27. AC circuits (power, resonance, transformers and filters)

AC source ⟺ 휋 휔퐿퐼 sin 휔푡 + The RLC Series Circuit; max 2 ∆푣퐿 1 휋 퐼 sin 휔푡 − 푅Imax sin 휔푡 휔퐶 max 2 ∆푣 휑 ∆푣푅

∆푣퐶 phasor diagram

2 1 ∆푣 = ΔVmax sin 휔푡 + 휑 Δ푉max = Imax 푅2 + 휔퐿 − 휔퐶 (∆푣 = ∆푣푅+ ∆푣퐿+ ∆푣퐶) 1 휔퐿 − 휑 = tan−1 휔퐶 푅 푖 = Imax sin 휔푡

Section 33.5 What happens if you interchange R and L (LRC)?

A.We get a different expression for the impedance B.We get the same expression for the impedance 휋 휔퐿퐼 sin 휔푡 + The RLC Series Circuit; max 2 ∆푣퐿 1 휋 퐼 sin 휔푡 − 푅Imax sin 휔푡 휔퐶 max 2 푣 휑 ∆푣푅

∆푣퐶 phasor diagram

2 1 푣 = ΔVmax sin 휔푡 + 휑 Δ푉max = Imax 푅2 + 휔퐿 − 휔퐶 (푣 = ∆푣푅+ ∆푣퐿+ ∆푣퐶) 1 휔퐿 − 휑 = tan−1 휔퐶 푅 푖 = Imax sin 휔푡

Section 33.5 6. Power in an AC circuit: Instant power: 푝 = 푖 ∙ 푣 Instant power:

푝 = 푖 ∙ 푣 = 퐼푚푎푥 sin 휔푡 ∙ 푉푚푎푥 sin 휔푡 + 휑 Instant power:

푝 = 푖 ∙ 푣 = 퐼푚푎푥 ∙ 푉푚푎푥 sin 휔푡 sin 휔푡 + 휑 Instant power: 1 푝 = 푖 ∙ 푣 = 퐼 ∙ 푉 sin 휔푡 sin 휔푡 + 휑 = 퐼 ∙ 푉 [cos 휑 −co푠 2휔푡 + 휑 ] 푚푎푥 푚푎푥 푚푎푥 푚푎푥 2 Instant power: 1 푝 = 푖 ∙ 푣 = 퐼 ∙ 푉 sin 휔푡 sin 휔푡 + 휑 = 퐼 ∙ 푉 [cos 휑 −co푠 2휔푡 + 휑 ] 푚푎푥 푚푎푥 푚푎푥 푚푎푥 2 average power: 1 푇 P = න 푖 ∙ 푣 푑푡 푇 0 Instant power: 1 푝 = 푖 ∙ 푣 = 퐼 ∙ 푉 sin 휔푡 sin 휔푡 + 휑 = 퐼 ∙ 푉 [cos 휑 −co푠 2휔푡 + 휑 ] 푚푎푥 푚푎푥 푚푎푥 푚푎푥 2 average power: 1 푇 1 푇 1 P = න 푖 ∙ 푣 푑푡 = 퐼푚푎푥 ∙ 푉푚푎푥 න [cos 휑 −co푠 2휔푡 + 휑 ] 푇 0 푇 0 2 Instant power: 1 푝 = 푖 ∙ 푣 = 퐼 ∙ 푉 sin 휔푡 sin 휔푡 + 휑 = 퐼 ∙ 푉 [cos 휑 −co푠 2휔푡 + 휑 ] 푚푎푥 푚푎푥 푚푎푥 푚푎푥 2 average power: 0 1 푇 1 푇 1 1 P = න 푖 ∙ 푣 푑푡 = 퐼푚푎푥 ∙ 푉푚푎푥 න [cos 휑 −co푠 2휔푡 + 휑 ] = cos 휑 퐼푚푎푥 ∙ 푉푚푎푥 푇 0 푇 0 2 2 Instant power: 1 푝 = 푖 ∙ 푣 = 퐼 ∙ 푉 sin 휔푡 sin 휔푡 + 휑 = 퐼 ∙ 푉 [cos 휑 −co푠 2휔푡 + 휑 ] 푚푎푥 푚푎푥 푚푎푥 푚푎푥 2 average power: 1 푇 1 푇 1 1 P = න 푖 ∙ 푣 푑푡 = 퐼푚푎푥 ∙ 푉푚푎푥 න [cos 휑 −co푠 2휔푡 + 휑 ] = cos 휑 퐼푚푎푥 ∙ 푉푚푎푥 푇 0 푇 0 2 2 1 = cos 휑 퐼푟푚푠 ∙ 푉푟푚푠 푉푟푚푠 ≡ 푉푚푎푥 2 (rms=root mean square) 1 퐼푟푚푠 ≡ 퐼푚푎푥 2 Notes About rms Values rms values are used when discussing alternating currents and voltages because ▪ AC ammeters and voltmeters are designed to read rms values. ▪ Many of the equations that will be used have the same form as their DC counterparts. If I take the same incandescent light bulb and connect it to an AC source with amplitude 170V and a DC source with amplitude 120V. Which light bulb would be brighter?

