From Ohm's Law to LC Resonance: The Circuit Math That Tunes Your Radio
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Chapter objectives: After finishing this chapter, you will be able to
- Apply Ohm's law and Kirchhoff's laws to compute voltage, current, and power, and derive the maximum power transfer theorem;
- Explain how capacitors and inductors work, compute time constants for RC and RL circuits, and estimate the inductance of an air-core coil using the Wheeler formula;
- Analyze AC circuits with complex impedance, distinguish real power from apparent power, and design an RC low-pass filter;
- Derive the resonant frequency, quality factor Q, and bandwidth for series and parallel resonance, and understand the "voltage magnification" of a resonant tank;
- Understand Q from an energy perspective, and solve the free oscillation of an RLC circuit numerically;
- Design the tuning circuit for a medium-wave crystal radio: choose a variable capacitor and work out how many coil turns are needed.
Prerequisites: Complex numbers, phasors, and decibels from Chapter 02. Sections 2, 3, and 7 use first- and second-order ordinary differential equations; you only need to verify that a function satisfies the equation.
The tuning knob on an old radio—what does it turn?
On a vintage radio, it turns a variable capacitor. The capacitor and a coil form a resonant tank: among the thousands of station signals arriving at the antenna simultaneously, only the one whose frequency equals the tank's resonant frequency gets through; all the others are blocked. Turning the knob changes the capacitance, which changes the resonant frequency, and thus changes the station.
That is the storyline of this chapter. We start with the most basic resistive circuits, gradually introduce capacitors and inductors, use the complex-number tools from Chapter 02 to analyze their behavior in AC circuits, and finally arrive at one of the most important circuits in radio: the LC resonant tank. Finally, we apply what we have learned to design the tuning circuit for the crystal radio we will build in Chapter 05.
1. Voltage, Current, and Resistance
1.1 Basic Physical Quantities
| Quantity | Symbol | Unit | Meaning |
|---|---|---|---|
| Charge | $q$ | Coulomb (C) | One electron carries $-1.602\times10^{-19}$ C |
| Current | $i = dq/dt$ | Ampere (A) | Charge passing through a cross-section per unit time |
| Voltage | $v$ | Volt (V) | Potential difference between two points, work done per unit charge |
| Power | $p = vi$ | Watt (W) | Energy consumed or delivered per unit time |
1.2 Ohm's Law
For a resistor, voltage and current are proportional:
$$ v = iR $$
The proportionality constant $R$ is the resistance, in ohms (Ω). The power consumed by a resistor is
$$ p = vi = i^2 R = \frac{v^2}{R} $$
All of this power becomes heat. A 100 W shortwave radio, fed from a 13.8 V DC supply, draws about 20 A while transmitting. If the power leads have only 0.04 Ω of resistance, that causes a 0.8 V drop and wastes 16 W as heat (Exercise 1). That is why radio power leads are so thick.
1.3 Kirchhoff's Laws
Two fundamental laws of circuit analysis:
- Kirchhoff's Current Law (KCL): The sum of currents entering a node equals the sum leaving it, because charge does not pile up at a node;
- Kirchhoff's Voltage Law (KVL): Along any closed loop, the algebraic sum of the voltage drops is zero, because electric potential is single-valued.
From these follow: series resistors $R = R_1 + R_2$; parallel resistors $\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2}$; and the voltage divider formula for series resistors $v_2 = v,\frac{R_2}{R_1 + R_2}$. All of these remain completely valid after we swap to complex impedance in Section 4.
1.4 Maximum Power Transfer Theorem
Any real signal source has internal resistance. Treat the source as an ideal voltage source $V_s$ in series with an internal resistance $R_s$, feeding a load $R_L$:
Figure 1: Left: a signal source with internal resistance driving a load. Right: load power and efficiency versus load resistance. Power is maximum when load equals internal resistance.
Loop current $I = \dfrac{V_s}{R_s + R_L}$, and load power is
$$ P_L = I^2 R_L = \frac{V_s^2,R_L}{(R_s + R_L)^2} $$
If $R_L$ is too small, current is high but the voltage across the load is small; if $R_L$ is too large, voltage is high but current is small. Somewhere in between the power is maximum. Differentiate with respect to $R_L$:
$$ \frac{dP_L}{dR_L} = V_s^2,\frac{(R_s + R_L)^2 - 2R_L(R_s + R_L)}{(R_s + R_L)^4} = V_s^2,\frac{R_s - R_L}{(R_s + R_L)^3} $$
Setting this to zero gives
$$ \boxed{\ R_L = R_s\quad\Longrightarrow\quad P_{L,\max} = \frac{V_s^2}{4R_s}\ } $$
Load resistance equal to the source internal resistance delivers maximum power to the load; this is called impedance matching. The appendix code uses $V_s = 100$ V, $R_s = 50$ Ω as an example:
| $R_L$ | 12.5 Ω | 25 Ω | 50 Ω | 100 Ω | 200 Ω |
|---|---|---|---|---|---|
| Load power | 32.00 W | 44.44 W | 50.00 W | 44.44 W | 32.00 W |
| Efficiency | 20.0 % | 33.3 % | 50.0 % | 66.7 % | 80.0 % |
Note two points:
- At matching, efficiency is only 50%, with half the power lost in the internal resistance. That is why power systems never aim for matching (they want efficiency). RF systems do aim for matching because signals are often very weak, every bit of power is precious: the signal picked up by the antenna must go as efficiently as possible into the receiver.
