Impedance Of A Series LCR Circuit

What Is Impedance?

Impedance is defined as a combination of resistance and reactance. It is basically an obstruction to the flow of electrons in an electric circuit. Z is the symbol used for describing the impedance. The unit of impedance is ohm.

The other way of defining impedance in terms of the phasor is that it is the ratio of the phasor voltage across the element to the phasor current through the element.

What is LCR Circuit?

The LCR circuit is defined as the circuit that consists of a resistor R, an inductor L, and a capacitor C. These three electrical components are either connected in series or in parallel. LCR circuit is also known as RLC circuit and this circuit gets its name from the symbols of the main electronic components that are used. The other names of the LCR circuit are resonant circuit and tuned circuit.

Similar read:

• Analysis of LCR circuit
• Difference between resistance and impedance

Derivation Of The Expression For The Impedance Of A Series LCR Circuit

In the below circuit diagram, let R, L and C be the resistance, inductance, and capacitance that is connected in series with an alternating current source.

The voltage applied across the LCR series circuit is given as:

v = v_o sinωt

Where,

• v is the instantaneous value
• v_o is the peak value
• ω = 2πf
• f is the frequency of alternating current

Let i be the instantaneous current at the time t such that the instantaneous voltage across R, L, and C are iR, iX_L, and iX_C, respectively.

The vector sum of the voltage amplitudes across R, L, C and the amplitude v_o of the applied voltage are equal.

The voltage amplitudes across R, L, and C are given as v_R, v_L, and v_C, respectively. The current amplitude is given as I_o.

Then, v_R = I_oR is in the phase with i_o.

v_L = i_oX_L = I_o(⍵L)

The above equation leads i_o by 90^o

v_C = i_oX_C = i_o(1/⍵C)

The above equation lags behind i_o by 90^o

We can notice that the current in a pure resistor is in phase with the voltage while the current in a pure inductor and the voltage lag by 90 degrees. The current in a pure capacitor and the voltage are led by 90 degrees.

When v_L > v_C, the phase angle (Φ) between the voltage and the current is positive.

OAP is the right-angled triangle from which we get,

OP² = OA² + AP² = OA² + OB² ( AP = OB)

v² = v²_R + (V_L – V_C)² = (iR)² + (iX_L – iX_C)²

= i²(R² + (X_L – X_C)²)

Therefore,

Here, Z is the impedance of the circuit

The phase angle between v and i is given as:

This gives us:

The below video is an explanation of alternating current and impedance. Please refer to the video for a better understanding of the concepts:

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