2.1 Complex Impedance, Phasors, and AC Circuit Analysis

2.1 Complex Impedance, Phasors, and AC Circuit Analysis

In direct current (DC) circuits, opposition to current flow is governed purely by resistance ($R$), where Ohm's law defines the relationship between potential difference and charge drift. In alternating current (AC) and radio frequency (RF) networks, current opposition becomes significantly more complex. Reactive components—inductors and capacitors—store and return energy to the circuit on alternating half-cycles, introducing a time-domain phase shift between current and voltage. To analyze such circuits rigorously, radio engineers employ complex impedance ($Z$), unifying dissipative resistance and energy-storing reactance into a single mathematical framework.

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The Nature of Impedance and Reactance

Impedance ($Z$), measured in ohms ($\Omega$), is the total opposition offered by an electrical circuit to the flow of alternating current at a specific frequency. Unlike simple resistance, which dissipates electrical energy irreversibly as heat, impedance encompasses two distinct components:

1. Resistance ($R$): The in-phase, dissipative component of impedance, independent of frequency in idealized conductors.
2. Reactance ($X$): The quadrature ($90^\circ$ out-of-phase), non-dissipative component of impedance, arising from energy stored alternately in magnetic fields (inductive reactance, $X_L$) or electric fields (capacitive reactance, $X_C$).

Reactance is frequency-dependent and defined by the fundamental AC relationships:

$$X_L = 2\pi f L = \omega L$$ $$X_C = \frac{1}{2\pi f C} = \frac{1}{\omega C}$$

where $f$ is frequency in hertz ($\text{Hz}$), $\omega = 2\pi f$ is angular frequency in radians per second ($\text{rad/s}$), $L$ is inductance in henrys ($\text{H}$), and $C$ is capacitance in farads ($\text{F}$).

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Complex Mathematical Representation: Rectangular and Polar Forms

Because resistive and reactive voltages are $90^\circ$ out of phase with one another, they cannot be summed algebraically. Instead, impedance is expressed as a complex quantity using the imaginary operator $j$ (where $j = \sqrt{-1}$, adopted in electrical engineering to avoid confusion with instantaneous current $i$).

Rectangular Form

In rectangular notation, impedance is partitioned into real and imaginary terms:

$$Z = R + jX$$

• A pure inductor has $Z_L=+jX_L$; current lags voltage by $90^\circ$.
• A pure capacitor has $Z_C=-jX_C$; current leads voltage by $90^\circ$.
• A series RLC branch has $Z=R+j(X_L-X_C)$. With resistance present, its phase depends on the reactance-to-resistance ratio, rather than being fixed at $90^\circ$.

Polar Form

Impedance can also be expressed in polar coordinates, defining its total scalar magnitude ($|Z|$) and phase angle ($\theta$):

$$Z = |Z| \angle \theta$$

The magnitude represents the absolute ratio of peak (or RMS) voltage to peak (or RMS) current:

$$|Z| = \sqrt{R^2 + X^2}$$

The phase angle $\theta$ represents the temporal phase displacement of the voltage waveform relative to the current waveform:

$$\theta = \arctan\left(\frac{X}{R}\right)$$

To convert from polar form back to rectangular components:

$$R = |Z| \cos(\theta)$$ $$X = |Z| \sin(\theta)$$

When $\theta > 0^\circ$, the circuit is net inductive and voltage leads current. When $\theta < 0^\circ$, the circuit is net capacitive and voltage lags current. When $\theta = 0^\circ$, the circuit is purely resistive ($X = 0$).

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Phasor Diagrams and Vector Analysis

A phasor is a complex vector representing a sinusoidal physical quantity whose length corresponds to the waveform's peak or RMS magnitude, and whose angular orientation relative to the horizontal positive real axis corresponds to its phase angle.