- Real transmitter output stages are not literally "a voltage source with a 50 Ω internal resistance". They are designed to deliver rated power and minimum distortion when loaded with 50 Ω. If the load deviates from 50 Ω, power drops, and in severe cases the output transistors can be damaged. This is why amateur stations require a low SWR on the antenna system; Chapter 08 discusses transmission lines in detail.
If the source impedance is complex $Z_s = R_s + jX_s$, the maximum power transfer condition becomes $Z_L = Z_s^* = R_s - jX_s$, known as conjugate matching: the load's reactance must cancel the source's reactance (Exercise 10 gives an example).
2. Capacitors
2.1 Definition and Parallel-Plate Capacitance
Two mutually insulated conductors, with a voltage $v$ applied, accumulate charges $+q$ and $-q$. Charge is proportional to voltage; the constant is the capacitance:
$$ C = \frac{q}{v} $$
Unit: Farad (F). 1 F is enormous; RF circuits typically use pF ($10^{-12}$ F) and nF ($10^{-9}$ F). For a parallel-plate capacitor with area $A$, separation $d$, and dielectric constant $\varepsilon = \varepsilon_r\varepsilon_0$:
$$ C = \frac{\varepsilon A}{d} $$
The appendix code calculates an air capacitor with area 10 cm² and separation 1 mm: only 8.854 pF. To get larger capacitance, increase area, reduce separation, or use a higher permittivity dielectric.
Parallel capacitors add directly; series capacitors combine as reciprocals (opposite of resistors): two capacitors of 100 pF and 220 pF in parallel give 320 pF, in series 68.75 pF.
2.2 Voltage-Current Relationship
Differentiating $q = Cv$:
$$ i = C,\frac{dv}{dt} $$
The current through a capacitor depends on the rate of change of voltage. DC voltage does not change, so a capacitor blocks DC; voltage cannot change abruptly (otherwise current would be infinite). The energy stored in a capacitor is
$$ W = \frac{1}{2}Cv^2 $$
2.3 RC Charging and Discharging
A voltage source $V$ charges an initially uncharged capacitor through a resistor $R$. By KVL, $V = iR + v_C = RC,\frac{dv_C}{dt} + v_C$. The solution to this first-order differential equation is
$$ v_C(t) = V\left(1 - e^{-t/\tau}\right),\qquad \tau = RC $$
$\tau$ is the time constant. Substituting back verifies the equation: $RC\cdot\frac{V}{\tau}e^{-t/\tau} + V - Ve^{-t/\tau} = V$. For discharging, $v_C = Ve^{-t/\tau}$.
Figure 2: RC charging and discharging. Each time constant, the remaining gap to the final value shrinks to 1/e of its previous size.
The appendix code computes (with $R = 10$ kΩ, $C = 100$ nF, $\tau = 1.0$ ms):
| Time | τ | 2τ | 3τ | 5τ |
|---|---|---|---|---|
| Charge level | 63.2 % | 86.5 % | 95.0 % | 99.3 % |
In engineering practice, we say a capacitor is fully charged after 5τ. Time constants are everywhere in radio: power supply filtering, the response speed of automatic gain control (AGC), and the rise time of a CW keyed waveform (as we'll see in Chapter 28, too steep an edge produces "key clicks" interference).
2.4 Capacitive Reactance
For sine signals, Chapter 02's phasor analysis already gave the capacitor's impedance:
$$ Z_C = \frac{1}{j\omega C} = -jX_C,\qquad X_C = \frac{1}{\omega C} $$
$X_C$ is the capacitive reactance. Higher frequency means lower reactance: capacitors "block DC and pass AC", and they impede high frequencies far less than low ones. The factor $-j$ means the capacitor's current leads the voltage by 90°. In RF circuits, coupling capacitors (letting the signal through while blocking DC bias) and bypass capacitors (shorting high-frequency interference on the power line to ground) both work on this principle.
3. Inductors
3.1 Definition
Current through a coil creates a magnetic field, and the magnetic flux is proportional to the current. When the current changes, the changing flux induces a voltage in the coil (Faraday's law), in a direction that always opposes the change (Lenz's law):
$$ v = L,\frac{di}{dt} $$
The constant $L$ is the inductance, in henries (H). RF circuits typically use μH and nH. The energy stored in an inductor is $W = \frac12 Li^2$. It is the dual of a capacitor: the current through an inductor cannot change abruptly. Abruptly breaking a circuit with an inductor produces a huge $di/dt$, hence a very high voltage; that is why a freewheeling diode is placed across a relay coil.
3.2 Air-Core Coil Inductance: the Wheeler Formula
Amateur radio operators often wind their own coils. The inductance of a single-layer, close-wound air-core coil can be estimated with the empirical formula given by Wheeler in 1928:
$$ L,[\mu\mathrm{H}] = \frac{r^2 N^2}{9r + 10,l} $$
where $N$ is the number of turns, $r$ is the coil radius, and $l$ is the winding length, all in inches (1 in = 25.4 mm). When $l > 0.8r$, the error is about 1%.
The appendix code gives: a 20 mm diameter, 20 mm long, 15-turn coil yields $L = 3.055$ μH. Winding 30 turns on the same former with the length doubled to 40 mm gives 7.231 μH, only 2.4 times larger, not the 4 times that $N^2$ alone would suggest, because the length increase enlarges the denominator as well. Only when the length stays fixed and turns are packed tighter does the inductance approach proportionality with $N^2$.
