Series RLC Circuit and RLC Series Circuit Analysis

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s RLC Circuit and RLC Series Circuit Analysis

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RLC Cir cuit

Capacit or in Series

Series C ircuit

Series RLC Circuit Analysis

The Series RLC Circuit

Thus far we have seen that the three basic passive components, R , L and C have very different phase relationships to each

ther when connected to a sinusoidal AC supply. In a pure ohmic resistor the voltage waveforms are “ in -phase” with the

urrent. In a pure inductance the voltage waveform “ leads” the current by 90 , giving us the expression of ELI . In a pure

apacitance the voltage waveform “lags” the current by 90 , giving us the expression of ICE .

This Phase Difference, Φ depends upon the reactive value of the components being used and hopefully by now we know thateactance, ( X ) is zero if the circuit element is resistive, positive if the circuit element is inductive and negative if it is capacitive thus

iving their resulting impedances as:

Element Impedance

Circuit Element Resistance, (R) Reactance, (X) Impedance, (Z)

Resistor R 0

o

o

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Inductor 0 ω L

Capacitor 0

nstead of analysing each passive element separately, we can combine all three together into a series RLC circuit. The analysis of aeries RLC circuit is the same as that for the dual series R and R circuits we looked at previously, except this time we need toake into account the magnitudes of both X and X to find the overall circuit reactance. Series RLC circuits are classed as second-rder circuits because they contain two energy storage elements, an inductance L and a capacitance C. Consider the RLC circuitelow.

Series RLC Circu it

The series RLC circuit above has a single loop with the instantaneous current flowing through the loop being the same for eachircuit element. Since the inductive and capacitive reactance’s X and X are a function of the supply frequency, the sinusoidalesponse of a series RLC circuit will therefore vary with frequency, ƒ. Then the individual voltage drops across each circuit elementf R, L and C element will be “out-of-phase” with each other as defined by:

i = I sin( ω t)

The instantaneous voltage across a pure resistor, V is “in-phase” with the current.

The instantaneous voltage across a pure inductor, V “leads” the current by 90

The instantaneous voltage across a pure capacitor, V “lags” the current by 90

Therefore, V and V are 180 “out-of-phase” and in opposition to each other.

For the series RLC circuit above, this can be shown as:

L CL C

L C

(t) max

R

L o

C o

L C o


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The amplitude of the source voltage across all three components in a series RLC circuit is made up of the three individualomponent voltages, V , V and V with the current common to all three components. The vector diagrams will therefore have theurrent vector as their reference with the three voltage vectors being plotted with respect to this reference as shown below.

ndividual Voltage Vectors

This means then that we can not simply add together V , V and V to find the supply voltage, V across all three components asll three voltage vectors point in different directions with regards to the current vector. Therefore we will have to find the supply

voltage, V as the Phasor Sum of the three component voltages combined together vectorially.

Kirchoff’s voltage law ( KVL ) for both loop and nodal circuits states that around any closed loop the sum of voltage drops aroundhe loop equals the sum of the EMF’s. Then applying this law to the these three voltages will give us the amplitude of the source

voltage, V as.

nstantaneous Voltages for a Series RLC Circu it

R L C

R L C S

S

S


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The phasor diagram for a series RLC circuit is produced by combining together the three individual phasors above and adding thesevoltages vectorially. Since the current flowing through the circuit is common to all three circuit elements we can use this as theeference vector with the three voltage vectors drawn relative to this at their corresponding angles.

The resulting vector V is obtained by adding together two of the vectors, V and V and then adding this sum to the remainingvector V . The resulting angle obtained between V and i will be the circuits phase angle as shown below.

Phasor Diagram for a Series RLC Circu it

We can see from the phasor diagram on the right hand side above that the voltage vectors produce a rectangular triangle,omprising of hypotenuse V , horizontal axis V and vertical axis V – V Hopefully you will notice then, that this forms our oldavourite the Voltage Triangle and we can therefore use Pythagoras’s theorem on this voltage triangle to mathematically obtain the

value of V as shown.

Voltage Triangle for a Series RLC Circu it

S L CR S

S R L C

S


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Please note that when using the above equation, the final reactive voltage must always be positive in value, that is the smallestvoltage must always be taken away from the largest voltage we can not have a negative voltage added to V so it is correct to haveV – V or V – V . The smallest value from the largest otherwise the calculation of V will be incorrect.

We know from above that the current has the same amplitude and phase in all the components of a series RLC circuit. Then thevoltage across each component can also be described mathematically according to the current flowing through, and the voltagecross each element as.

By substituting these values into Pythagoras’s equation above for the voltage triangle will give us:

So we can see that the amplitude of the source voltage is proportional to the amplitude of the current flowing through the circuit.This proportionality constant is called the Impedance of the circuit which ultimately depends upon the resistance and the inductivend capacitive reactance’s.

