RL Circuit Transfer Function Time Constant RL Circuit as Filter

The resistor and inductor are the most fundamental linear (element having linear relationship between voltage and current) and passive (which consume energy) elements. When resistor and inductor are connected across voltage supply, the circuit so obtained is called RL circuit.

Types of RL Circuit

1. RL series circuit - When resistance and inductor are connected in series with voltage supply. The circuit is called series RL circuit.

2. RL parallel circuit - When resistance and inductor are connected in parallel with each other and is driven by voltage source , the circuit so obtained is called parallel RL circuit.

Transfer Function of Series RL Circuit

A Transfer function is used to analysis RL circuit. It is defined as the ratio of the output of a system to the input of a system, in the Laplace domain. Consider a RL circuit in which resistor and inductor are connected in series with each other. Let V_in be the input supply voltage, V_L is the voltage across inductor, L, V_R is the voltage across resistor, and I is the current flowing through the circuit.
Now for finding transfer function apply voltage or potential divider rule. The voltage divider rule is a simplest rule used for determine the output voltage across any element in circuit. It states that the voltage divided between the resistors is in direct proportion to their respective resistance. Using voltage divider rule, the voltage across inductor V_L is: The voltage across the resistor V_R is: The transfer function, H_L for the inductor is: Similarly, the transfer function, H_R for the resistor is,
Current Since the circuit is in series so the current in resistor and inductor are the same and is given by:

Time Constant in RL Circuit

The time constant of an RL circuit is defined as the time taken by the current to reach its maximum value that had maintained during its initial rate of rise. The time constant of a series RL circuit equal to the ratio of value of inductor to the value of resistance: Where, T = time constant in seconds, L = inductor in

Henry, R = resistance in ohms. In RL circuit due to presence of inductor the current in the circuit does not build up at a steady rate because inductor has a property to oppose the change in current flowing through it. So rate of increase in current is initially rapid but it slows down as it approaches its maximum value. During each time constant, the current build up 63.2 % of its remaining distance. As shown in graph it takes 5 times constant to build up a current in RL circuit.

RL Circuit as Filter

Low Pass RL Filter

Consider a RL circuit is supplying with a voltage source of varying frequency and the circuit output voltage is taken across resistor R₁. The resistor, R₁ is independent of frequency but the inductive reactance is directly proportional to frequency (as X_L = 2πfL). At low or zero (as in case of DC) frequency, the inductive reactance X_L is very small as compared to resistance because when frequency is low, inductive reactance is also low so, it act as a short circuit. As there is no voltage drop across inductor the output voltage is almost same as that of input voltage both in magnitude and the phase and it acts as low pass filter. Now when frequency is increases, inductive reactance, X_L also increases and this causes increase in magnitude of voltage drop across inductor and hence reduce the output voltage across resistor. This increase in inductive reactance creates a phase shift between input and output voltage.

High Pass RL Filter

Consider a RL circuit is supplying with a voltage source of varying frequency and the circuit output voltage is taken across inductor, L₁. At very low or zero frequency, inductive impedance is zero so, inductor acts as short circuit and the output voltage across it is zero. As the frequency increases, inductive reactance also increases causing more voltage to drop across it and it act as high pass filter.

RL Parallel Circuit

In RL parallel circuit resistor andinductor are connected in parallel with each other and this combination is supplied by a voltage source, V_in. The output voltage of circuit is V_out. Since the resistor and inductor are connected in parallel, the input voltage is equal to output voltage but the currents flowing in resistor and inductor are different. The parallel RL circuit is not used as filter for voltages because in this circuit, the output voltage is equal to input voltage and for this reason it is not commonly used as compared to series RL circuit.

Let us say: I_T = the total current flowing from voltage source in amperes. I_R = the current flowing in the resistor branch in amperes.

Three Phase Circuit | Star and Delta System

There are two types of system available in electric circuit, single phase and three phase system. In single phase circuit, there will be only one phase, i.e the current will flow through only one wire and there will be one return path called neutral line to complete the circuit. So in single phase minimum amount of power can be transported. Here the generating station and load station will also be single phase. This is an old system using from previous time. In 1882, new invention has been done on polyphase system, that more than one phase can be used for generating, transmitting and for load system. Three phase circuit is the polyphase system where three phases are send together from the generator to the load. Each phase are having a phase difference of 120°, i.e 120° angle electrically. So from the total of 360°, three phases are equally divided into 120° each. The power in three phase system is continuous as all the three phases are involved in generating the total power. The sinusoidal waves for 3 phase system is shown below-


The three phases can be used as single phase each. So if the load is single phase, then one phase can be taken from the three phase circuit and the neutral can be used as ground to complete the circuit.

Why Three Phase is preferred Over Single Phase?