3.3 RL Circuits and Inductive Reactance
When an inductor and resistor are connected in series to a voltage source, the current rises as $i = \frac{V}{R}\left(1 - e^{-t/\tau}\right)$, with time constant $\tau = L/R$, following exactly the same pattern as RC charging (Figure 2). For example, 100 μH and 10 Ω give $\tau = 10.0$ μs.
For sine signals, the inductor's impedance is
$$ Z_L = j\omega L = jX_L,\qquad X_L = \omega L $$
$X_L$ is the inductive reactance. It increases with frequency: an inductor "passes DC and blocks AC". The factor $j$ means the inductor's current lags the voltage by 90°. A radio-frequency choke (RFC) is simply an inductor that passes the DC supply while blocking high-frequency signals.
Figure 3: Left: inductive reactance of 1 μH and capacitive reactance of 100 pF versus frequency (log-log). Right: phase relationship between voltage and current for an inductor and a capacitor.
The appendix code lists reactances at a few frequencies:
| Frequency | 1 MHz | 7.1 MHz | 30 MHz | 145 MHz |
|---|---|---|---|---|
| $X_L$ (1 μH) | 6.28 Ω | 44.61 Ω | 188.50 Ω | 911.06 Ω |
| $X_C$ (100 pF) | 1591.55 Ω | 224.16 Ω | 53.05 Ω | 10.98 Ω |
The two lines intersect at 15.915 MHz, where both reactances are 100 Ω. This intersection is the resonant frequency of Section 5.
4. Complex Impedance and AC Circuits
4.1 Impedance
Resistors, capacitors, and inductors are unified using complex impedance:
$$ Z = R + jX $$
The real part $R$ is resistance, the imaginary part $X$ is reactance: $X > 0$ is inductive, $X < 0$ is capacitive. Phasor Ohm's law $\mathbf{V} = Z\mathbf{I}$, as well as the series/parallel formulas and the voltage divider, all work exactly as in DC circuits, just with complex arithmetic. The reciprocal of impedance is admittance $Y = 1/Z = G + jB$; it is often more convenient for parallel circuits.
4.2 Two Examples
50 Ω resistor in series with 1 μH inductor, at 7.1 MHz:
$$ Z = 50 + j\omega L = 50 + j44.61\ \Omega = 67.01\angle 41.74°\ \Omega $$
50 Ω resistor in parallel with 100 pF capacitor, at 7.1 MHz:
$$ Z = \frac{50\cdot(-j224.16)}{50 - j224.16} = 47.63 - j10.62\ \Omega = 48.8\angle -12.57°\ \Omega $$
Figure 4: The impedance plane. Horizontal axis is resistance, vertical is reactance; the upper half-plane is inductive, the lower is capacitive.
The impedance plane is the basis for understanding matching networks and Smith charts (Chapter 08). For example, if an antenna analyzer measures an antenna's impedance as $50 + j44.61$ Ω, you know it is "inductive": inserting a suitable capacitor to cancel the $j44.61$ turns it into a pure 50 Ω (Exercise 10).
4.3 AC Power
Apply 10 V (RMS) to the series circuit above; the current is $\mathbf{I} = \mathbf{V}/Z = 0.1492\angle -41.74°$ A, lagging the voltage by 41.74°. The appendix code computes:
- Real power $P = |I|^2 R = 1.114$ W: only the resistor consumes energy;
- Apparent power $S = |V||I| = 1.492$ VA;
- Power factor $\cos\varphi = P/S = 0.746$.
Inductors and capacitors consume no energy. They absorb energy during half a cycle and return it in the next half; the power that sloshes back and forth is called reactive power. So an ideal inductor and capacitor do not heat up. Real components always have some loss, which we will quantify with the quality factor Q below.
4.4 The First Filter: RC Low-Pass
Figure 5: RC low-pass filter and its Bode plot. At the cutoff frequency, attenuation is −3 dB and phase is −45°; above cutoff, attenuation is −20 dB per decade.
Place the resistor and capacitor in series and take the output across the capacitor. From the voltage divider:
$$ H(j\omega) = \frac{\mathbf{V}{out}}{\mathbf{V}{in}} = \frac{1/(j\omega C)}{R + 1/(j\omega C)} = \frac{1}{1 + j\omega RC} $$
$|H| = 1/\sqrt{1 + (\omega RC)^2}$. At low frequencies $|H| \approx 1$, the signal passes directly; at high frequencies the capacitor approximates a short, and the signal is attenuated. When $\omega RC = 1$, $|H| = 1/\sqrt2$, i.e. −3.01 dB, phase −45°, and the corresponding frequency is the cutoff frequency:
$$ f_c = \frac{1}{2\pi RC} $$
The appendix code uses $R = 1$ kΩ, $C = 47$ nF, so $f_c = 3386.3$ Hz:
| Frequency | 338.6 Hz ($f_c/10$) | 3000 Hz | 3386.3 Hz ($f_c$) | 33863 Hz ($10f_c$) |
|---|---|---|---|---|
| $\lvert H\rvert$ | −0.04 dB | −2.52 dB | −3.01 dB | −20.04 dB |
| Phase | −5.71° | −41.54° | −45.00° | −84.29° |
Such a filter can sit on the audio output of a receiver to suppress "hiss" above 3 kHz. But you also see its limitation: the roll-off is gentle. At 3 kHz it has already lost 2.5 dB, and only reaches −20 dB at ten times the cutoff. To get a steep filter, you need higher-order LC filters (Chapter 16) or digital filters (Chapter 20).
