Series LCR circuit: impedance, resonance, power and power factor

Derivation with the phasor diagram

R, L and C are joined in series to an AC source. The **same current** flows through all three, so we draw the current phasor first.

1. 1
   Draw the peak current I₀ as a horizontal phasor.
2. 2
   V_R = I₀R is in phase with the current: draw it along I₀.
3. 3
   V_L = I₀X_L leads the current by 90°: draw it straight up.
4. 4
   V_C = I₀X_C lags the current by 90°: draw it straight down. V_L and V_C are opposite, so their resultant is (V_L − V_C).
5. 5
   The source voltage is the phasor sum: E₀² = V_R² + (V_L − V_C)² = I₀²[R² + (X_L − X_C)²].
6. 6★
   So, I₀ = E₀/Z, with impedance Z = √(R² + (X_L − X_C)²), and tan φ = (X_L − X_C)/R.
7. 7
   Impedance is the total opposition of the circuit to AC. It is measured in ohm.

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Resonance and tuning

X_L = 2πνL rises with frequency, while X_C = 1/2πνC falls. At one frequency they become equal. This is resonance. Then V_L and V_C cancel, the impedance is least, Z = R, and the current is largest, I₀ = E₀/R. The current is in phase with the voltage (φ = 0). From 2πν_rL = 1/2πν_rC we get ν_r = 1/(2π√(LC)). The resonant frequency does not depend on R. But a smaller R gives a taller, sharper peak of current. Radio and TV sets use this to tune. Turning the knob changes a capacitor, and so changes ν_r. When ν_r matches a station's frequency, that station's signal produces the largest current and is received.

Power in an AC circuit, and the power factor

1. 1
   The instantaneous power is P = VI. When V and I differ in phase by φ, P changes all through the cycle.
2. 2★
   Averaged over a cycle, P = V_rms I_rms cos φ. The factor cos φ is the power factor.
3. 3
   For a series LCR circuit, cos φ = R/Z. So P = I_rms²R: only the resistor uses energy.
4. 4
   Pure resistor, or LCR at resonance: φ = 0, cos φ = 1, and the power is largest, V_rms I_rms.
5. 5
   Pure inductor or pure capacitor: φ = 90°, cos φ = 0, and the average power is zero. Such a current is called wattless current.
6. 6
   So, a low power factor needs more current for the same power, and that wastes energy in the supply lines.

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Activity: inductance and resistance of a coil, using a series resistance and an AC voltmeter

You need a coil (for example the secondary of a step-down transformer), a resistance box, a 12 V, 50 Hz AC supply from a step-down transformer, an AC voltmeter (0 to 15 V), a key and wires.

1. 1
   Join the coil (inductance L, resistance r) in series with the resistance box R, the key and the 12 V AC supply.
2. 2
   Take out about 200 Ω from the box and close the key. Measure V_R across the box, V_L across the coil and V across both.
3. 3
   Notice that V is less than V_R + V_L. The two voltages are out of phase, so they add as phasors.
4. 4
   Draw a line AB whose length stands for V_R (say 1 cm per volt). From B, draw an arc of radius V_L; from A, an arc of radius V. They meet at C.
5. 5
   Extend AB and drop a perpendicular CD onto it. Then BD stands for Ir and CD stands for IX_L.
6. 6★
   Since AB stands for IR: r = R × BD/AB and X_L = R × CD/AB. Then L = X_L/2πν, with ν = 50 Hz.
7. 7
   Repeat for 5 values of R, and take the mean of r and of L.
8. 8
   Result: write the mean inductance L (in henry) and the mean resistance r (in ohm) of the coil.

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Worked example: an LCR circuit, and the coil activity

How to draw the phasor diagram and the resonance curve

• Draw the current phasor I₀ as a horizontal arrow from a point O, and draw V_R along it.
• Draw V_L straight up and V_C straight down from O, with V_L longer when X_L > X_C.
• Draw (V_L − V_C) upwards. Complete the rectangle with V_R to get the source voltage E₀, and mark the angle φ between E₀ and I₀.
• Resonance curve: draw the horizontal axis for frequency ν and the vertical axis for peak current I₀.
• Mark ν_r on the horizontal axis. Draw a curve that rises from near zero, reaches a peak at ν_r, and falls again on the other side.
• Draw a second curve, for a larger resistance, with its peak at the same ν_r but lower. Label the taller curve 'small R' and the lower curve 'large R'.