Ch24 AC Circuits

ELI the ICE man

L à Voltage leads the Current by 90°

C à Current leads the Voltageby 90°

Alternating Current Circuits

For Resistors, Voltage and current are in phase.

V = V_max sin(ω t)

I = I_max sin(ω t) 

(Ohms law à V = IR)

instantaneous values of the voltage or current are represented by the phasor is its projection on the y axis.

Alternating Current and Voltage

Obviously if you take the average of a sinusoidal wave, it is ZERO.

For sinusoidal waves, we refer to the ROOT MEAN SQUARE (rms), x_ave² = ½ x_max²

x_rms = x_max/√2 = 0.707 x_max

Example

Capacitors in AC Circuits

Just for fun, let’s look at the true definition.

We know

I = Δq/ Δt

I = dq/ dt

I = d[CV_maxsin(ωt)] / dt

I = C ω V_maxcos(ωt)

I = C V_maxωsin(ωt + 90°)

I = q_maxω sin(ωt + 90°)

I = I_maxsin(ωt + 90°)

So we know current, I, leads V, voltage, by 90°

V = C ω V_maxsin(ωt)

q = C V

q = C V_max sin(ωt)

What is the maximum value of sin(ωt)?

Sine varies between -1 to 0 to +1

So V_maxsin(ωt) = 1

sin(ωt + 90°) = cos (ωt)

*******Let’s test

sin(120°) = cos(30°) = 0.866

sin(150°) = cos(60°) = ½

sin(180°) = cos(90°) = 0

· We know capacitors charge (or discharge) proportional to the rate of voltage change.

· The faster the voltage changes the more charge flows.

· The slower the voltage changes the less current.

o As a result we can conclude the reactance of an AC capacitor is “inversely proportional” to the frequency of the supply

Capacitance Reactance has the units of and analogous to Resistance

X_C = 1 / 2πfC

X_C = 1/ ωC

· What happens when freqà infinity?

o Ans:  this is Short Circuit

· What happens when freqàzero?

o Ans:  DC circuits, this is why capacitors block direct current

Example

RC Circuits

As we saw above the current in a resistor and capacitor are not in phase. This means that the maximum current is not the sum of the maximum resistor current and the maximum capacitor current; they do not peak at the same time.

V_max² = (I_max R)² + (I_maxX_C)²

V_max = I_max √(R² + X_C²)

Impedance, Z = √(R² + X_C²)

Remember

X_C = 1 / 2πfC

X_C = 1/ ωC

cosφ	=	Imax R	=	R
		Imax Z		Z

P_ave = I_rmsV_rmscosφ;   as a result cosφ is called                      the Power Factor

Example

Inductors in AC Circuits

Inductive Reactance = X_L = ωL  with units of Ohms

Impedance, Z = √(R² + X_L²)

cosφ	=	R	=	R
		Z		√(R^2 + (ωL)^2)

Example

RLC Circuits

Z = [(R² + (X_L – X_C)²)]^½

tanφ	=	Imax (XL - XC)	=	XL - XC
		Imax R		R

If X_L = X_C the phase angle is zero

At high frequencies, the capacitive reactance is very small, while the inductive reactance is very large

At low frequencies, the inductive reactance is very small, while the capacitive reactance is very large

Example

Resonance in Electric Circuits

In an RLC circuit with an ac power source, the impedance is a minimum at the resonant frequency

Resonance occurs at Z = R

ω_o = 1 / √(LC) = 2πf_o

Example

Your microwave oven uses a magnetron that resonates at 2.7 GHz approximating an RLC circuit.  If the capacitance is of the magnetron is 5×10−13 F, what is the value of its inductance? 

ω = 1 / √(LC) = 2 π  f

L = 1 /    C     (2 π      f  )²

L = 1 / 5e-13 (2 π  2.7e9)²

L = 6 x 10^-9 Henry’s