5. Series Resonance
5.1 Resonant Frequency
With a resistor, inductor, and capacitor in series, the total impedance is
$$ Z = R + j\left(\omega L - \frac{1}{\omega C}\right) $$
Inductive reactance rises with frequency, capacitive reactance falls (Figure 3). There is always a frequency where the two are equal and cancel each other, leaving zero reactance:
$$ \omega_0 L = \frac{1}{\omega_0 C}\quad\Longrightarrow\quad \boxed{\ f_0 = \frac{1}{2\pi\sqrt{LC}}\ } $$
This is the resonant frequency. At resonance the impedance is minimum, equal to the pure resistance $R$, and the current is maximum, in phase with the voltage. Below resonance, capacitive reactance dominates and the circuit is capacitive; above resonance, inductive reactance dominates and it is inductive.
For the 40-meter band, take $L = 2.2$ μH; to resonate at 7.1 MHz,
$$ C = \frac{1}{\omega_0^2 L} = 228.40\ \mathrm{pF} $$
5.2 Quality Factor and Bandwidth
How sharp the resonance curve is, is measured by the quality factor $Q$:
$$ Q = \frac{\omega_0 L}{R} = \frac{1}{\omega_0 C R} = \frac{1}{R}\sqrt{\frac{L}{C}} $$
$\sqrt{L/C}$ is the magnitude of the inductive (and capacitive) reactance at resonance, called the characteristic impedance of the circuit. With $R = 5$ Ω, the appendix code computes $\sqrt{L/C} = 98.14$ Ω, so $Q = 19.63$.
Figure 6: Series RLC resonance curves (top) and current phase (bottom). Smaller R gives higher Q and a sharper curve.
Bandwidth is defined as the difference between the two frequencies where the current drops to $1/\sqrt2$ of its maximum (power drops to half, i.e. −3 dB). This happens when $|Z| = \sqrt2 R$, i.e. when the magnitude of the reactance equals the resistance:
$$ \omega L - \frac{1}{\omega C} = \pm R\quad\Longrightarrow\quad \omega^2 L \mp R\omega - \frac{1}{C} = 0 $$
Taking the positive root:
$$ \omega_{1,2} = \frac{\mp R + \sqrt{R^2 + 4L/C}}{2L} $$
Subtracting the two roots cancels the square root term:
$$ \omega_2 - \omega_1 = \frac{R}{L}\quad\Longrightarrow\quad \boxed{\ B = f_2 - f_1 = \frac{R}{2\pi L} = \frac{f_0}{Q}\ } $$
For a series RLC circuit, this result is exact. Multiplying the two roots gives $\omega_1\omega_2 = 1/(LC) = \omega_0^2$, meaning the resonant frequency is the geometric mean of the two half-power frequencies. The appendix code scans 2 million points between 6 and 8.2 MHz and finds the half-power points at 6.9214 MHz and 7.2832 MHz, giving a bandwidth of 361.7 kHz, exactly matching $f_0/Q = 361.7$ kHz.
Higher Q means better selectivity: with Q = 19.6 the bandwidth is 362 kHz, nearly the whole 40-meter band; with Q = 98 the bandwidth is only about 72 kHz.
5.3 Voltage Magnification
At resonance, the current is $I = V_s/R$, so the voltage across the inductor is
$$ |V_L| = I\cdot\omega_0 L = \frac{V_s}{R},\omega_0 L = Q,V_s $$
The capacitor also has $QV_s$ across it, but with opposite phase, so the two cancel and the total voltage remains $V_s$. The appendix code shows that with a 1 V source, the inductor and capacitor each have 19.63 V across them.
This is both a blessing and a hazard. The good side: a resonant tank has natural "voltage gain", and the tuned circuit at the receiver front-end can amplify a weak antenna signal by Q times. The hazard: components in a high-Q circuit have to withstand high voltage. A magnetic loop antenna (Chapter 11) commonly used by amateurs can have Q in the hundreds; when fed with 100 W, the tuning capacitor sees several kilovolts (Exercise 6). It must be a vacuum capacitor with sufficient rating or an air capacitor with wide spacing, and you must never touch it during tuning.
6. Parallel Resonance
6.1 Ideal Case
With an inductor and capacitor in parallel, the total admittance is
$$ Y = j\omega C + \frac{1}{j\omega L} = j\left(\omega C - \frac{1}{\omega L}\right) $$
At the same $\omega_0 = 1/\sqrt{LC}$, $Y = 0$, so the impedance is infinite. This is the opposite of series resonance: at series resonance the impedance is minimum, at parallel resonance it is maximum.
6.2 Parallel Tank with Loss
A real inductor coil always has resistance $r$ (wire resistance, plus skin effect at high frequencies). The inductor branch $r + j\omega L$ is in parallel with the capacitor:
$$ Z = \frac{(r + j\omega L)\cdot\dfrac{1}{j\omega C}}{r + j\omega L + \dfrac{1}{j\omega C}} = \frac{r + j\omega L}{1 - \omega^2 LC + j\omega Cr} $$
At $\omega_0 = 1/\sqrt{LC}$, the real part of the denominator is $1 - \omega_0^2 LC = 0$; if $Q \gg 1$, the $r$ in the numerator can also be neglected:
$$ Z(\omega_0) \approx \frac{j\omega_0 L}{j\omega_0 Cr} = \frac{L}{Cr} = Q^2 r $$
This impedance is called the resonant impedance (dynamic resistance) of the parallel tank. Using the same $L = 2.2$ μH, $C = 228.40$ pF, $r = 5$ Ω, the appendix code finds the formula gives 1926.4 Ω and the numerical maximum is 1928.9 Ω, a difference under 0.2%. A coil with only 5 Ω of loss, when paralleled with a capacitor, presents nearly 2 kΩ at resonance. The code also shows that exactly at $1/(2\pi\sqrt{LC})$, the impedance has a phase of −2.92°, not purely resistive: with loss, the three conditions "maximum impedance", "pure resistance", and "$1/\sqrt{LC}$" no longer coincide exactly. Higher Q brings them closer together.
