# SIMPLE A.C. CIRCUITS

**
**

**Subject: **

## PHYSICS

Table of Contents

**Term:**

FIRST TERM

**Week:**

WEEK 9

**Class:**

SS 3

**Topic:**

**SIMPLE A.C. CIRCUITS**

CONTENT

- Alternating Current Circuits
- A.C. in Resistors
- A.C. Through a Capacitor
- A.C. Through an Inductor
- Series Circuits
- Power in A.C. Circuits

**
**

**Alternating Current Circuits**

An a.c. circuit is one in which the magnitude of the current changes periodically with time. The a.c. is produced by an alternating voltage supply. The pattern of the a.c. voltage is sinusoidal in nature, that is it varies like the sine curve with constant amplitude and frequency.

*Image to be added*

V_{0 }is the maximum or peak voltage, which represent the maximum displacement (amplitude).

V is the instantaneous voltage, representing the displacement.

Now, *sinθ*=*VV*0

Hence, *V*=*V*0*sinθ*=*V*0*sinωt*

The rms voltage is defined as the steady voltage which would produce the same heating effect per second in a given resistor.

The rms value of current is defined as the steady current which would dissipate at the same rate in a given resistor.

*V*0=2*V**rms*−−−−−√∴*V*=2*V**rms**sinωt*−−−−−−−−−√

Thus, *V*=2(*V**rms**sin*2*πft*)−−−−−−−−−−−−√

Where ω = 2πf

Similarly,

*I*=*I*0*sinθ*=2(*I**rms**sinωt*)−−−−−−−−−−√*I*=2(*I**rms**sin*2*πft*)−−−−−−−−−−−−√*I*0=*I**rms*2–√

Where:

V : Instantaneous value

V_{0}: Peak value

V_{rms }: root mean square value.

Also,

I: Instantaneous value

I_{0 }: peak value

I_{rms }: root mean square value.

θ: phase angle between voltage and current

ω: angular velocity

**A.C. in Resistors**

The ability of a resistor to restrict the flow of current in an a.c circuit is called its resistance R.

When an a.c is applied to a resistor, both current and the voltage attain maximum and minimum at the same time. Hence, they are in phase.

According to Ohm’s law,

*R*=*VI**I*0=*V*0*R**I**rms*=*V**rms**R*

**EVALUATION**

- Define an alternating current.
- Describe the path of an a.c.
- State the mathematical relationship between I
_{0 }and I_{rms}.

**A.C. Through a Capacitor**

When an a.c is applied through a capacitor, current leads by 90° or *π*2radians. They are out of phase.

*Image to be added*

The ability of a capacitor to resist the flow of current in an a.c circuit is called its ‘Capacitive Reactance X_{c}’.

The reactance of a capacitor, X_{c }is given by:

*X**C*=1*ωC*=12*πfC**I**rms*=*V**rms**X**C**andI*0=*V*0*X**C*

The capacitance of a capacitor is measured in Farad F.

**A.C. Through an Inductor**

When an a.c is applied through an inductor, voltage leads by 90° or *π*2radians. They are out of phase.

An inductor has an inductance (L), measured in Henri (H). The ability of an inductor to restrict the flow of current in an a.c circuit is called its ‘Inductive Reactance X_{l}’. The reactance of an inductor, X_{l }is given by:

*X**I*=*ωL*=2*πfLI**rms*=*V**rms**X**L**andI*0=*V*0*X**L*

**Worked Examples**

**Example 1:**

A capacitor of 1μF is used in a radio circuit where the frequency is 1000H_{z }and the current is 2mA. Calculate the voltage across the capacitor.

**Solution:**

*V*=*IX**C*

But, *X**C*=12*πfC*

Also, *I*=2*mA*=21000=0.002*AV*=*I*2*πfC*=0.0022*π*×1000×1×10−6=0.32*V*

(NB: μ = 10^{– 6}, m= 10^{– 3})

**Example 2:**

An inductor of 2H and a negligible resistor is connected to 12V mains supply. If the frequency is 50H_{z}, find the current flowing.

