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 other when connected to a sinusoidal AC supply. In a pure ohmic resistor the voltage waveforms are “in-phase” with the current. In a pure inductance the voltage waveform “leads” the current by 90^o, giving us the expression of ELI. In a pure capacitance the voltage waveform “lags” the current by 90^o, 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 that reactance, ( X ) is zero if the element is resistive, positive if the element is inductive and negative if the element is capacitive giving the resulting impedance values as:

Element Impedance

Circuit Element	Resistance, (R)	Reactance, (X)	Impedance, (Z)
Resistor	R	0
Inductor	0	ωL
Capacitor	0

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

Series RLC Circuit

 

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

• i_(t) = I_max sin(ωt)
•       The instantaneous voltage across a pure resistor, V_R is “in-phase” with the current.
•       The instantaneous voltage across a pure inductor, V_L “leads” the current by 90^o
•       The instantaneous voltage across a pure capacitor, V_C “lags” the current by 90^o
•       Therefore, V_L and V_C are 180^o “out-of-phase” and in opposition to each other.

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

Individual Voltage Vectors

 

This means then that we can not simply add together V_R, V_L and V_C to find the supply voltage, V_S across all three components as all three voltage vectors point in different directions with regards to the current vector. Therefore we will have to find the supply voltage, V_S 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 around the 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_S as.

Instantaneous Voltages for a Series RLC Circuit

 

The phasor diagram for a series RLC circuit is produced by combining together the three individual phasors above and adding these voltages vectorially. Since the current flowing through the circuit is common to all three circuit elements we can use this as the reference vector with the three voltage vectors drawn relative to this at their corresponding angles.
The resulting vector V_S is obtained by adding together two of the vectors, V_L and V_C and then adding this sum to the remaining vector V_R. The resulting angle obtained between V_S and i will be the circuits phase angle as shown below.

Phasor Diagram for a Series RLC Circuit

 

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_S, horizontal axis V_R and vertical axis V_L – V_C  Hopefully you will notice then, that this forms our old favourite the Voltage Triangle and we can therefore use Pythagoras’s theorem on this voltage triangle to mathematically obtain the value of V_S 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_R so it is correct to have V_L – V_C or  V_C – V_L. The smallest value from the largest otherwise the calculation of V_S 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 the voltage across each component can also be described mathematically according to the current flowing through, and the voltage across 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 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_L, X_C and R with the reactance, X_T of any series RLC circuit being defined as: X_T = X_L – X_C or  X_T = X_C – X_L  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 with each other, X_L, X_C and R must also be “out-of-phase” with each other with the relationship between R, X_L and X_C 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

 

The impedance Z of a series RLC circuit depends upon the angular frequency, ω as do X_L and X_C  If the capacitive reactance is greater than the inductive reactance, X_C > X_L then the overall circuit reactance is capacitive giving a leading phase angle. Likewise, if the inductive reactance is greater than the capacitive reactance, X_L > X_C then the overall circuit reactance is inductive giving the series circuit a lagging phase angle. If the two reactance’s are the same and X_L = X_C then the angular frequency at which this occurs 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 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, θ between the source voltage, V_S and the current, i is the same as for the angle between Z and R in 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:

 

Series RLC Circuit Example No1

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

 

Inductive Reactance, X_L.

 

Capacitive Reactance, X_C.

 

Circuit Impedance, Z.

 

Circuits Current, I.

 

Voltages across the Series RLC Circuit, V_R, V_L, V_C.

 

Circuits Power factor and Phase Angle, θ.

 

Phasor Diagram.

Since the phase angle θ is calculated as a positive value of 51.8^o the overall reactance of the circuit 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^o so we can say that the phase angle is lagging as confirmed by our mnemonic expression “ELI”.

Series RLC Circuit Summary

In a series RLC circuit containing a resistor, an inductor and a capacitor the source voltage V_S is the phasor sum made up of three components, V_R, V_L and V_C with the current common to all three. 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, and 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 reactive elements is I x X = I x X_L – I x X_C while the source voltage is equal to I x Z. The angle between V_S and 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 can be all added together to form a single component. For example all resistances are added together, R_T = ( R₁ + R₂ + R₃ )…etc or all the inductance’s L_T = ( L₁ + L₂ + L₃ )…etc this way a circuit containing many elements can be easily reduced to a single impedance.

 

 

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