Figure 7: Impedance of a parallel LC tank with loss. At resonance the impedance is maximum, about L/(Cr); higher Q gives a sharper peak.
6.3 Series vs. Parallel Resonance
The below table compares the two:
| Series Resonance | Parallel Resonance | |
|---|---|---|
| Impedance at resonance | Minimum, equal to $R$ | Maximum, approximately $Q^2 r$ |
| Magnified quantity | Voltage on components ($Q$ times) | Circulating current in the tank ($Q$ times) |
| Off resonance | Impedance increases rapidly | Impedance decreases rapidly |
| Typical uses | In series with the signal path, passing only the resonant frequency; a trap to ground | Across signal and ground, showing high impedance only at resonance; the "tank" of oscillators, IF transformers, and tuned circuits |
A parallel resonant circuit is also called a tank circuit, because energy sloshes back and forth between the inductor's magnetic field and the capacitor's electric field, like water sloshing in a tank. A trap in an amateur antenna is a parallel LC: placed in series with the element of a dipole, it presents a high impedance at its resonant frequency, effectively "cutting" the antenna, so one antenna can work on two bands.
7. Energy Meaning of Q: Free Oscillation
7.1 Energy Definition of Q
The more fundamental definition of Q is
$$ Q = 2\pi\times\frac{\text{energy stored in the circuit}}{\text{energy lost per cycle}} $$
To verify it matches $\omega_0 L/R$: in a series resonance, let the peak current be $I_p$. The stored energy transfers between the inductor and capacitor, and the total equals the inductor's maximum stored energy $\frac12 L I_p^2$. The resistor dissipates $I_{\mathrm{rms}}^2 R,T = \frac12 I_p^2 R\cdot\frac{2\pi}{\omega_0}$ per cycle. The ratio multiplied by $2\pi$ is exactly $\omega_0 L/R$.
A high Q means only a small fraction of stored energy is lost each cycle, and the circuit can "ring" by itself for a long time.
7.2 Free Oscillation
Charge a capacitor to 1 V, then disconnect the source and let the RLC circuit oscillate freely. By KVL:
$$ L\frac{d^2q}{dt^2} + R\frac{dq}{dt} + \frac{q}{C} = 0 $$
When $Q > \frac12$, the solution is a damped oscillation:
$$ v_C(t) = e^{-\alpha t}\left(\cos\omega_d t + \frac{\alpha}{\omega_d}\sin\omega_d t\right),\qquad \alpha = \frac{R}{2L} = \frac{\omega_0}{2Q},\quad \omega_d = \omega_0\sqrt{1 - \frac{1}{4Q^2}} $$
The envelope decays as $e^{-\omega_0 t/2Q}$. Each cycle, the amplitude is multiplied by $e^{-\pi/Q}$. So after about Q cycles, the amplitude decays to $e^{-\pi} \approx 4.3%$. This gives Q an intuitive meaning: a high-Q circuit is like a good bell; one strike rings for a long time.
Figure 8: Free oscillation of an RLC circuit (7.1 MHz, Q = 19.63).
7.3 Numerical Experiment
The appendix code does not use the analytic solution above; instead it integrates the differential equation directly with a fourth-order Runge-Kutta (RK4) method, taking 400 steps per cycle, and then "measures" Q from the resulting waveform: the logarithm of the ratio of two adjacent positive peaks is the logarithmic decrement $\delta$, from which $\delta \approx \pi/Q$ gives Q. Results:
| Quantity | Value |
|---|---|
| Logarithmic decrement δ | 0.16010 |
| Q measured from decay = π/δ | 19.62 |
| Formula $\omega_0 L/R$ | 19.63 |
| Peak after 5 cycles (predicted by $e^{-n\pi/Q}$) | 0.4491 V (0.4492 V) |
| Peak after 10 cycles | 0.2017 V (0.2018 V) |
| Peak after 20 cycles | 0.0407 V (0.0407 V) |
The numerical solution agrees with theory to four significant digits. This method of "write the differential equation, integrate numerically, then measure a parameter from the result" will appear again in later chapters (for example, Chapter 07 uses FDTD to solve Maxwell's equations).
7.4 Typical Q Values for Components
| Component | Approximate Q |
|---|---|
| Air-core and ferrite-core coils on HF bands | 100～300 |
| High-quality ceramic capacitors, air variable capacitors | 1000 or more |
| Quartz crystals | $10^4$～$10^6$ |
| Microwave cavities | $10^3$～$10^4$ |
The Q of an LC tank is usually set by the coil. Quartz crystals have Q values several orders of magnitude higher than LC tanks, so oscillators made with them are extremely frequency-stable, and filters made with them can have bandwidths as narrow as a few hundred hertz, which is exactly what is needed to receive CW and SSB signals.
8. In Practice: Designing the Tuning Circuit of a Medium-Wave Crystal Radio
In Chapter 05 we will build a crystal radio. Its core is a parallel LC tuning tank. Here we design it now.
Requirements: cover the medium-wave broadcast band from 530 to 1600 kHz.