**Solution:**

*I*=*VX**L*=122*πfL*=122*π*×50×2=628.3*A*

**Series Circuits**

**Capacitor and resistor in series (RC circuit):**

When a capacitor is connected in series with a resistor, the total opposition to the current flowing through the circuit is called ‘**Impedance, Z**’. Here, current leads voltage by 90°. From the vector diagram above,

*V*2=*V*2*C*+*V*2*R*∴*V*2=*I*2*X*2*C*+*I*2*R*2*V*2*I*2=*X*2*C*+*R*2

But *Z*=*VI*→*Z*2=*V*2*I*2∴*Z*2=*X*2*C*+*R*2∴*Z*=*X*2*C*+*R*2−−−−−−−√

For the phase angle,

*Tanθ*=*V**C**V**R*=*IX**C**IR*=*X**C**R*

**Inductor and resistor in series:**

When an inductor is connected in series with a resistor, voltage leads by 90° on the current.

Considering the vector diagram,

*V*2=*V*2*L*+*V*2*R*∴*V*2=*I*2*X*2*L*+*I*2*R*2*V*2*I*2=*X*2*L*+*R*2

But *Z*=*VI*→*Z*2=*V*2*I*2∴*Z*2=*X*2*L*+*R*2∴*Z*=*X*2*L*+*R*2−−−−−−−√

For the phase angle,

*Tanθ*=*V**L**V**R*=*IX**L**IR*=*X**L**R*

**Capacitor, inductor and resistor in series (RLC Circuit):**

Now, *V*2=*V*2*R*+(*V**L*−*V**C*)2*V*2=*I*2*R*2+(*IX**L*−*IX**C*)2*V*2=*I*2*R*2+*I*2(*X**L*−*X**C*)2*V*2*I*2=*R*2+(*X**L*−*X**C*)2*Z*=(*R*2+(*X**L*−*X**C*)2)−−−−−−−−−−−−−−−√

Note that the two reactance must be subtracted before squaring.

For the phase angle,

*Tanθ*=*V**L*−*V**C**V**R*=*X**L*−*X**C**R*

**Worked Examples**

**Example 3:**

A 2.5μH inductor is connected in series with a non-inductive resistor of 300Ω across a 50V alternating at 160H_{z}. Calculate the r.m.s value of the current in the circuit.

**Solution:**

*V**rms*=*I**rms**ZI**rms*=*V**rms**Z*

Now, *Z*=*X*2*L*+*R*2−−−−−−−√

But, *X**L*=2*πfL*=2*π*160×2.5×10−6=2.5×10−3Ω*Z*=3002+(2.5×10−3)2−−−−−−−−−−−−−−−−√=300Ω*I**rms*=50300=0.17*A*

**Example 4:**

A 2.0μF capacitor is connected in series with a resistor of 300Ω across a 240V a.c alternating at 160H_{z}. Find the rms value of the current in the circuit.

**Solution:**

*I**rms*=*V**rms**Z*

Now, *Z*=*X*2*C*+*R*2−−−−−−−√

But, *X**C*=12*πfL*=12*π*160×2.0×10−6=497.4Ω∴*Z*=3002+(497.4)2−−−−−−−−−−−−√=580.8Ω*I**rms*=240580.8=0.41*A*

**EVALUATION**

- State the mathematical relationship between Z, X
_{C }and X_{L}. - Define an impedance Z.
- Define the reactance of a capacitor and an inductor.

**Power in A.C. Circuits**

The average power dissipated in an A.C. circuit per cycle is given as:

*P*=12*I*0*V*0

But, *I*0=2(*I**rms*)−−−−−−√ and

*V*0=2(*V**rms*)−−−−−−√∴*P*=12(2(*I**rms*)−−−−−−√×2(*V**rms*)−−−−−−√)∴*P*=2*I**rms**V**rms*2∴*P*=*I**rms**V**rms*

According to Ohm’s law, V = IR

∴*P*=*I*2*R*=*V*2*R*

Power is always dissipated in a resistor. This because the current and the voltage are in phase. In an inductor, power dissipation is zero due to the fact that power is positive when energy is stored in the magnetic field of the coil and negative when the energy is given back to the voltage supply on the next part of the cycle.

In a capacitor, power dissipation is also zero due to the fact that power is positive when energy is stored in the electric field between the capacitor’s plates and negative when the energy is given back to the voltage supply.

In essence, power dissipation is zero both in an inductor and a capacitor because the voltage and current are out of phase by90°.

Therefore, in an a.c. circuit containing inductor, capacitor and a resistor in series, power is only dissipated in the resistor.