There are various reasons for this question because there are numbers of advantages over single phase circuit. The three phase system can be used as three single phase line so it can act as three single phase system. The three phase generation and single phase generation is same in the generator except the arrangement of coil in the generator to get 120° phase difference. The conductor needed in three phase circuit is 75% that of conductor needed in single phase circuit. And also the instantaneous power in single phase system falls down to zero as in single phase we can see from the sinusoidal curve but in three phase system the net power from all the phases gives a continuous power to the load.Till now we can say that there are three voltage source connected together to form a three phase circuit. And actually it is inside the generator. The generator is having three voltage source s which are acting together in 120° phase difference. If we can arrange three single phase circuit with 120° phase difference, then it will become a three phase circuit. So 120° phase difference is must otherwise the circuit will not work, the three phase load will not be able to get active and it may also cause damage to the system. The size or metal quantity of three phase devices is not having much difference. Now if we consider the transformer, it will be almost same size for both single phase and three phase because transformer will make only the linkage of flux. So the three phase system will have higher efficiency compared to single phase because for the same or little difference in mass of transformer, three phase line will be out whereas in single phase it will be only one. And losses will be minimum in three phase circuit. So overall in conclusion the three phase system will have better and higher efficiency compared to the single phase system. In three phase circuit, connections can be given in two types:

1. Star connection
2. Delta connection

Star Connection

In star connection, there is four wire, three wires are phase wire and fourth is neutral which is taken from the star point. Star connection is preferred for long distance power transmission because it is having the neutral point. In this we need to come to the concept of balanced and unbalanced current in power system.When equal current will flow through all the three phases, then it is called as balanced current. And when the current will not be equal in any of the phase, then it is unbalanced current. In this case, during balanced condition there will be no current flowing through the neutral line and hence there is no use of the neutral terminal. But when there will be unbalanced current flowing in the three phase circuit, neutral is having a vital role. It will take the unbalanced current through to the ground and protect the transformer. Unbalanced current affects transformer and it may also cause damage to the transformer and for this star connection is preferred for long distance transmission. The star connection is shown below- In star connection, the line voltage is √3 times of phase voltage. Line voltage is the voltage between two phases in three phase circuit and phase voltage is the voltage between one phase to the neutral line. And the current is same for both line and phase. It is shown as expression below

Delta Connection

In delta connection, there is three wires alone and no neutral terminal is taken. Normally delta connection is preferred for short distance due to the problem of unbalanced current in the circuit. The figure is shown below for delta connection. In the load station, ground can be used as neutral path if required. In delta connection, the line voltage is same with that of phase voltaage. And the line current is √3 times of phase current. It is shown as expression below, In three phase circuit, star and delta connection can be arranged in four different ways-

1. Star-Star connection
2. Star-Delta connection
3. Delta-Star connection
4. Delta-Delta connection

But the power is independent of the circuit arrangement of the three phase system. The net power in the circuit will be same in both star and delta connection. The power in three phase circuit can be calculated from the equation below, Since, there is three phases, so the multiple of 3 is made in the normal power equation and the PF is power factor. Power factor is a very important factor in three phase system and some times due to certain error, it is corrected by using capacitors.

RLC 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, Let V_R be the voltage 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 in-phase 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 phase 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_L and capacitive reactance, X_C. If the inductive reactance is greater than the capacitive reactance i.e X_L > X_C, then the RLC circuit has lagging phase angle and if the capacitive reactance is greater than the inductive reactance i.e X_C > X_L then, the RLC circuit have leading phase angle and if both inductive and capacitive are same i.e X_L = X_C then circuit will behave as purely resistive circuit. We know that Where, Substituting the values

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_R is the current flowing in the resistor, R in amps. I_C is the current flowing in the capacitor, C in amps. I_L is the current flowing in the inductor, L in amps. I_s 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 I_R, I_C, I_L are drawn relative to this voltage vector. The current through each element can be found using Kirchhoff'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.

Resonance in RLC Circuit

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

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 oscillation s to die out over period of time and it is called as the damping effect of resistor.

Formula for Resonant Frequency

During resonance, at certain frequency called resonant frequency, f_r.
When resonance occurs, the inductive reactance of the circuit becomes equal to capacitive reactance, which causes the circuit impedance to be minimum in case of series RLC circuit; but when resistor, inductor and capacitor are connected in parallel, the circuit impedance becomes maximum, so the parallel RLC circuit is sometimes called as anti resonator.

