Planar and Non Planar Graphs of Circuit

Graph theory plays very crucial role in understanding of complicated electrical circuits. Now what that actually means is a circuit consisting of more than six loops are very complicated to handle manually with pen and paper. If we have computer with multisim then it becomes a easy to handle task but without computer task become too complicated so to simply analysis of such circuit we often prefer Graph Theory.

How are Graphs Formed ?

Each element present in circuit act as single branch of graph whereas Active sources like Voltage source and Current source are replace with their internal impedance (replace voltage source with short circuit whereas replacing current source with open circuit).


Oriented Graph is one which contains direction on each branch which are direction of flow of current in electric circuit to be analysed.

Planar Graph

Planar graph is graph which can be represented on plane without crossing any other branch. Below figure show an example of graph that is planar in nature since no branch cuts any other branch in graph. At first sight it looks as non planar graph since two resistor cross each other but it is planar graph which can be drawn as shown below. It is not always that graph which looks non-planar is always non-planar with some modification it can be made planar. See example below for explanation.

Non-Planar Graph

Non planar graph is one which cannot be represented on paper without crossing other branch.

Resonance in Series RLC Circuit

Consider a RLC circuit in which resistor, inductor and capacitor are connected in series across a voltage supply. This series RLC circuit has a distinguishing property of resonating at a specific frequency called resonant frequency. In this circuit containing inductor and capacitor, the energy is stored in two different ways.

1. When a current flows in an inductor, energy gets stored in magnetic field.
2. When a capacitor is charged, energy gets stored in static electric field.

The magnetic field in the inductor is built by the current, which is 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 to die out over period of time and is called damping effect of resistor.

Variation in Inductive Reactance and Capacitive Reactance with Frequency

Variation of Inductive Reactance Vs Frequency

We know that inductive reactance X_L = 2πfL means inductive reactance is directly proportional to frequency ( X_L and prop ƒ ). When the frequency is zero or in case of DC, inductive reactance is also zero, the circuit acts as a short circuit; but when frequency increases; inductive reactance also increases. At infinite frequency, inductive reactance becomes infinity and circuit behaves as open circuit. It means that, when frequency increases inductive reactance also increases and when frequency decreases, inductive reactance also decreases. So, if we plot a graph between inductive reactance and frequency, it is a straight line linear curve passing through origin as shown in the figure above.

Variation of Capacitive Reactance Vs Frequency

It is clear from the formula of capacitive reactance X_C = 1 / 2πfC that, frequency and capacitive reactance are inversely proportional to each other. In case of DC or when frequency is zero, capacitive reactance becomes infinity and circuit behaves as open circuit and when frequency increases and becomes infinite, capacitive reactance decreases and becomes zero at infinite frequency, at that point the circuit acts as short circuit, so the capacitive reactance increases with decease in frequency and if we plot a graph between capacitive reactance and frequency, it is an hyperbolic curve as shown in figure above.

Inductive Reactance and Capacitive Reactance Vs Frequency

From the above discussion, it can be concluded that the inductive reactance is directly proportional to frequency and capacitive reactance is inversely proportional to frequency, i.e at low frequency X_L is low and X_C is high but there must be a frequency, where the value of inductive reactance becomes equal to capacitive reactance. Now if we plot a single graph of inductive reactance vs frequency and capacitive reactance vs frequency, then there must occur a point where these two graphs cut each other. At that point of intersection, the inductive and capacitive reactance becomes equal and the frequency at which these two reactances become equal, is called resonant frequency, f_r. At resonant frequency, X_L = X_L At resonance f = f_r and on solving above equation we get,

Variation of Impedance Vs Frequency

At resonance in series RLC circuit, two reactances become equal and cancel each other. So in resonant series RLC circuit, the opposition to the flow of current is due to resistance only. At resonance, the total impedance of series RLC circuit is equal to resistance i.e Z = R, impedance has only real part but no imaginary part and this impedance at resonant frequency is called dynamic impedance and this dynamic impedance is always less than impedance of series RLC circuit. Before series resonance i.e before frequency, f_r capacitive reactance dominates and after resonance, inductive reactance dominates and at resonance the circuit acts purely as resistive circuit causing a large amount of current to circulate through the circuit.

