series rlc circuit analysis and rlc series circuits

8/10/2019 Series RLC Circuit Analysis and RLC Series Circuits

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We can see from the phasor diagram on the right hand side above that the voltage vectors produce a rectangular triangle,

comprising of hypotenuse V , horizontal axis V and vertical axis V - V Hopefully you will notice then, that this forms our

old favourite the Voltage Triangleand 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 Circuit

Please note that when using the above equation, the final reactive voltage must always be positive in value, that is the

smallest voltage must always be taken away from the largest voltage we can not have a negative voltage added to V so it is

correct to have V - V or V - V . The smallest value from the larges t otherwise the calculation of V will be incorrect.

We know from above that the current has the same amplitude and phas e in all the components of a series RLC circuit. Then

the voltage across each component can also be des cribed mathematically according to the current flowing through, and the

voltage across each elem ent as.

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

S R L C

S

R

L C C L S


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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 Impedanceof the circuit which ultimately depends upon the resistance and

the inductive and 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 and R with the reactance, X of any series RLC circuit being defined as:X = X - X or X = X - X with the total impedance of the circuit 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 wi th each other, X , X and Rm ust also be "out-of-phase" with each other with

the relationship between R, X and X being the vector sum of these three components thereby giving us the circuits overall

impedance, Z. These circuit impedances can be drawn and represented by an Impedance Triangleas s hown below.

The Impedance Triangle for a Series RLC Circuit

The impedance Zof a series RLC circuit depends upon the angular frequency, as do X and X If the capacitive reactanceis greater than the inductive reactance, X > X then the overall circuit reactance is capacitive giving a leading phase ang le.

Likewise, if the inductive reactance is greater than the capacitive reactance, X > X then the overall circuit reactance is

inductive giving the series circuit a lagging phas e angle. If the two reactance's are the sam e and X = X then the angular

frequency at which this occurs is called the resonant frequency and produces the effect of resonancewhich 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 im pedance, Zis at its

maximum, the current is a minimum and likewise, when Z is at its minimum, the current is at maximum. So the above

equation for impedance can be re-written as:

The phase angle, between the source voltage, V and the current, iis the same as for the angle between Zand Rin the

impedance triangle. This phase angle may be positive or negative in value depending on whether the source voltage leads

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

Example No1

A series RLC circuit containing a resistance of 12, an inductance of 0.15Hand a capacitor of 100uFare connected in

series across a 100V, 50Hz supply. Calculate the total circuit impedance, the circuits current, power factor and draw the

voltage phasor diagram.

L C TT L C T C L

L C

L C

L C

C L

L C

L C

S


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Inductive 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, .

Phasor Diagram.

L

C

R L C


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Resonance Resistor Electrical DC Voltage
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series rlc circuit analysis and rlc series circuits

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