Unit 5: Electrical Circuits

1. AC Circuits: Kirchhoff's Laws, Reactance, Impedance

Kirchhoff's Laws for AC Circuits

Kirchhoff's laws, fundamental to DC circuits, also apply to AC circuits. However, we must use phasors and complex impedances, as the voltages and currents have both magnitude and phase.

• Kirchhoff's Current Law (KCL): The phasor sum of currents entering any node is zero. (ΣI = 0)
• Kirchhoff's Voltage Law (KVL): The phasor sum of voltage drops around any closed loop is zero. (ΣV = 0)

Complex Reactance and Impedance

To analyze AC circuits, we use complex numbers (j, where j² = -1) to represent phase shifts. A sinusoidal voltage V(t) = V₀cos(ωt) is represented as the real part of V = V₀e^jωt.

Impedance (Z) is the "complex resistance" of a component. It is defined as the ratio of the phasor voltage (V) to the phasor current (I): Z = V / I.

Reactance (X) is the imaginary part of impedance. It represents the opposition to current flow that arises from phase shifts in inductors and capacitors.

Impedances of Basic Components:

• Resistor (R):

  Z_R = R

  (Voltage and current are in phase).

• Inductor (L):

  Z_L = jωL

  (Reactance: X_L = ωL. Voltage leads current by 90°).

• Capacitor (C):

  Z_C = 1 / (jωC) = -j / (ωC)

  (Reactance: X_C = -1/(ωC). Voltage lags current by 90°).

2. Series LCR Circuit

A resistor (R), inductor (L), and capacitor (C) are connected in series to an AC voltage source V.

The total impedance Z is the sum of the individual impedances:

Total Impedance (Z): Z = Z_R + Z_L + Z_C = R + jωL - j/(ωC)

Z = R + j(ωL - 1/(ωC))

The magnitude of the impedance is |Z| = √[R² + (ωL - 1/(ωC))²].

The current in the circuit is I = V / Z. Its magnitude |I| = |V| / |Z|.

(i) Resonance

Resonance occurs when the total reactance is zero (X_L + X_C = 0).

ωL - 1/(ωC) = 0 => ωL = 1/(ωC)

This happens at the resonant angular frequency (ω₀):

Resonant Frequency: ω₀ = 1 / √(LC)

At resonance:

• Impedance Z is minimum (|Z| = R).
• Current |I| is maximum (|I| = |V|/R).
• The circuit is purely resistive (current is in phase with voltage).

(ii) Power Dissipation

Only the resistor dissipates power. The average power (P_avg) dissipated by the circuit is:

Average Power: P_avg = I_rms²R

This can also be written as P_avg = V_rmsI_rms cos(φ), where cos(φ) = R/|Z| is the Power Factor. At resonance, |Z|=R, so cos(φ)=1 and power is maximum.

(iii) Quality Factor (Q)

The Q Factor is a dimensionless parameter that measures the "sharpness" or "quality" of the resonance peak.

A high Q factor means a very sharp, narrow resonance (high selectivity). A low Q factor means a broad, flat resonance.

Q Factor: Q = ω₀L / R = 1 / (ω₀CR)

(iv) Bandwidth (Δω)

The bandwidth is the range of frequencies for which the power dissipated is at least half the maximum power. It is the width of the resonance curve at the "half-power points".

Δω = ω_high - ω_low

The bandwidth is related to the Q factor:

Bandwidth: Δω = ω₀ / Q = R / L

A high Q factor corresponds to a small bandwidth.

3. Parallel LCR Circuit

R, L, and C are connected in parallel. It is easier to analyze this using admittance (Y), which is the reciprocal of impedance (Y = 1/Z).

Y = Y_R + Y_L + Y_C = 1/R + 1/(jωL) + jωC

Total Admittance (Y): Y = 1/R + j(ωC - 1/(ωL))

Resonance occurs when the imaginary part is zero, at the same ω₀ = 1/√(LC).

At resonance:

• Admittance Y is minimum (Y = 1/R).
• Impedance Z is maximum (Z = R).
• Total current |I| = |V|/|Z| is minimum.

This is often called a "rejector" circuit because it has minimum current (maximum impedance) at the resonant frequency.

4. Network Theorems

These are powerful tools for simplifying complex linear circuits.

Thevenin's Theorem

Statement: Any linear, two-terminal network can be replaced by a single equivalent voltage source (V_th) in series with a single equivalent resistor (R_th).

• V_th (Thevenin Voltage): The open-circuit voltage measured across the two terminals.
• R_th (Thevenin Resistance): The equivalent resistance "looking back" into the terminals, with all independent voltage sources shorted and all independent current sources opened.

Norton's Theorem

Statement: Any linear, two-terminal network can be replaced by a single equivalent current source (I_N) in parallel with a single equivalent resistor (R_N).

• I_N (Norton Current): The short-circuit current that flows between the two terminals.
• R_N (Norton Resistance): This is exactly the same as the Thevenin resistance (R_N = R_th).

Relation: V_th = I_N * R_th

Maximum Power Transfer Theorem

Statement: For a given source (with a Thevenin equivalent V_th, R_th), maximum power is delivered to a load (R_L) when the load resistance is equal to the source resistance.

Condition for Max Power: R_L = R_th

When this condition is met, the maximum power transferred to the load is:

Max Power: P_max = V_th² / (4R_th)

Note for AC: In AC circuits, max power is transferred when the load impedance Z_L is the complex conjugate of the source impedance Z_th. (Z_L = Z_th*).

5. Ballistic Galvanometer

A ballistic galvanometer is a special type of galvanometer designed to measure the total charge (Q) that passes in a short burst or pulse, rather than a steady current (I).

Working Principle

A normal galvanometer has low inertia and reacts quickly to current. A ballistic galvanometer is built with a high moment of inertia (I_m) and a long period of oscillation.

1. A pulse of charge Q = ∫ I dt is passed through the coil. The pulse must be very fast, finishing before the coil has time to move significantly.
2. This pulse delivers an angular impulse to the coil, giving it an initial angular velocity.
3. The coil then swings, twisting its suspension. The kinetic energy is converted into potential energy in the suspension.
4. The coil swings to a maximum deflection angle (θ_max) (the "throw").

It can be shown that the total charge (Q) is directly proportional to the first maximum deflection (θ_max):

Formula: Q = k * θ_max

Where 'k' is the ballistic constant of the instrument.

Sensitivity

• Current Sensitivity: Deflection per unit current (θ/I). This is high for a normal galvanometer.
• Charge Sensitivity (Ballistic): Deflection per unit charge (θ/Q). This is what is maximized for a ballistic galvanometer.