       +j (Imaginary Axis - Inductive Reactance / Voltage)
               ^
               |
         +jX_L |         * Z = |Z| ∠ θ = R + jX_L
         (+jV_L)        /|
               |       / |
               |      /  |
               |     /   | +jX (Net Inductive)
               |    /    |
               |   / θ   |
   -Real ------+----------> +Real (Resistance R / In-phase Voltage V_R)
               |   \ -θ  |
               |    \    |
               |     \   | -jX (Net Capacitive)
               |      \  |
         -jX_C |       \ |
         (-jV_C)        \|
               |         * Z = |Z| ∠ -θ = R - jX_C
               v
       -j (Imaginary Axis - Capacitive Reactance / Voltage)

In AC circuit analysis:

• The reference vector is placed along the positive real axis ($0^\circ$). In series circuits, current $I$ is identical throughout every element and serves as the reference phasor ($I = |I| \angle 0^\circ$).
• The voltage drop across the resistor ($V_R = I R$) lies in phase with current along the real axis.
• The voltage drop across the inductor ($V_L = I X_L$) leads current by $+90^\circ$ and is plotted along the positive imaginary ($+j$) axis.
• The voltage drop across the capacitor ($V_C = I X_C$) lags current by $-90^\circ$ and is plotted along the negative imaginary ($-j$) axis.
• Total applied generator voltage is the vector resultant:

$$V_{\text{total}} = V_R + j(V_L - V_C) = \sqrt{V_R^2 + (V_L - V_C)^2} \angle \arctan\left(\frac{V_L - V_C}{V_R}\right)$$

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Series AC Circuit Analysis: R-L, R-C, and R-L-C Networks

In a series circuit, impedances add directly in rectangular form:

$$Z_{\text{total}} = Z_1 + Z_2 + Z_3 + \dots + Z_n$$

For a series R-L-C network:

$$Z_{\text{total}} = R + jX_L - jX_C = R + j(X_L - X_C)$$

Three operational regimes exist:

1. $X_L > X_C$ (Inductive regime): Net reactance is positive ($+jX$). Circuit voltage leads circuit current by phase angle $\theta = \arctan[(X_L - X_C)/R]$.
2. $X_C > X_L$ (Capacitive regime): Net reactance is negative ($-jX$). Circuit voltage lags circuit current by phase angle $\theta = -\arctan[(X_C - X_L)/R]$.
3. $X_L = X_C$ (Resonant condition): Net reactance cancels completely ($X = 0$). Impedance is purely resistive ($Z = R \angle 0^\circ$), current reaches its maximum possible value, and voltage is in phase with current.

Step-by-Step Worked Example: Series R-L-C Network at HF

An amateur radio transmitter output filter section contains a $40\ \Omega$ non-inductive resistor, a $1.50\ \mu\text{H}$ inductor, and a $600\text{ pF}$ mica capacitor connected in series across an RF source operating at $f = 7.10\text{ MHz}$ with an RMS terminal voltage of $50.0\text{ V}$.

Step 1: Calculate Inductive Reactance ($X_L$):

$$X_L = 2\pi f L = 2\pi \times (7.10 \times 10^6\text{ Hz}) \times (1.50 \times 10^{-6}\text{ H}) \approx 66.92\ \Omega$$

Step 2: Calculate Capacitive Reactance ($X_C$):

$$X_C = \frac{1}{2\pi f C} = \frac{1}{2\pi \times (7.10 \times 10^6\text{ Hz}) \times (600 \times 10^{-12}\text{ F})} \approx 37.36\ \Omega$$

Step 3: Determine Net Reactance ($X$):

$$X = X_L - X_C = 66.92\ \Omega - 37.36\ \Omega = +29.56\ \Omega \quad (\text{net inductive})$$

Step 4: Formulate Rectangular and Polar Impedance:

$$Z = 40 + j29.56\ \Omega$$ $$|Z| = \sqrt{R^2 + X^2} = \sqrt{40^2 + 29.56^2} = \sqrt{1600 + 873.79} = \sqrt{2473.79} \approx 49.74\ \Omega$$ $$\theta = \arctan\left(\frac{29.56}{40}\right) = \arctan(0.7390) \approx 36.46^\circ$$

The impedance in polar notation is $Z \approx 49.74\ \Omega \angle 36.46^\circ$.

