(a) (i) Show that the Smith chart constructed for a lossless transmission line gives a family of r-circles, having a radius of…

(a)(i)

Let the normalized load impedance be z = r + jx and Γ = Γᵣ + jΓᵢ. For a lossless line, Γ = (z − 1)/(z + 1), so z = (1 + Γ)/(1 − Γ).

Thus r + jx = (1 + Γᵣ + jΓᵢ)/(1 − Γᵣ − jΓᵢ).

Equating real parts, r[(1 − Γᵣ)² + Γᵢ²] = 1 − Γᵣ² − Γᵢ².

Expanding and collecting terms: (1 + r)(Γᵣ² + Γᵢ²) − 2rΓᵣ + (r − 1) = 0.

Divide by (1 + r): Γᵣ² + Γᵢ² − [2r/(1 + r)]Γᵣ + (r − 1)/(1 + r) = 0.

Completing the square in Γᵣ: (Γᵣ − r/(1 + r))² + Γᵢ² = 1/(1 + r)².

Hence each constant-r locus is a circle with centre Γᵣ = r/(1 + r), Γᵢ = 0 and radius 1/(1 + r). This proves the Smith-chart r-circle family.

(a)(ii)

For normal incidence between two lossless dielectrics, use the intrinsic-impedance method.

η₀ = √(μ₀/ε₀) = √[(4π × 10⁻⁷)/((1/(36π)) × 10⁻⁹)] = 120π Ω.

Region 1: η₁ = η₀/√εᵣ₁ = 120π/√7.5 Ω = 137.66 Ω.

Region 2 is free space: η₂ = η₀ = 120π Ω = 376.99 Ω.

Reflection coefficient for the electric field: Γ = (η₂ − η₁)/(η₂ + η₁) = (√7.5 − 1)/(√7.5 + 1) = (17 − 2√30)/13 ≈ 0.4650.

Transmission coefficient: τ = 1 + Γ = 2η₂/(η₁ + η₂) = (30 − 2√30)/13 ≈ 1.4650.

Given E₀ⁱ = 1.2 × 10⁻³ V/m: E₀ʳ = ΓE₀ⁱ = 0.4650 × 1.2 × 10⁻³ = 5.5805 × 10⁻⁴ V/m.

E₀ᵗ = τE₀ⁱ = 1.4650 × 1.2 × 10⁻³ = 1.7581 × 10⁻³ V/m.

Magnetic-field amplitudes: H₀ⁱ = E₀ⁱ/η₁ = (1.2 × 10⁻³)/(137.66) = 8.717 × 10⁻⁶ A/m.

H₀ʳ = E₀ʳ/η₁ = (5.5805 × 10⁻⁴)/(137.66) = 4.054 × 10⁻⁶ A/m.

H₀ᵗ = E₀ᵗ/η₂ = (1.7581 × 10⁻³)/(376.99) = 4.663 × 10⁻⁶ A/m.

These are valid for σ₁ = 0 and normal incidence.

(b)(i)

An ideal envelope detector removes the carrier and follows the positive envelope. Hence x(t) = A_c|1 + 2 cos(2πfₘt)|.

Let θ = 2πfₘt and f(θ) = |1 + 2 cos θ|. Since f is even, only cosine terms appear: x(t) = a₀ + Σ aₙ cos(nθ).

The DC term is a₀ = (A_c/2π)∫₀²π |1 + 2 cos θ| dθ = A_c(1/3 + 2√3/π).

For n ≥ 1, aₙ = (A_c/π)∫₀²π |1 + 2 cos θ| cos nθ dθ.

Splitting where 1 + 2 cos θ is positive and negative: positive intervals: [0, 2π/3] ∪ [4π/3, 2π]; negative interval: [2π/3, 4π/3].

This gives a₁ = A_c(2/3 + √3/π), a₂ = A_c√3/π.

For n ≥ 2:

• if n ≡ 0 mod 3, aₙ = −4A_c√3/[π(n² − 1)];
• if n ≡ 1 mod 3, aₙ = 2A_c√3/[πn(n + 1)];
• if n ≡ 2 mod 3, aₙ = 2A_c√3/[πn(n − 1)].

Therefore, x(t) = A_c(1/3 + 2√3/π) + A_c(2/3 + √3/π) cos(2πfₘt) + (A_c√3/π) cos(4πfₘt) − (A_c√3/2π) cos(6πfₘt) + (A_c√3/10π) cos(8πfₘt) + (A_c√3/10π) cos(10πfₘt) + ⋯.

(b)(ii)

Second harmonic amplitude = A_c√3/π. Fundamental amplitude = A_c(2/3 + √3/π).

Ratio = (√3/π)/(2/3 + √3/π) = 3√3/(2π + 3√3) ≈ 0.45265.

So the second harmonic is approximately 45.27 % of the fundamental amplitude.

(c)

Voltage control in three-phase inverters is achieved mainly by:

• Variable DC-link voltage: a controlled rectifier, chopper, or PWM rectifier changes V_dc; the AC output varies in proportion. It is simple but slow and needs a controlled front end.
• PWM control: vary the modulation index in sinusoidal PWM. The fundamental output voltage is approximately proportional to the modulation index.
• Third-harmonic injection PWM: adds a third-harmonic component to the reference, extending the linear modulation range and improving DC utilisation.
• Space Vector PWM: controls the inverter voltage vector by dwell-time selection; it gives better DC utilisation and lower harmonic distortion.
• Selective harmonic elimination PWM: switching angles are chosen to eliminate selected harmonics while setting the desired fundamental.
• Phase-shift control: two or more inverter bridges feed a common transformer; varying their phase displacement changes the resultant output voltage.
• Multilevel or transformer-tap/star-delta switching: connection changes or module count changes give stepped voltage control.
• AC-side voltage control: thyristor phase control or AC chopper between inverter and load can vary voltage, but harmonics and power factor are poorer.
• Across the PWM family, the switching (carrier) frequency sets the harmonic sidebands: a higher switching frequency moves harmonics to higher orders for easier filtering, at the cost of increased switching losses.