Resonant Current

In series RLC circuit, the total voltage is the phasor sum of voltage across resistor, inductor and capacitor. At resonance in series RLC circuit, both inductive and capacitive reactance cancel each other and we know that in series circuit, the current flowing through all the elements is same, So the voltage across inductor and capacitor is equal in magnitude and opposite in direction and thereby they cancel each other. So, in a series resonant circuit, voltage across resistor is equal to supply voltage i.e V = V_r.In series RLC circuit current, I = V / Z but at resonance current I = V / R , therefore the current at resonant frequency is maximum as at resonance in impedance of circuit is resistance only and is minimum.
The above graph shows the plot between circuit current and frequency. At starting, when the frequency increases, the impedance Z_c decreases and hence the circuit current increases. After some time frequency becomes equal to resonant frequency, at that point inductive reactance becomes equal to capacitive reactance and the impedance of circuit reduces and is equal to circuit resistance only. So at this point, the circuit current becomes maximum I = V / R. Now when the frequency is further increased, Z_L increases and with increase in Z_L, the circuit current reduces and then the current drops finally to zero as frequency becomes infinite.

Power Factor at Resonance

At resonance, the inductive reactance is equal to capacitive reactance and hence the voltage across inductor and capacitor cancel each other. The total impedance of circuit is resistance only. So, the circuit behaves like a pure resistive circuit and we know that in pure resistive circuit, voltage and the circuit current are in same phase i.e V_r , V and I are in same phase direction. Therefore, the phase angle between voltage and current is zero and the power factor is unity.

Application of Series RLC Resonant Circuit

Since resonance in series RLC circuit occurs at particular frequency, so it is used for filtering and tuning purpose as it does not allow unwanted oscillations that would otherwise cause signal distortion, noise and damage to circuit to pass through it. Summary For a series RLC circuit at certain frequency called resonant frequency, the following points must be remembered. So at resonance:

1. Inductive reactance X_L is equal to capacitive reactance X_C.
2. Total impedance of circuit becomes minimum which is equal to R i.e Z = R.
3. Circuit current becomes maximum as impedance reduces, I = V / R.
4. Voltage across inductor and capacitor cancels each other, so voltage across resistor V_r = V, supply voltage.
5. Since net reactance is zero, circuit becomes purely resistive circuit and hence the voltage and the current are in same phase, so the phase angle between them is zero.
6. Power factor is unity.
7. Frequency at which resonance in series RLC circuit occurs is given by

Resistances in Series and Resistances in Parallel

More than one electrical resistance can be connected either in series or in parallel in addition to that, more than two resistances can also be connected in combination of series and parallel both. Here we will discuss mainly about series and parallel combination.

Resistances in Series

Suppose you have, three resistors, R₁, R₂ and R₃ and you connect them end to end as shown in the figure below, then it would be referred as resistances in series. In case of series connection, the equivalent resistance of the combination, is sum of these three electrical resistances. That means, resistance between point A and D in the figure below, is equal to the sum of three individual resistances. The current enters in to the point A of the combination, will also leave from point D as there is no other parallel path provided in the circuit.


Now say this current is I. So this current I will pass through the resistance R₁, R₂ and R₃. Applying Ohm’s law , it can be found that voltage drops across the resistances will be V₁ = IR₁, V₂ = IR₂ and V₃ = IR₃. Now, if total voltage applied across the combination of resistances in series, is V.
Then obviously Since, sum of voltage drops across the individual resistance is nothing but the equal to applied voltage across the combination. Now, if we consider the total combination of resistances as a single resistor of electric resistance value R, then according to Ohm’s law , V = IR ………….(2) Now, comparing equation (1) and (2), we get So, the above proof shows that equivalent resistance of a combination of resistances in series is equal to the sum of individual resistance. If there were n number of resistances instead of three resistances, the equivalent resistance will be

Resistances in Parallel

Let’s three resistors of resistance value R₁, R₂ and R₃ are connected in such a manner, that right side terminal of each resistor are connected together as shown in the figure below, and also left side terminal of each resistor are also connected together. This combination is called resistances in parallel. If electric potential difference is applied across this combination, then it will draw a current I (say).As this current will get three parallel paths through these three electrical resistances, the current will be divided into three parts. Say currents I₁, I₁ and I₁ pass through resistor R₁, R₂ and R₃ respectively. Where total source current Now, as from the figure it is clear that, each of the resistances in parallel, is connected across the same voltage source, the voltage drops across each resistor is same, and it is same as supply voltage V (say). Hence, according to Ohm’s law , Now, if we consider the equivalent resistance of the combination is R. Then, Now putting the values of I, I₁, I₂ and I₃ in equation (1) we get, The above expression represents equivalent resistance of resistor in parallel. If there were n number of resistances connected in parallel, instead of three resistances, the expression of equivalent resistance would be