Step 1: Capacitance range. Pick a common inductance value $L = 240$ μH. From $C = 1/(\omega_0^2 L)$, the appendix code computes the required capacitance: from 41.2 pF (at 1600 kHz) to 375.7 pF (at 530 kHz). The ratio is 9.11, exactly $(1600/530)^2$: since $f_0 \propto 1/\sqrt{C}$, the frequency coverage ratio equals the square root of the capacitance change ratio.
Step 2: Choose a variable capacitor. The common single-section variable capacitor in radios has a maximum of about 365 pF. Its minimum capacitance, plus the stray capacitance of the coil and wiring, totals between 15 and 25 pF. The appendix code shows that with a 240 μH coil, the tuning range is 538 to 2055 kHz (with 25 pF minimum) or 538 to 2653 kHz (with 15 pF minimum). There is plenty of headroom at the high end; the low end at 538 kHz is slightly above 530 kHz, and adding about 10 pF of stray capacitance is enough to fix it (Exercise 8).
Figure 9: Resonant frequency of a 240 μH coil with a variable capacitor. The frequency coverage ratio equals the square root of the capacitance ratio.
Step 3: Wind the coil. On a 50 mm diameter former (for example, a piece of PVC pipe), use 0.3 mm enameled wire close-wound; each turn takes about 0.35 mm of length. The coil length $l = 0.35N$ mm changes with the number of turns. Rearrange the Wheeler formula into a quadratic equation in $N$:
$$ r^2 N^2 - 10,L,p,N - 9,L,r = 0 $$
where $p$ is the length per turn, all lengths in inches, and $L$ in μH. The appendix code solves this to $N = 66.9$ turns, a winding length of 23.4 mm, and substituting back into Wheeler's formula gives $L = 240.0$ μH. In practice, wind 67 turns. The actual thickness of the enameled wire and how tightly it is wound will shift the inductance by a few percent; fortunately the variable capacitor's range has enough headroom to cover that error.
Thus we have designed a tuning tank that can receive the entire medium-wave broadcast band. In Chapter 05 we will add the antenna, ground, detector diode, and headphones to make a radio that needs no battery.
Summary
- Ohm's law $v = iR$, and power $p = i^2R = v^2/R$. Kirchhoff's current and voltage laws are the foundation of circuit analysis.
- Maximum power transfer: when the load resistance equals the source internal resistance, the load gets maximum power, $V_s^2/4R_s$, at 50% efficiency. For complex impedances, conjugate matching $Z_L = Z_s^*$ is required.
- Capacitor $i = C,dv/dt$; voltage cannot change abruptly; it blocks DC and passes AC; capacitive reactance $X_C = 1/\omega C$; current leads voltage by 90°. Parallel-plate capacitance $C = \varepsilon A/d$.
- Inductor $v = L,di/dt$; current cannot change abruptly; it passes DC and blocks AC; inductive reactance $X_L = \omega L$; current lags voltage by 90°. A single-layer air-core coil can be estimated with the Wheeler formula $L = r^2N^2/(9r + 10l)$ (inches, μH).
- Time constant: RC circuit $\tau = RC$, RL circuit $\tau = L/R$; after τ it reaches 63.2% of the final value, after 5τ it reaches 99.3%.
- Complex impedance $Z = R + jX$ lets AC circuits be computed like DC circuits. Only resistors consume real power; the RC low-pass has cutoff $f_c = 1/(2\pi RC)$, then attenuates at 20 dB per decade.
- Resonance: $f_0 = 1/(2\pi\sqrt{LC})$. Series resonance has minimum impedance (= R) and magnifies component voltages by Q; parallel resonance has maximum impedance (≈ L/Cr = Q²r). Bandwidth $B = f_0/Q$.
- Quality factor $Q = \omega_0L/R = 2\pi\times$stored energy/energy lost per cycle. A free oscillation decays to $e^{-\pi}$ after about Q cycles.
- Frequency coverage ratio equals the square root of the tuning capacitance ratio; a 240 μH coil with a 365 pF variable capacitor covers the medium-wave broadcast band.
Next chapter, we change perspective on signals: from "waveforms that change over time" to "which frequencies compose them" — that is Fourier analysis and spectra.
Exercises
- (Power lead) A shortwave radio draws 20 A from a 13.8 V supply when transmitting 100 W. (a) What is the overall efficiency of the radio? (b) If the power leads are 2 m each way and have 0.01 Ω/m, what is the voltage drop and heat loss in the leads?
- (Mismatch loss) A 50 Ω source is connected to a 75 Ω load (for example, mistakenly using a TV 75 Ω cable). What fraction of the maximum available power reaches the load? How many dB are lost?
- (Inductive reactance) At what frequency does a 10 μH inductor have a reactance of 50 Ω?
- (Series resonance) Use a 1.5 μH inductor to make a series resonant circuit at 14.2 MHz. What capacitance is needed? What is the characteristic impedance of the circuit?
- (Selectivity) A parallel tuned circuit resonates at 3.6 MHz with Q = 150. (a) What is its bandwidth? (b) A nearby station 10 kHz away will be attenuated by how many dB? Can a single such circuit separate the two stations?
- (Voltage magnification) A series resonant circuit with Q = 200 is driven by 10 V (RMS). What is the RMS and peak voltage across the capacitor at resonance?
- (Programming) Change the resistor in part 9 of the appendix code from 5 Ω to 1 Ω and increase the number of oscillation cycles to 120. What Q do you measure? After 98 cycles, what remains of the peak?