A. AC current brighter than DC current B. DC current brighter than AC current C. Both would be equally bright Instant power: 1 푝 = 푖 ∙ 푣 = 퐼 ∙ 푉 sin 휔푡 sin 휔푡 + 휑 = 퐼 ∙ 푉 [cos 휑 −co푠 2휔푡 + 휑 ] 푚푎푥 푚푎푥 푚푎푥 푚푎푥 2 average power: 1 푇 1 푇 1 1 P = න 푖 ∙ 푣 푑푡 = 퐼푚푎푥 ∙ 푉푚푎푥 න [cos 휑 −co푠 2휔푡 + 휑 ] = cos 휑 퐼푚푎푥 ∙ 푉푚푎푥 푇 0 푇 0 2 2 1 = cos 휑 퐼푟푚푠 ∙ 푉푟푚푠 푉푟푚푠 ≡ 푉푚푎푥 2 (rms=root mean square) 1 170V/ ퟐ=120V 퐼푟푚푠 ≡ 퐼푚푎푥 2 I connect an incandescent light bulb to an AC voltage source, will the average power through the light bulb depend on the frequency?

A.Yes, the light bulb will be brighter at higher frequency B.Yes the light bulb will be fainter at higher frequency C.No change of brightness Light bulb as a function of frequency demo I add a capacitor in series to the incandescent light bulb and the AC voltage source, will the average power through the light bulb depend on the frequency?

A. Yes, the light bulb will be brighter at lower frequency B. Yes the light bulb will be fainter at lower frequency C. No change of brightness Light bulb + capacitor as a function of frequency demo 휋 휔퐿퐼 sin 휔푡 + The RLC Series Circuit; max 2 ∆푣퐿 1 휋 퐼 sin 휔푡 − 푅Imax sin 휔푡 휔퐶 max 2 푣 휑 ∆푣푅

∆푣퐶 phasor diagram

2 1 푣 = ΔVmax sin 휔푡 + 휑 Δ푉max = Imax 푅2 + 휔퐿 − 휔퐶 (푣 = ∆푣푅+ ∆푣퐿+ ∆푣퐶) 1 휔퐿 − 휑 = tan−1 휔퐶 푅 푖 = Imax sin 휔푡

Section 33.5 휋 휔퐿퐼 sin 휔푡 + The RLC Series Circuit; max 2 ∆푣퐿 1 휋 퐼 sin 휔푡 − 푅Imax sin 휔푡 휔퐶 max 2 푣 휑 ∆푣푅

∆푣퐶 phasor diagram

∆Vmax 퐼max = 푣 = ΔVmax sin 휔푡 + 휑 1 2 (푣 = ∆푣푅+ ∆푣퐿+ ∆푣퐶) 푅2 + 휔퐿 − 휔퐶 1 휔퐿 − 휑 = tan−1 휔퐶 푖 = Imax sin 휔푡 푅 Section 33.5 The RC Series Circuit; 0 1 휋 퐼 sin 휔푡 − 푅Imax sin 휔푡 휔퐶 max 2

휑 ∆푣푅 푣 ∆푣퐶 phasor diagram

∆Vmax 퐼max = 푣 = ΔVmax sin 휔푡 + 휑 1 2 (푣 = ∆푣푅+ ∆푣퐶) 푅2 + 휔퐶 1 − 휑 = tan−1 휔퐶 푖 = Imax sin 휔푡 푅 Section 33.5 The RC Series Circuit; 0 1 휋 퐼 sin 휔푡 − 푅Imax sin 휔푡 휔퐶 max 2

휑 ∆푣푅 푣 ∆푣퐶 phasor diagram

∆Vmax ∆Vmax휔퐶 퐼max = = 푣 = ΔV sin 휔푡 + 휑 1 + 푅휔퐶 2 max 1 2 푅2 + (푣 = ∆푣푅+ ∆푣퐶) 휔퐶

