# Alternating Currents

#### Characteristics of alternating currents

Peak current, I_{0} = 3 A

Peak-to-peak current, I_{p-p} = 6 A

Period, T = 20 ms

Frequency, f = 1 / T = 50 Hz

Angular Frequency, ω = 2 π f = 314 rad s^{-1}

Instantaneous current: the current at a particular instant.

- Since this A.C. signal can be described by the equation:
- I = I0 sin (ω t)

or V = V0 sin (ω t)

the instantaneous current I or voltage V at time t is given by*I*or_{0}sin (ωt)*V*._{0}sin (ωt)

Note: Both the period and amplitude of a sinusoidal A.C should be **constant**.

Root-mean-square current of an alternating current is defined as that__ steady {NOT direct}__ current that produces the

__same heating effect__{ie I

^{2}R} as the alternating current

__in a given resistor__.

(Instantaneous) sinusoidal current: I = I_{0}sinωt , { Similarly, V = V_{0} sinωt }

I_{rms} = I_{o} / √2, V_{rms} = v_{o} / √2, {for **sinusoidal** ac only}

Relationship between Peak, & RMS values of PD & Current: V_{0} = I_{0}R , V_{rms} = I_{rms}R

**Mean/Ave Power, P _{ave} = I_{rms}^{2} R = V_{rms}^{2} / R = I_{rms} / V_{rms}** = ½ x Maximum Instantaneous Power =

**½ I**{for sinusoidal AC}

_{0}V_{0} Max (Instantaneous) Power, P_{max} = I_{0}V_{0} = I_{0}^{2} R

The **root-mean-square** (R.M.S.) value, I_{rms}, of an A.C. is the magnitude of the direct current that produces the same **average** heating effect as the alternating current in a given resistance whereas peak value is the maximum current of an AC.

Ideal transformer: V_{p} I_{p} = V_{s} I_{s} → N_{S} / N_{P} = V_{S} / V_{P} = I_{P} / I_{S}

{Mean power in the primary coil = Mean power in the secondary coil )

{Values of I & V may be either R.M.S. or peak but not instantaneous values; N_{S} / N_{P}: turns ratio}

#### Power Loss during Transmission of Electrical Power

Power Generated at power station P_{gen} = V_{i} I,

where I: current in the transmission, Vi: Voltage at which power is transmitted

I = P_{gen} / V_{i}

Power Loss in Transmission Cables, P_{L} = I^{2} R_{C} = (P_{gen} / V_{i})^{2} R_{C} ;R_{C} = cable resistance

Thus to reduce power loss, for a given amt of power generated, electricity is transmitted at __high voltage__ V_{i} {ie low current}. {V_{i} is NOT the pd across the cables}

#### Rectification with a diode

If a single diode is connected to an A.C. circuit as shown, a **half-wave rectification** occurs.

The graphs for the input and output voltages, and the output current, are shown below.

In the regions A and C, the diode is forward biased, allowing current to flow. When the input voltage becomes negative, the diode prevents the current flow, because it is reverse biased.