- (Tuning range) In the design in Section 8, if the variable capacitor has a maximum of 365 pF plus about 10 pF of stray capacitance, what is the low end of the tuning range?
- (Design an RC low-pass) With a 47 nF capacitor, what resistor is needed for a 3000 Hz cutoff RC low-pass filter?
- (Conjugate matching preview) The $50 + j44.61$ Ω circuit in Section 4 (at 7.1 MHz) is to be made a pure 50 Ω resistor. What series capacitance is needed?
Answers
- (a) Input power $13.8 \times 20 = 276$ W, efficiency $100/276 = $ 36.2%, the other about 176 W becomes heat, so the radio needs a heat sink and fan. (b) Round trip total 4 m, resistance 0.04 Ω, voltage drop $20 \times 0.04 = $ 0.8 V, heat $20^2 \times 0.04 = $ 16 W. The radio actually gets only 13.0 V, which may be below its rated operating voltage.
- Load power as a fraction of maximum is $\dfrac{4R_sR_L}{(R_s + R_L)^2} = \dfrac{4\times50\times75}{125^2} = $ 0.96%? Wait, that is 96%. The loss is 0.18 dB. A slight mismatch loses little; severe mismatch (like the extremes in Figure 1) is what matters.
- $f = \dfrac{X_L}{2\pi L} = \dfrac{50}{2\pi\times10^{-5}} = $ 795.8 kHz.
- $C = \dfrac{1}{(2\pi\times14.2\times10^6)^2\times1.5\times10^{-6}} = $ 83.75 pF, characteristic impedance $\sqrt{L/C} = $ 133.8 Ω.
- (a) $B = 3.6\ \mathrm{MHz}/150 = $ 24 kHz. (b) For a high-Q circuit, the relative response at offset $\Delta f$ from resonance is about $1/\sqrt{1 + (2Q\Delta f/f_0)^2}$. $2\times150\times10/3600 = 0.833$, so attenuation is $10\lg(1 + 0.833^2) = $ 2.29 dB (the full parallel circuit formula gives 2.29 dB). That is a reduction of only 2 dB, far from enough to separate the two stations. This is why a simple tuned circuit has poor selectivity; a superheterodyne receiver converts the signal to a fixed intermediate frequency and then uses crystal or ceramic filters (Chapter 15).
- $V_C = QV_s = $ 2000 V (RMS), peak $2000\sqrt2 = $ 2828 V.
- Measured $Q = $ 98.14 (formula value $\omega_0L/R = 98.14$). After 98 cycles the peak is 0.0434 V, close to $e^{-\pi} = 0.0432$.
- With $C_{\max} = 375$ pF, $f = \dfrac{1}{2\pi\sqrt{240\times10^{-6}\times375\times10^{-12}}} = $ 530.5 kHz, just covering the low end of the medium wave band.
- $R = \dfrac{1}{2\pi f_c C} = \dfrac{1}{2\pi\times3000\times47\times10^{-9}} = $ 1129 Ω. Use a standard value of 1.1 kΩ (cutoff about 3.08 kHz) or 1.2 kΩ (about 2.82 kHz).
- Need $X_C = 44.61$ Ω, so $C = \dfrac{1}{2\pi\times7.1\times10^6\times44.61} = $ 502.5 pF. After adding this in series, $Z = 50 + j44.61 - j44.61 = 50$ Ω. Note this only holds at 7.1 MHz; at a different frequency, the reactances will no longer balance. Chapter 08 will treat impedance matching systematically.
References
- W. H. Hayt, J. E. Kemmerly, J. D. Phillips, S. M. Durbin. Engineering Circuit Analysis. 9th ed. McGraw-Hill, 2018. Chapters 2–10 and 16 (frequency response and resonance).
- J. B. Hagen. Radio-Frequency Electronics: Circuits and Applications. 2nd ed. Cambridge University Press, 2009. Sections on resonant circuits and Q.
- P. Horowitz, W. Hill. The Art of Electronics. 3rd ed. Cambridge University Press, 2015. Chapter 1.
- W. Hayward, R. Campbell, B. Larkin. Experimental Methods in RF Design. ARRL, 2003. Relevant sections on filters and resonant circuits.
- H. A. Wheeler. Simple Inductance Formulas for Radio Coils. Proceedings of the IRE, 16(10): 1398–1400, 1928.
- ARRL. The ARRL Handbook for Radio Communications. Latest edition. Section "Electrical Fundamentals".
Appendix: Full Code
The following code reproduces every number in this chapter: maximum power transfer, parallel-plate capacitance and RC time constant, Wheeler's formula, reactance, complex impedance and AC power, RC low-pass, Q and bandwidth of series resonance (with numerical verification), parallel resonance impedance, free oscillation solved with RK4 and Q measured from it, and the medium-wave crystal radio tuning circuit design. It then plots the series RLC resonance curve. It depends only on numpy and matplotlib, save as example.py and run python example.py.
"""Radio Self-Study 03: Circuit Foundations — Appendix Code
1. Ohm's law and maximum power transfer: 50 Ω source with different loads;
2. Capacitors: parallel-plate capacitance, RC charging time constant;
3. Inductors: Wheeler formula for single-layer air-core coil, RL time constant;
4. Reactance: 1 μH and 100 pF at 7.1 MHz, and the frequency where they are equal;
5. Complex impedance: 50 Ω series 1 μH, 50 Ω parallel 100 pF, AC power and power factor;
6. RC low-pass: cutoff frequency, -3 dB and -45°;
7. Series resonance: resonant frequency, Q, bandwidth (numerical verification), voltage magnification;
8. Parallel resonance: numerical verification of resonant impedance L/(Cr);
9. Free oscillation: numerically solve RLC circuit decay, measure Q from logarithmic decrement;
10. Medium-wave crystal radio tuning circuit: variable capacitor range and coil turns.
Finally plot the series RLC resonance curve.