Parallel RLC Circuit

Consider a RLC circuit in which resistor, inductor and capacitor are connected in parallel to each other. This parallel combination is supplied by voltage supply, V_S. This parallel RLC circuit is exactly opposite to series RLC circuit. In series RLC circuit, the current flowing through all the three components i.e the resistor, inductor and capacitor remains the same, but in parallel circuit, the voltage across each element remains the same and the current gets divided in each component depending upon the impedance of each component. That is why parallel RLC circuit is said to have dual relationship with series RLC circuit.

The total current, I_S drawn from the supply is equal to the vector sum of the resistive, inductive and capacitive current, not the mathematic sum of the three individual branch currents, as the current flowing in resistor, inductor and capacitor are not in same phase with each other; so they cannot be added arithmetically.


Apply 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 we get,

Phasor Diagram of Parallel RLC Circuit

Let V is the supply voltage. I_S is the total source current. I_R is the current flowing through the resistor. I_C is the current flowing through the capacitor. IL is the current flowing through the inductor. θ is the phase angle difference between supply voltage and current.
For drawing the phasor diagram of parallel RLC circuit, voltage is taken as reference because voltage across each element remains the same and all the other currents i.e I_R, I_C, I_L are drawn relative to this voltage vector. We know that in case of resistor, voltage and current are in same phase; so draw current vector I_R in same phase and direction to voltage. In case of capacitor, current leads the voltage by 90° so, draw I_C vector leading voltage vector, V by 90°. For inductor, current vector I_L lags voltage by 90° so draw I_L lagging voltage vector, V by 90°. Now draw the resultant of I_R, I_C, I_L i.e current I_S at a phase angle difference of θ with respect to voltage vector, V. Simplifying the phasor diagram, we get a simplified phasor diagram on right hand side. On this phasor diagram, we can easily apply Pythagoras's theorem and we get,

Impedance of Parallel RLC Circuit

From the phasor diagram of parallel RLC circuit we get, Substituting the value of I_R, I_C, I_L in above equation we get, On simplifying, 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. For solving parallel RLC circuit it is convenient if we find admittance of each branch and the total admittance of the circuit can be found by simply adding each branch's admittance.

Admittance Triangle of Parallel RLC Circuit

In series RLC circuit, impedance is considered, but as stated in introduction on parallel RLC circuit, it is exactly opposite to that of series RLC circuit; so in Parallel RLC circuit, we will consider admittance. The impedance Z has two components; resistance, R and reactance, X. Similarly, admittance also has two components such as conductance , G (reciprocal of resistance, R) and suspceptance, B (reciprocal of reactance, X). So admittance triangle of parallel RLC circuit is completely opposite to that of series impedance triangle.

Resonance in Parallel RLC Circuit

Like series RLC circuit, parallel RLC circuit also resonates at particular frequency called resonance frequency i.e. there occurs a frequency at which inductive reactance becomes equal to capacitive reactance but unlike series RLC circuit, in parallel RLC circuit the impedance becomes maximum and the circuit behaves like purely resistive circuit leading to unity electrical power factor of the circuit.

Series RLC Circuit

When resistor, inductor and capacitor are connected in series across a voltage supply, the circuit so obtained is called series RLC circuit.

Phasor Diagram of Series RLC Circuit

The phasor diagram of series RLC circuit is drawn by combining the phasor diagram of resistor, inductor and capacitor. Before doing so, one should understand the relationship between voltage and current in case of resistor, capacitor and inductor.

1. Resistor In case of resistor, the voltage and the current are in same phase or we can say that the phase angle difference between voltage and current is zero.
2. Inductor In inductor, the voltage and the current are not in phase. The voltage leads that of current by 90° or in the other words, voltage attains its maximum and zero value 90° before the current attains it.
3. Capacitor In case of capacitor, the current leads the voltage by 90° or in the other words, voltage attains its maximum and zero value 0° after the current attains it i.e the phasor diagram of capacitor is exactly opposite of inductor.