Hence, *P*=*IV*=*I*2*R*=*V*2*R*

**Example 5:**

A circuit consists of a resistor 500ohms and a capacitor of 5μF connected in series. If an alternating voltage of 10v and frequency 50 Hz is applied across the series circuit, calculate

(a) the reactance of the capacitor

(b) the current flowing in the circuit

(c) the voltage across the capacitor

(d) If the capacitor is replaced with an inductor of 150mH, calculate the impedance and voltage across the inductor

**Solution:**

(a) *X**C*=12*πfL*=12*π*×50×5.0×10−6=636.62*ohms*

(b) *Z*=*R*2+*X*2*C*−−−−−−−√ since there is no L

*Z*=5002+636.622−−−−−−−−−−−−√=809.5*ohmsI*=*VZ*=10809.5*A*=12.35×10−3*A*

(c) *V**C*=*IX**C*=12.35×10−3×636.62=7.86*V*

(d) *X**L*=*ωL*=2*πfL*=2×*π*×50×10−3*ohms*=47.12*ohmsZ*=*R*2+*X*2*L*−−−−−−−√=5002+47.122−−−−−−−−−−−√=502.2*ohmsI*=*VZ*=10502.2=19.9×10−3*AV*=*IX**L*=19.9×10−3×47.12=938×10−3*V*

** **

**EVALUATION**

- Why is power only dissipated in the resistor in a series circuit containing resistor, capacitor and an inductor?
- State the formula for power.

- Tell the meaning of the following denotations: V
_{0}, V_{rms}, I_{0}, I - In what component of the series circuit is voltage and current in step with each other?
- Critically examine the passage of an a.c through an inductor.
- Define the reactance and the resistance of a capacitor and a resistor respectively.
- Discuss why power is not dissipated in an inductor when current passes through it.

One component of a series circuit where voltage and current are in step with each other is called the inductor. This component typically consists of a coil of wire that stores energy in a magnetic field. To critically examine the passage of alternating current (AC) through an inductor, we need to consider factors such as the reactance and resistance of the inductor, as well as the power that is dissipated in it. Typically, an inductor does not dissipate power when current passes through it because its resistance is very low compared to its reactance. This means that energy from the AC input is temporarily stored in the magnetic field of the inductor and then released back into the circuit, as the current and voltage in the circuit are shifted out of phase with each other. Overall, an inductor is a key component in AC circuits that helps to regulate and control the flow of electricity through devices such as motors or speakers.

## In what component of the series circuit is voltage and current in step with each other?

The component of a series circuit that exhibits voltage and current that are in step with each other is commonly referred to as the inductor. This component typically consists of a coiled wire or conductor that stores energy in an electric or magnetic field, and it plays an important role in regulating and controlling the flow of electricity through devices such as motors, speakers, or other electrical components. Additionally, the resistance of an inductor is typically very low compared to its reactance, which means that energy from an alternating current (AC) input is temporarily stored in the magnetic field of the inductor and then released back into the circuit as current and voltage shift out of phase with each other. Overall, the inductor is a key component in AC circuits, and it plays an important role in controlling the flow of electricity through electrical systems and devices.

### Critically examine the passage of an a.c through an inductor.

One of the key challenges when passing alternating current (AC) through an inductor is that its resistance is typically very low compared to its reactance. This means that energy from the AC input is temporarily stored in the magnetic field of the inductor, and then released back into the circuit as current and voltage shift out of phase with each other. Additionally, in order to critically examine the passage of AC through an inductor, it is important to consider factors such as the reactance and resistance of the inductor, as well as power dissipation in different parts of the circuit. Overall, there are a number of challenges and considerations when dealing with an inductor in a series circuit, and understanding these factors is critical for effectively and efficiently controlling the flow of electricity in electrical systems and devices.

### Define the reactance and the resistance of a capacitor and a resistor respectively.

The reactance of a capacitor refers to its ability to oppose changes in voltage, while the resistance of a resistor refers to its ability to dissipate electrical energy as heat. Both of these properties are determined by the physical characteristics of the capacitor or resistor, including factors such as material composition, size, and shape. Additionally, both capacitors and resistors typically exhibit different reactance and resistance values depending on the frequency and amplitude of the electrical current passing through them, which means that they have different properties at different points in an AC circuit. Ultimately, understanding the reactance and resistance of capacitors and resistors is essential for effectively controlling the flow of electricity through electrical devices and systems.

**GENERAL EVALUATION**

- Tell the meaning of the following denotations: V
_{0}, V_{rms}, I_{0}, I - In what component of the series circuit is voltage and current in step with each other?
- Critically examine the passage of an a.c through an inductor.
- Define the reactance and the resistance of a capacitor and a resistor respectively.
- Discuss why power is not dissipated in an inductor when current passes through it.