Then in the series RLC circuit above, it can be seen that the opposition to current flow is made up of three components, X , X anR with the reactance, X of any series RLC circuit being defined as: X = X – X or X = X – X with the total impedance of theircuit being thought of as the voltage source required to drive a current through it.

The Impedance of a Series RLC Circuit

As the three vector voltages are out-of-phase with each other, X , X and R must also be “out-of-phase” with each other with theelationship between R, X and X being the vector sum of these three components thereby giving us the circuits overall impedance,

Z. These circuit impedance’s can be drawn and represented by an Impedance Triangle as shown below.

The Impedance Triangle for a Series RLC Circuit

RL C C L S

L CT T L C T C L

L CL C


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The impedance Z of a series RLC circuit depends upon the angular frequency, ω as do X and X If the capacitive reactance isreater than the inductive reactance, X > X then the overall circuit reactance is capacitive giving a leading phase angle. Likewise,f the inductive reactance is greater than the capacitive reactance, X > X then the overall circuit reactance is inductive giving theeries circuit a lagging phase angle. If the two reactance’s are the same and X = X then the angular frequency at which thisccurs is called the resonant frequency and produces the effect of resonance which we will look at in more detail in another tutorial.

Then the magnitude of the current depends upon the frequency applied to the series RLC circuit. When impedance, Z is at imaximum, the current is a minimum and likewise, when Z is at its minimum, the current is at maximum. So the above equation for mpedance can be re-written as:

The phase angle, θ between the source voltage, V and the current, i is the same as for the angle between Z and R in thmpedance triangle. This phase angle may be positive or negative in value depending on whether the source voltage leads or lags

he circuit current and can be calculated mathematically from the ohmic values of the impedance triangle as:

Series RLC Circuit Example No1

A series RLC circuit containing a resistance of 12 Ω, an inductance of 0.15H and a capacitor of 100uF are connected in series across100V, 50Hz supply. Calculate the total circuit impedance, the circuits current, power factor and draw the voltage phasor diagram.

L CC L

L CL C

S


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nductive Reactance, X .

Capacitive Reactance, X .

Circuit Impedance, Z.

Circuits Current, I.

Voltages across the Series RLC Circuit, V , V , V .

Circuits Power factor and Phase Angle, θ .

L

C

R L C


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Electrical Circuit Theory andTechnology

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Phasor Diagram.

Since the phase angle θ is calculated as a positive value of 51.8 the overall reactance of theircuit must be inductive. As we have taken the current vector as our reference vector in a series

RLC circuit, then the current “lags” the source voltage by 51.8 so we can say that the phase angles lagging as confirmed by our mnemonic expression “ELI”.

Series RLC Circui t Summary

n a series RLC circuit containing a resistor, an inductor and a capacitor the source voltage V ishe phasor sum made up of three components, V , V and V with the current common to allhree. Since the current is common to all three components it is used as the horizontal reference

when constructing a voltage triangle.

The impedance of the circuit is the total opposition to the flow of current. For a series RLC circuit,nd impedance triangle can be drawn by dividing each side of the voltage triangle by its current, I.

The voltage drop across the resistive element is equal to I x R, the voltage across the two reactivelements is I x X = I x X – I x X while the source voltage is equal to I x Z. The angle between V

nd I will be the phase angle, θ .

When working with a series RLC circuit containing multiple resistances, capacitance’s or inductance’s either pure or impure, they cane all added together to form a single component. For example all resistances are added together, R = ( R + R + R ) …etc or ahe inductance’s L = ( L + L + L )…etc this way a circuit containing many elements can be easily reduced to a singlempedance.

o

o

SR L C

L C S

T 1 2 3T 1 2 3

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n the next tutorial about Parallel RLC Circuits we will look at the voltage-current relationship of the three components connectedogether this time in a parallel circuit configuration when a steady state sinusoidal AC waveform is applied along with theorresponding phasor diagram representation. We will also introduce the concept of Admi tt ance for the first time.

« AC Capacitance and Capacitive Reactance | Parallel RLC Circuit Analysis »

Other Good Tutorials in this Category AC Capacitance and Capacitive Reactance

AC Inductance and Inductive Reactance

AC Resistance and Impedance

AC Waveform and AC Circuit Theory

Average Voltage Tutorial

Complex Numbers and Phasors

Harmonics

Parallel Resonance Circuit

Parallel RLC Circuit Analysis

Passive Components in AC Circuits

Phase Difference and Phase Shift

Phasor Diagrams and Phasor Algebra

Reactive Power

RMS Voltage Tutorial

Series Resonance Circuit

Series RLC Circuit Analysis

Sinusoidal Waveforms

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Dickson

Thanks for your clarity in RLC circilts analysis and indeed your example,it was of great help.

October 4th, 2015Reply

nada

thanks a ton! it really helped a lot..!!explained in brief,clearly and easy 2 understand..:)))

September 29th, 2015Reply

Viet

Thanks a lot

September 4th, 2015Reply

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Series RLC Circuit and RLC Series Circuit Analysis