NOTE: For remembering the phase relationship between voltage and current, learn this simple word called 'CIVIL', i.e in capacitor current leads voltage and voltage leads current in inductor.RLC Circuit For drawing the phasor diagram of series RLC circuit, follow these steps:

Construction and Working of AC Circuits

Bridge circuit is nothing but the electrical circuit configuration which is used to measure unknown values of the resistance, impedance, induction, and capacitance. Many bridges like Wheatstone bridge, Maxwell Bridge, Kelvin Bridge, and many more are very useful to measure quantities with accuracy and working on the same principle. Here is a brief description of functioning of some of the bridges given below:

Wheatstone Bridge

A Wheatstone bridge is an electrical circuit developed by Charles Wheatstone, and it is used to determine the value of an unknown electrical resistance in the circuit. Wheatstone bridge is highly capable in calculating very low valued resistances which other instruments like multimeter does not calculate accurately. The Wheatstone bridge circuit is a diamond-shaped arrangement of four resistors. It has two parallel legs and each leg having two resistors in series.

A third leg connected between the two parallel legs at some point within the legs, as drawn in figure. Among the four resistors, one resistance value can be determined by balancing the two legs. Out of four resistors, the value of two resistors R₁ and R₃ are known, the value of R₂ is adjustable, and the value of R_x is to be calculated. Then this adjustment is connected to electric supply and a galvanometer between terminal D and terminal B. Now the value of an adjustable resistor is adjusted until the ratio of the two branches resistances become equal i.e. (R₁/ R₂) = (R₃/R_x), and galvanometer reads zero as current stop flowing through the circuit. Now the circuit is balanced and the value of the unknown resistor could be measured easily. The reading of the R₃ decides the

direction of the flow of current.

Maxwell’s Bridge

The working principle of the Maxwell’s inductance bridge is same as the Wheatstone bridge. Only little modifications have been done in Wheatstone bridge. In this bridge, the four branches consist of unknown inductance (L₁), a variable capacitor (C₄), four resistors and detector instead of galvanometer as shown in the figure. It is used to measure the value of inductance by comparing the unknown value with the standard variable capacitance.The basic principle of the bridge is to compensate the positive angle phase of the unknown impedance with the negative phase of a capacitance by putting it in opposite branch. By doing so, the potential difference across the detector will become zero and no current will flow through it. The capacitor C₄ and resistor R₄ are connected in parallel and the value of both are adjusted so that bridge get balanced.

Kelvin Bridge

Kelvin Bridge is another modification of the Wheatstone bridge which is used to measure low resistance in the range of 1mΩ to 1kΩ with great accuracy. For precise measurement of low resistance, high voltage supply and a sensitive galvanometer are required in Kelvin Bridge. While measuring low resistance, the resistance of connecting wires plays an important role. Wheatstone bridge is used which has two additional resistors as shown in the figure. The resistors R₁ and R₂ are connected to the second set of ratio-arm and constructed four terminal resistors. Here R is unknown and S is the standard resistor. A galvanometer is placed between c and d so that resistance of connecting wire r can be neglected and does not affect the measurement value. Under the balance condition, galvanometer shows zero and no current flows through the circuit. The equation at balance condition is:

Hay’s Bridge Circuit

Hay’s bridge is another variation of Maxwell’s bridge circuit. In Maxwell’s circuit resistance is kept parallel to the capacitor where as, in Hay’s circuit, the resistor is connected in series with the standard capacitor as shown in the figure. It is very useful if the phase angle of inductive impedance is very large, which could be overcome by taking a low resistance in series.

Anderson’s Bridge

Anderson Bridge is modified version of Maxwell’s inductor capacitance bridge. It is mainly used for measuring self-inductance in a coil by using standard capacitor and resistors. The main advantage of this bridge is that it does not require the frequent balancing of the bridge.To balance the bridge by steady current, variable resistance r is adjusted and AC source is replaced by battery and headphone by moving coil galvanometer. Once the bridge is balanced the potential at the terminal D is similar to the potential at E. the flow of current in respective branches are denoted by I₁, I₂,and I₃ as shown in the figure.

Diode Bridge Circuit

It is a bridge circuit having an arrangement of four diodes that gives the same output polarity for every input polarity. Diode bridge circuit which also called bridge rectifier is used where ever there is a need to change alternating current into direct current. It is also used to detect the amplitude of radio signals. When the positive terminal of the input is connected to the upper left and negative to the lower right, the current flows from upper supply terminal to the output flowed by red path and returns back to the lower supply terminal through the blue path as shown in the figure.