AC Circuits

Chapter 21

Emf and current values always changing according to a sinusoidal wave function
Vary from a positive maximum to a negative minimum so average value of both is zero
Instantaneous values depend on angle θ that the generator coil makes with the magnetic field and thus on the angular portion of one complete cycle of the alternating current
Use lower case symbols to designate instantaneous values: e, i, v
e = E_maxsinθ and i = I_maxsinθ

Value of ac voltage or current that would have the same effect on a resistance as dc
Not the average value (which would be 0) but a special average called the root mean squared value (rms)
To find rms value for current, square the maximum value (I_max), divide that by 2, then take the square root of the result.
Similar equation for E & V
So it turns out that the rms value = .707 max value (for I, E, or V)

Phasor - rotating vectors denoting quantities related in time or phase
Can be used to represent current, emf, and voltages across ac circuit elements
Shows phase relationship between voltage and current by the angle between phasors. This angle is called the phase angle, represented by the Greek letter phi
i = I_maxsinθ and e = E_maxsin(θ + φ)
Add phasors like vectors

Instantaneous power of ac is product of e and i
Average power is product of I_rms and E _rms
In circuit with only resistance, v and i are in phase: both positive or negative at same time P = I²R = VI (same as dc)
In this case, power is always positive; it varies from 0 to ( I_maxE _max) with a frequency twice the ac frequency
If a capacitor or inductor is present, the phase relationship of v and i changes and I²R does not equalVI due to impedance in the circuit.

DC - current only flows while capacitor is charging or discharging; current is at a maximim when voltage across the capacitor is 0; current is 0 when voltage across the capacitor is at a maximim
AC - As the capacitor charges, voltage across the capacitor builds and current decreases; then the current changes direction and the capacitor discharges; when the current reaches 0 again, the capacitor is charged with the opposite polarity
Because of this, the voltage lags behind the current by 90°; this current to voltage relationship is called the phase angle, represented by φ
φ = - 90° for circuit with only capacitance
If resistance and capacitance are both present in the circuit, the phase angle will be somewhere between 0° and -90°
Energy is stored in electric field of capacitor as current goes from max value to 0 and returned as current rises from 0 to max
No electrons actually flow through the capacitor: therefore it blocks dc but the effect of the ac passes through

Inductor resists any change in current, producing an opposing emf (Lenz's law)
Maximum emf is induced when the current changes most, which is when it crosse the zero point
So voltage is at max when current is zero
Therefore V leads I by 90° called phase angle φ
For a theoretical circuit with only inductor, φ = 90°; if resistance and inductance are both present, the phase angle will be somewhere between 0° and 90°

P = ei but now e and iare not positive or negative at the same time, so the power is sometimes negative
As current rises, energy is stored in the inductor; when current drops, this energy is released
Since inductors have some resistance, no circuit purely inductive so voltage leads current by angle between 0° and 90°
Power is only dissipated in the resistor, not in the inductor so P = V_RI; but V_R = Vcosφ
So power consumption P = VI cosφ
cosφ = V_R /V ; is called power factor, gives idea of available power in circuit

Inductors and capacitors have voltage drops across them due to their opposition to changing ac voltage
but this voltage drop is non-resistive (no power consumed)
The property that creates this voltage drop is called reactance; symbol X ( X_L or X_C ); unit is ohm
X_L= V_L/ I = 2π f L ; X_C= V_C/ I = 1/(2π f C) where f is the frequency of the ac
Inductive reactance directly proportional to inductance and frequency
Capacitive reactance inversely proportional to capacitance and frequency
Total reactance X is vector sum of X_L and X_C ; Since phase angles are 180 degrees apart, X = X_L - X_C if both C and L are in circuit

Combined effect of resistance and reactance in ac circuit: total opposition to current; symbol: Z unit: ohm
Ohm's Law for ac ; V = IZ or E = IZ
Phase angle for impedance φ = tan ^-1( X/ R)

In RLC circuit, if X_L > X_C circuit is inductive and voltage leads current; if X_C >X_L circuit is capacitive and voltage lags current
Frequency determines the reactance of the circuit: at high f, X_L is large and X_C is small so circuit will be inductive; at low f, X_C is large and X_L is small so circuit will be capacitive. At some frequency, X_L =X_C , and total reactance = 0; therefore, Z = R and impedance is a minimum
This is resonant frequency ( f_R) when current will be a maximum, impedance minimum and power maximum (power factor = 1)
At f_R , X_L =X_C so 2πf_R L=1/ (2π)f_R C) or f_R = 1/[2π(LC)^½]
Since current is high at resonant frequency, circuit discriminates among applied frequencies - called selectivity
Basis of radio and TV tuner circuits
Can change the resonant frequency with a variable capacitor or inductor and tune the circuit to one specific frequency

Often a resonant circuit is an inductor and capacitor in series. The only resistance is in the inductor coil.
The circuit depends on the ratio of inductance and resistance.
X_L/ R is called the Q of the circuit (short for quality factor) and expresses the selectivity of the circuit.
A high Q means the circuit has a sharp resonant peak, and is a highly selective circuit.

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