1 − 휑 = tan−1 휔퐶 푖 = Imax sin 휔푡 푅 Section 33.5 The RC Series Circuit; 0 1 휋 퐼 sin 휔푡 − 푅Imax sin 휔푡 휔퐶 max 2

휑 ∆푣푅 푣 ∆푣퐶 phasor diagram

∆Vmax ∆Vmax휔퐶 퐼max = = 푣 = ΔV sin 휔푡 + 휑 1 + 푅휔퐶 2 max 1 2 푅2 + (푣 = ∆푣푅+ ∆푣퐶) 휔퐶 The current will decay to zero at zero frequency! 1 − 휑 = tan−1 휔퐶 푖 = Imax sin 휔푡 푅 Section 33.5 I add a capacitor and an inductor in series to the incandescent light bulb and the AC voltage source, will the average power through the light bulb depend on the frequency?

A. Yes, the light bulb will be brighter at much lower frequency B. Yes the light bulb will be fainter at much lower frequency C. No change of brightness D. Yes, the light bulb will be brighter at much higher frequency E. Yes the light bulb will be fainter at much higher frequency Light bulb + capacitor + loudspeaker (inductor) as a function of frequency demo AC circuit 4 (AC source L+R+C):

1 푍 = 푅2 + (휔퐿 − )2 휔퐶 1 휔퐿 − R=1 tan 휑 = 휔퐶 푅

R=3 V 푡 = 푉0 cos 휔푡 R=10 푉 퐼 푡 = 0 cos 휔푡 − 휑 푍 RLC circuit

=> Band pass filter AC circuit 4 (AC source L+R+C):

1 푍 = 푅2 + (휔퐿 − )2 휔퐶 1 휔퐿 − R=1 tan 휑 = 휔퐶 푅

R=3 V 푡 = 푉0 cos 휔푡 R=10 푉 퐼 푡 = 0 cos 휔푡 − 휑 푍 RLC circuit ퟏ ퟏ 흎푳 − = ퟎ => Resonance at 흎 = 흎푪 ퟎ 푳푪 Power as a Function of Frequency

Power can be expressed as a function of frequency in an RLC circuit.

2 2 (VRrms ) ω Pav = 2 2 2 2 2 2 R ω+− L( ω ωo ) This shows that at resonance, the average power is a maximum.

Section 33.7 A useful application of RLC: the loudspeaker

• The woofer (low tones) and the tweeter (high tones) are connected in parallel across the amplifier output. 8. Transformer: Transformers

An AC transformer consists of two coils of wire wound around a core of iron. The side connected to the input AC voltage source is called the primary and has N1 turns. The other side, called the secondary, is connected to a resistor and has N2 turns. The core is used to increase the magnetic flux and to provide a medium for the flux to pass from one coil to the other.

Section 33.8 Transformers High voltage transformer:

9. Filters: High-Pass Filter

The circuit shown is one example of a high-pass filter. A high-pass filter is designed to preferentially pass signals of higher frequency and block lower frequency signals.

Section 33.9 High-Pass Filter, cont

At low frequencies, ΔVout is much smaller than Δvin. ▪ At low frequencies, the capacitor has high reactance and much of the applied voltage appears across the capacitor. At high frequencies, the two voltages are equal. ▪ At high frequencies, the capacitive reactance is small and the voltage appears across the resistor.

Section 33.9 Low-Pass Filter

At low frequencies, the reactance and voltage across the capacitor are high. As the frequency increases, the reactance and voltage decrease. This is an example of a low-pass filter.

Section 33.9 If I take an LED and connect it to an AC source with amplitude 2 Vrms and a DC source with amplitude 2 V. Which LED would be brighter?

A. AC voltage brighter than DC voltage B. DC voltage brighter than AC voltage C. Both would be equally bright DC vs AC LED brightness 퐼푚푎푥

퐼푟푚푠 Current through a light bulb

퐼 푡 = 퐼푚푎푥 cos 휔푡 −퐼푟푚푠 −퐼0

퐼푚푎푥 Current through a LED

퐼 푡 = 퐼푚푎푥 cos 휔푡 ∙ Θ(cos 휔푡 ) Theta function (is zero for a negative argument) 퐼푚푎푥

퐼푟푚푠 Current through a light bulb

퐼 푡 = 퐼푚푎푥 cos 휔푡 −퐼푟푚푠 Resistance is a linear element −퐼0

퐼푚푎푥 Current through a LED

퐼 푡 = 퐼푚푎푥 cos 휔푡 ∙ Θ(cos 휔푡 ) Theta function (is zero for a negative argument)

LED (Diode) is a non-linear element