Series And Parallel Circuit Analysis With Resistive, Capacitive And Inductive Network

SERIES AND PARALLEL CIRCUIT ANALYSIS WITH RESISTIVE, CAPACITIVE AND INDUCTIVE NETWORK

Series DC Circuit

When all the resistive components ents of a DC circuit are connected end to end to form a single path for flowing current, then the circuit is referred as series DC circuit. The manner of connecting components end to end is known as series connection. Suppose we have n number of resistors R₁, R₂, R₃, ..., R_a and they are connected in end to end manner, means they are series connected. If this series combination is connected across a voltage source, end current starts flowing through that series combination is connected across a voltage source, the current starts flowing through that single path. As the resistors are connected in end to end manner, the current first enters in to R,, then this same current comes in R2, then R, and at last it reaches R,, from which the current enters into the negative terminals of the voltage source. In this way, the same current circulates through every resistor connected in series. Hence, it can be concluded that in a series DC circuit, the same current flows through all parts of the electrical circuit. Again according to Ohm's law, the voltage drop across a resistor is the product of its electrical resistance and the current flow through it. Here, current through every resistor is the same, hence the voltage drop across each resistor's proportional to its electrical resistance value. If the resistances of the resistors are not equal then the voltage dro across them would also not be equal. Thus, every resistor has its individual voltage rop in a series DC circuit.

An Example of Series in DC Circuit

Suppose three resistors R1, R, and R, are connected in series across a voltage source of V (quantified as volts) are shown in the Figure 1.11. Let current I (quantified as Ampere) flow through the series circuit. Now according to Ohm's law,

Voltage drop across resistor R₁, V₁ = IR₁

Voltage drop across resistor R₂, V₂ = IR₂

Voltage drop across resistor R₃, V₃ = IR₃

Voltage drop across whole series DC circuit,

Voltage drop across resistor R₁ + Voltage drop across resistor R₂ + Voltage drop across resistor R₃

According to Ohm's law, the electrical resistance of an electrical circuit is given by V/I and that is R.

Therefore, V/I =R=R₁ + R₂ + R₃

So, effective resistance of the series DC circuit is R=R₁ + R₂ + R₃. From the above expression it can be concluded, that when a number of resistors are connected in series, the equivalent resistance of the series combination is the arithmetic sum of their individual resistances.

From the above discussion, the following points come out:

1. When a number of electrical components are connected in series, the same current flows through all the components of the circuit.

2. The applied voltage across a series circuit is equal to the sum total of voltage drops across each components.

3. The voltage drop across individual components is directly proportional to it s resistance value.

Parallel DC Circuit

When two or more electrical components are connected in a way that one end of each component is connected to a common point and the other end is connected to another common point, then the electrical components are said to be connected in parallel, and such an electrical DC circuit is referred as a parallel DC circuit. In this circuit every component will have the same voltage drop across them, and it will be exactly equal to the voltage which occurs between the two drop across them, and it will be exactly equal to the voltage which occurs between the two common points where the components are connected. Also in a parallel DC circuit, the current has several parallel paths through these parallel connected components, so the circuit current will be divided into as many paths as the number of components. Here, in this electrical circuit, the voltage drop across each component is equal. Again as per Ohm's law, voltage drop across any resistive component is equal to the product of its electrical resistance an current through it. As the voltage drop across every component connected in parallel is the same, the current through them is inversely proportional to its resistance value.

An Example of Parallel DC Circuit

Suppose three resistors R., R., and R. are connected in parallel across R1, R2 and R, are connected in parallel across a voltage source of V (volt) as shown in the Figure 1.13. Let I (Ampere) be the total circuts current which is divided into current I, I, and I, flowing through R1, R2 and R, respectively. Now according to Ohm's law: 

Thus when a number of resistors are connected in parallel, the reciprocal of the equivalent resistance is given by the arithmetic sum of the reciprocals of their individual resistances. From the above discussion of the parallel DC circuit, we can come to the following conclusion:

1. Voltage drops are the same across all the components connected in parallel.

2. Current through individual components connected in parallel, is inversely proportional to their resistances.

3. Total circuit current is the arithmetic sum of the currents passing through individual components connected in parallel.

4. The reciprocal of equivalent resistance is equal to the sum of the reciprocalss the resistances of individuals components connected in parallel.

Series and Parallel Circuit

So far we have discussed series DC circuit and parallel DC circuit separately, but in practice, the electrical circuit is generally a combination of both series circuits and parallel circuits. Such combined series and parallel circuits can be solved by proper application of Ohm's law and the rules for series and parallel circuits to the various parts of the complex circuit.