Step 5: Determine Circuit Current and Individual Component Voltages:

$$I = \frac{V}{|Z|} = \frac{50.0\text{ V}}{49.74\ \Omega} \approx 1.005\text{ A RMS}$$

The voltage drops across each element are:

• $V_R = I \times R = 1.005\text{ A} \times 40\ \Omega = 40.20\text{ V}$
• $V_L = I \times X_L = 1.005\text{ A} \times 66.92\ \Omega = 67.25\text{ V}$
• $V_C = I \times X_C = 1.005\text{ A} \times 37.36\ \Omega = 37.55\text{ V}$

Verifying vector addition of voltages:

$$V_{\text{total}} = \sqrt{V_R^2 + (V_L - V_C)^2} = \sqrt{40.20^2 + (67.25 - 37.55)^2} = \sqrt{1616.04 + 882.09} \approx 50.0\text{ V}$$

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Parallel AC Circuits, Admittance, Conductance, and Susceptance

Analyzing parallel AC circuits using impedance requires reciprocal summation ($1/Z_{\text{total}} = 1/Z_1 + 1/Z_2 + \dots$), which becomes unwieldy when working with complex numbers. To streamline calculations, electrical engineers define Admittance ($Y$), the reciprocal of impedance:

$$Y = \frac{1}{Z} = G + jB$$

Admittance is measured in siemens ($\text{S}$, historically mhos, $\mho$). In rectangular form:

• Conductance ($G$): The real part of admittance, representing the ease with which in-phase current flows.
• Susceptance ($B$): The imaginary part of admittance, representing the ease with which alternating reactive current flows.

Converting Between Impedance and Admittance

Given an arbitrary branch impedance $Z = R + jX$, its admittance is:

$$Y = \frac{1}{R + jX} = \frac{R - jX}{(R + jX)(R - jX)} = \frac{R - jX}{R^2 + X^2} = \frac{R}{R^2 + X^2} - j\frac{X}{R^2 + X^2}$$

Equating real and imaginary parts:

$$G = \frac{R}{R^2 + X^2}$$ $$B = \frac{-X}{R^2 + X^2}$$

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Parallel-to-Series Equivalent Transformations

In RF filter design and impedance matching networks (such as Antenna Tuning Units and Pi-networks), engineers frequently transform a physical series combination ($R_s + jX_s$) into an equivalent parallel combination ($R_p \parallel jX_p$), or vice versa, at a single specified frequency.

Defining the circuit quality factor $Q$:

$$Q = \frac{|X_s|}{R_s} = \frac{R_p}{|X_p|}$$

The exact conversion relationships are:

$$R_p = R_s \left(1 + Q^2\right)$$ $$X_p = X_s \left(\frac{1 + Q^2}{Q^2}\right) = X_s \left(1 + \frac{1}{Q^2}\right)$$

Conversely, converting from parallel to series:

$$R_s = \frac{R_p}{1 + Q^2}$$ $$X_s = X_p \left(\frac{Q^2}{1 + Q^2}\right)$$

When $Q \ge 10$ (a standard high-Q circuit approximation):

• $R_p \approx Q^2 R_s$
• $X_p \approx X_s$

This approximation proves invaluable when determining the loaded impedance presented by tuned LC tanks to RF power amplifier tubes or transistors.

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Practical RF Engineering: Complex Antenna Feedpoint Impedance

In amateur radio transmission systems, the load presented to a coaxial transmission line by an antenna is rarely a pure $50\ \Omega$ resistance across an entire operating band. The antenna feedpoint exhibits complex impedance:

$$Z_{\text{ant}} = R_{\text{ant}} + jX_{\text{ant}} = (R_r + R_{\text{loss}}) + jX_{\text{ant}}$$

where:

• $R_r$ is radiation resistance, representing the equivalent resistance that accounts for power radiated into space as electromagnetic waves.
• $R_{\text{loss}}$ represents ohmic heat loss in wire conductors, dielectric losses in antenna insulators, and earth ground resistance.
• $X_{\text{ant}}$ is the residual reactive component.

When an antenna analyser reads $Z = 50 + j25\ \Omega$, the real part ($50\ \Omega$) matches standard coaxial characteristic impedance ($Z_0 = 50\ \Omega$), but the $+j25\ \Omega$ inductive component causes standing waves on the line, producing an SWR of approximately $1.6:1$. Inserting a series capacitor having $X_C = 25\ \Omega$ (where $C = 1 / (2\pi f \times 25)$) adds $-j25\ \Omega$, yielding $Z = 50 + j25 - j25 = 50 + j0\ \Omega$. This cancellation establishes a conjugate match, eliminating reflected power at the feedline junction.

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Comparison Table: Series vs. Parallel AC Circuit Formulations