Equation of RLC Circuit Consider a RLC circuit having resistor R, inductor L, and capacitor C connected in series and are driven by a voltage source V. Let Q be the charge on the capacitor and the current flowing in the circuit is I. Apply Kirchhoff's voltage law In this equation; resistance, inductance, capacitance and voltage are known quantities but current and charge are unknown quantities. We know that an current is a rate of electric charge flowing, so it is given by Differentiating again I'(t) = Q’’ (t) Differentiating the above equation with respect to ’t’ we get, Now at time t = 0 , V(0) = 0 and at time t = t , V(t) = E_osinωt Differentiating with respect to ’t’ we get V'(t) = ωE_ocosωt Substitute the value of V'(t) in above equation Let us say that the solution of this equation is I_P(t) = Asin(ωt - ǿ) and if I_P(t) is a solution of above equation then it must satisfy this equation, Now substitute the value of I_P(t) and differentiate it we get, Apply the formula of cos (A + B) and combine similar terms we get, Match the coefficient of sin(ωt - φ ) and cos(ωt - φ ) on both sides we get, Now we have two equations and two unknowns i.e φ and A, and by dividing the above two equations we get, Squaring and adding above equation, we get

Analysis of RLC Circuit Using Laplace Transformation

Step 1 : Draw a phasor diagram for given circuit. Step 2 : Use Kirchhoff's voltage law in RLC series circuit and Kirchhoff's current law in RLC parallel circuit to form differential equations in the time-domain. Step 3 : Use Laplace transformation to convert these differential equations from time-domain into the s-domain. Step 4 : For finding unknown variables, solve these equations. Step 5 : Apply inverse Laplace transformation to convert back equations from s-domain into time domain.

Applications of RLC Circuit

It is used as low-pass filter, high-pass filter, band-pass filter, band-stop filter, voltage multiplier and oscillator circuit . It is used for tuning radio or audio receiver.

Series Parallel Battery Cells

Battery Cells

Battery is an electrical element where electrical potential is produced due to chemical reaction. Every electrochemical reaction has its limit of producing electric potential difference between two electrodes. Battery cells are those where these electro-chemical reactions take place to produce the limited electric potential difference. For achieving desired electric potential difference across the battery terminals multiple numbers of cells are to be connected in series. Hence it can be concluded like that, a battery is a combination of several cells where a cell is a unit of a battery. For example, Nickel-cadmium battery cells normally develop about 1.2 V per cell while lead acid battery develop about 2 V per cell. So a 12 volt battery will have total 6 number of cells connected in series.

EMF of Battery

If anyone just measures the electric potential difference between two terminals of a battery when load is not connected with the battery , he or she will get the voltage developed in the battery when there is no current flowing through it. This voltage is generally referred as electromotive force or emf of battery . It is also referred as no-load voltage of battery.

Terminal Voltage of Battery

Terminal voltage of battery is the potential difference across its terminals when the current is being drawn from it. Actually when load is connected with the battery, there will be load current flowing through it. As a battery is an electrical equipment, it must have some electrical resistance inside it. Because of this internal resistance of battery , there will be some voltage drops across it. So, if any one measures the terminal voltage of the load i.e. terminal voltage of battery when load is connected, he or she will get the voltage which is less than emf of the battery by internal voltage drop of the battery. If E is the emf or no – load voltage of the battery and V is the terminal voltage of load voltage of the battery , then E – V = internal voltage drop of the battery.As per Ohm’s law , this internal voltage drop is nothing but the product of electrical resistance offered by the battery and the current flows through it.

Internal Resistance of Battery

The entire resistance encountered by a current as if it flows through a battery from the negative terminal to the positive terminal is known as internal resistance of battery .

Series Parallel Batteries

Battery cells can be connected in series, in parallel and as well as a mixture of both the series and parallel.

Series Batteries

When in a battery, positive terminal of one cell is connected with the negative terminal of succeeding cell, then the cells are said to be series connected or simply series battery . Here, overall emf of the battery is algebraic sum of all individual cells connected in series. But overall discharged current of the battery does not exceed the discharged current of individual cells.


If E is the overall emf of the battery combined by n number cells and E₁, E₂, E₃, …………… E_n are the emfs of individual cells. Similarly, if r₁, r₂, r₃, …………… r_n are the internal resistances of individual cells, then the internal resistance of the battery will be equal to the sum of the internal resistance of the individual cells i.e.

Parallel Batteries

When positive terminals of all cells are connected together and similarly negative terminals of these cells are connected together in a battery, then the cells are said to be connected in parallel. These combinations are also referred as parallel batteries . If emf of each cell is identical, then the emf of the battery combined by n numbers of cells connected in parallel, is equal to the emf of each cell. The resultant internal resistance of the combination is, The current delivered by the battery is sum of currents delivered by individual cells.

Mixed Grouping of Batteries or Series Parallel Batteries

As we said earlier, the cells in a battery can also be connected in mixture of both series and parallel. These combinations are some time referred as series parallel battery . A load can require both voltage and current more than that of an individual battery cell. For achieving the required load voltage, the desired numbers of battery cells can be combined in series and for achieving the required load current, desired numbers of these series combinations are connected in parallel. Let m, numbers of series, each containing n numbers of identical cells, are connected in parallel.




Again assume emf of each cell is E and internal resistance of each cell is r. As n numbers of cells are connected in each series, the emf of each series as well as the battery will be nE. The equivalent resistance of the series is nr. As, m number of series connected in parallel equivalent internal resistance of that series and parallel battery is nr / m.

Video Presentation about Series Parallel Batteries