In RLC circuit, the most fundamental elements like resistor, inductor and capacitor are connected across a voltage supply. All these elements are linear and passive in nature; i.e., they consume energy rather than producing it and these elements have a linear relationship between voltage and current. There are number of ways of connecting these elements across voltage supply, but the most common method is to connect these elements either in series or in parallel. The RLC circuit exhibits the property of resonance in same way as LC circuit exhibits, but in this circuit the oscillation dies out quickly as compared to LC circuit due to the presence of resistor in the circuit.

Series RLC Circuit

When a resistor, inductor and capacitor are connected in series with the voltage supply, the circuit so formed is called series RLC circuit. Since all these components are connected in series, the current in each element remains the same.

I_R = I_L = I_C = I(t) where I(t)= I_M sin ωt

Let V_R be the votlage across resistor, R.

V_L be the voltage across inductor, L.

V_C be the voltage across capacitor, C.

X_L be the inductive reactance.

X_C be the capacitive reactance.

The total voltage in RLC circuit is not equal to algebraic sum of voltages across the resistor, the inductor and the capacitor, but it is a vector sum because, in case of resistor the voltage is inphase with the current, for inductor the voltage leads the current by 90° and for capacitor, the voltage lags behind the current by 90°. So, voltages in each component are not in phae with each other; so they cannot be added arithmetically. The figure below shows the phasor diagram of series RLC circuit. For drawing the phasor diagram for RLC series circuit, the current is taken as reference because, in series circuit the current in each element remains the same and the corresponding voltage vectors for each component are drawn in reference to common current vector.

The Impedance for a Series RLC Circuit

The impedance Z of a series RLC circuit is defined as opposition to the flow of current due circuit resistance R, inductive reactance, X, and capacitive reactance, X. If the inductive reactance is greater than the capacitive reactance i.e., X1 > Xc, then the RLC circuit has lagging phase angle and if the capacitive reactance is greater than the inductive reactance i.e., XXL then, the RLC circuit have leading phase angle and if both inductive and capacitive are same. i.e., X = Xc then circuit will behave as purely resistive circuit.

Parallel RLC Circuit

In parallel RLC Circuit the resistor, inductor and capacitor are connected in parallel across a voltage supply. The parallel RLC circuit is exactly opposite to the series RLC circuit. The applied voltage remains the same across all components and the supply current gets divided. The total current drawn from the supply is not equal to mathematical sum of the current flowing in the individual component, but it is equal to its vector sum of all the currents, as the current flowing in resistor, inductor and capacitor are not in the same phase with each other; so they cannot be added arithmetically.

Phasor diagram of parallel RLC circuit, I, is the current flowing in the resistor, R in amps. I is the current flowing in the capacitor, C in amps. It is the current flowing in the inductor, L in amps. I, is the supply current in amps. In the parallel RLC circuit, all the components are connected in parallel; so the voltage across each element is same. Therefore, for drawing phasor diagram, take voltage as reference vector and all the other currents i.e., IR, IC. I are drawn relative to this voltage vector. The current through each element can be found using Kirchoff's Current Law, which states that the sum of currents entering a junction or node is equal to the sum of current leaving that node.

As shown above in the equation of impedance, Z of a parallel RLC circuit; each element has reciprocal of impedance (1/Z) i.e., admittance, Y. So in parallel RLC circuit, it is convenient to use admittance instead of impedance. mols ont le

Resonance in RLC Circuit

In a circuit containing inductor and capacitor, the energy is stored in two different

1. When a current flows in a inductor, energy is stored in magnetic field.

2. When a capacitor is charged, energy is stored in static electric field.

The magnetic field in the inductor is built by the current, which gets provided by the discharging capacitor. Similarly, the capacitor is charged by the current produced by collapsing magnetic field of inductor and this process continues on and on, causing electrical energy to oscillate between the magnetic field and the electric field. In some cases at certain frequency called resonant frequency, the inductive reactance of the circuit becomes equal to capacitive reactance which causes the electrical energy to oscillate between the electric field of the capacitor and magnetic field of the inductor. This forms a harmonic oscillator for current. In RLC circuit, the presence of resistor causes these oscillations to die out over period of time an it is called as the damping effect of resistor.

Formula for Resonant Frequency

During resonance, at certain frequency called resonant frequency, f.

When resonance occurs, the inductive reactance of the circuit become equal to capacitive reactance, which causes the circuit impedance to be minimum in case of series RLC circuit; but then resistor, inductor and capacitor are connected in parallel, the circuit impedance becomes maximum, so the parallel RLC circuit is sometimes called as anit resonator.