will be exactly equal and opposite of the supply voltage.

Here in the figure, the rate of change of current with respect to time is maximum at points 1, 3 and 5. Add that instances, the value of induced EMF is maximum across the inductor. On the other hand, the value of the current is maximum at points 2 and 4. But at those instances, the value of the induced EMF is zero. The current is increased from zero to its maximum positive value from points 1 to 2. As per Lenz’s law, the induced EMF will oppositely oppose the supply voltage. Therefore the induced EMF will rise from its negative maximum value to zero from points 1 to 2. Again from points 2 to 3 the current will again decrease from its positive maximum value to zero. So, the value of induced EMF will rise from zero to its positive maximum value. It is clear that when the current is crossing zero the induced EMF gets its maximum value. Oppositely, when the current gets its maximum value, the induced EMF becomes zero. This is because the induced EMF is directly proportional to the rate of change of current in an inductor. The current reaches its maximum rate of change when it crosses zero. Therefore the induced EMF across the inductor becomes zero at that instant. In other words, we can say, there will be a phase difference of 90 degrees between current and voltages in a purely inductive circuit.

## Purely Capacitive Circuit

When voltage is applied across a capacitor it starts charging. Therefore it takes its charging current from the source. The charging current establishes the charge across the capacitor. On the other hand, the charge accumulated in the capacitor causes the voltage developed across the capacitor. There should be a simple relation between the rate of accumulation of charge and the rising voltage of a capacitor. In a capacitor, these two parameters are linearly proportional and the constant of proportionality is taken as the capacity of the capacitor denoted by capital C.

The rate of charge accumulation in a capacitor is equal to its charging current. Therefore we can write.

Here from the above equation, we can see that the charging current is directly proportional to the rate of change of voltage across the capacitor. Therefore the charging current is maximum when the rate of change of voltage across the capacitor is maximum. Again we know that a sinusoidal voltage waveform gets its maximum rate of change when it crosses zero. So vacancy so we can say that when voltage appears across the capacitor is zero the current gets its maximum value. Besides it, the rate of voltage variation across the capacitor is zero when the voltage reaches its maximum value. Therefore in that situation, the current through the capacitor will also become zero.

From the above explanation, it can be concluded that when the current becomes zero the voltage gets its maximum value and when the voltage becomes zero the current gets its maximum value. Ultimately we can say the phase difference between current and voltage of a capacitor is 90°.

When the voltage starts rising from its zero value to positive maximum value, the current reaches its maximum positive value. Again when the voltage starts rising from its zero value to the negative maximum value the current gets its negative maximum value.

Therefore it is also to be concluded that the current in a purely capacitive circuit, leads the voltage by 90°.

### Series RL Circuit or Series Resistance Inductance Circuit

In a series RL circuit, resistances and inductances are connected in series. There may be more than one resistance as well as more than one inductance but all of them are connected in series. We know that there is no phase difference between voltage and current across a pure resistance. Also, we know the current through an inductance lags 90° from the voltage. Since the resistance and inductance are connected in series the same current will flow through both of them. Therefore we need to draw the vector diagram of a series RL circuit by taking the current as the reference axis.

Let us consider that V is the supply voltage, VR is the voltage drop across the resistance and VL is the voltage developed across the inductance. ‘I’ is the current.

The voltage drop across the resistance will be in phase with the current. The voltage across the inductance will be perpendicular to the current axis. This is because, the phase difference between the current through the inductor is exactly 90°. Therefore the relation between supply voltage, resistive voltage drop and inductive voltage is,

The term Z is known as the impedance of the RL series circuit. The ratio of resistance to the impedance of a circuit is known as the power factor of the circuit. And this can be written as,

Here, φ is the angle between current and voltage vectors of the circuit or in other words we can say this is the phase difference between voltage and current of the circuit.

### Series RC Circuit or Resistance Capacitance Series Circuit

In an RC series circuit, the resistance and capacitance are connected in series. There may be one or more capacitors as well as resistors connected. This is the reason the same current will flow through the capacitors as well as resistors. We shall consider all the capacitors together as a single capacitor and all the resistors together as a single resistor. The current will lead the voltage developed across the capacitor by 90°.

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