Run: python example.py
"""
import numpy as np
import matplotlib.pyplot as plt
plt.rcParams["font.sans-serif"] = ["Microsoft YaHei", "SimHei", "DejaVu Sans"]
plt.rcParams["axes.unicode_minus"] = False
EPS0 = 8.8541878128e-12 # vacuum permittivity [F/m]
INCH = 25.4e-3 # 1 inch [m]
def polar_str(z, unit=""):
return f"{abs(z):.4g}{unit}∠{np.degrees(np.angle(z)):.2f}°"
def rect_str(z, unit=""):
sign = "+" if z.imag >= 0 else "-"
return f"{z.real:.4g} {sign} j{abs(z.imag):.4g}{unit}"
def wheeler_uh(d, length, n):
"""Inductance of a single-layer air-core coil [μH]. d: coil diameter [m], length: winding length [m], n: turns."""
r, l = d / 2 / INCH, length / INCH
return r**2 * n**2 / (9 * r + 10 * l)
def resonant_c(f, L):
return 1 / ((2 * np.pi * f) ** 2 * L)
# ---------------- 1. Ohm's law and maximum power transfer ----------------
print("=== 1. Maximum power transfer: open-circuit voltage 100 V (RMS), internal resistance 50 Ω ===")
VS, RS = 100.0, 50.0
for rl in (12.5, 25.0, 50.0, 100.0, 200.0):
i = VS / (RS + rl)
p_load = i**2 * rl
eff = rl / (RS + rl)
print(f"R_L = {rl:6.1f} Ω: current {i:.3f} A, load power {p_load:6.2f} W, efficiency {eff * 100:5.1f} %")
print(f"Theoretical maximum V^2/(4R_s) = {VS**2 / (4 * RS):.2f} W")
# ---------------- 2. Capacitors ----------------
print("\n=== 2. Capacitors ===")
A, d = 10e-4, 1e-3
print(f"Parallel plate: area 10 cm^2, gap 1 mm, air: C = εA/d = {EPS0 * A / d * 1e12:.3f} pF")
R, Cc = 10e3, 100e-9
tau = R * Cc
print(f"RC charging: R = 10 kΩ, C = 100 nF, τ = RC = {tau * 1e3:.1f} ms")
for k in (1, 2, 3, 5):
print(f" t = {k}τ: capacitor voltage reaches {(1 - np.exp(-k)) * 100:.1f} % of supply")
c1, c2 = 100e-12, 220e-12
print(f"100 pF and 220 pF: parallel {(c1 + c2) * 1e12:.1f} pF, series {c1 * c2 / (c1 + c2) * 1e12:.2f} pF")
# ---------------- 3. Inductors ----------------
print("\n=== 3. Inductors ===")
L_coil = wheeler_uh(20e-3, 20e-3, 15)
print(f"Wheeler formula: diameter 20 mm, length 20 mm, 15 turns air-core coil L = {L_coil:.3f} μH")
print(f"Same coil wound with 30 turns (length also doubled): L = {wheeler_uh(20e-3, 40e-3, 30):.3f} μH")
print(f"RL circuit: L = 100 μH, R = 10 Ω, τ = L/R = {100e-6 / 10 * 1e6:.1f} μs")
# ---------------- 4. Reactance ----------------
print("\n=== 4. Reactance X_L = ωL, X_C = 1/(ωC) ===")
L1, C1 = 1e-6, 100e-12
for f in (1e6, 7.1e6, 30e6, 145e6):
w = 2 * np.pi * f
print(f"f = {f / 1e6:6.1f} MHz: X_L(1 μH) = {w * L1:8.2f} Ω, X_C(100 pF) = {1 / (w * C1):8.2f} Ω")
print(f"Frequency where they are equal f = 1/(2π sqrt(LC)) = {1 / (2 * np.pi * np.sqrt(L1 * C1)) / 1e6:.3f} MHz")
# ---------------- 5. Complex impedance and AC power ----------------
print("\n=== 5. Complex impedance (f = 7.1 MHz) ===")
w = 2 * np.pi * 7.1e6
z_series = 50 + 1j * w * L1
print(f"50 Ω series 1 μH: Z = {rect_str(z_series, ' Ω')} = {polar_str(z_series, ' Ω')}")
V = 10.0
I = V / z_series
print(f" Driven by 10 V (RMS): I = {polar_str(I, ' A')}")
print(f" Real power P = |I|^2 R = {abs(I) ** 2 * 50:.3f} W, "
f"apparent power S = |V||I| = {V * abs(I):.3f} VA, power factor cos φ = {np.cos(np.angle(z_series)):.3f}")
zc = 1 / (1j * w * C1)
z_par = 50 * zc / (50 + zc)
print(f"50 Ω parallel 100 pF: Z = {rect_str(z_par, ' Ω')} = {polar_str(z_par, ' Ω')}")
# ---------------- 6. RC low-pass ----------------
print("\n=== 6. RC low-pass: R = 1 kΩ, C = 47 nF ===")
R6, C6 = 1e3, 47e-9
fc = 1 / (2 * np.pi * R6 * C6)
print(f"Cutoff frequency f_c = 1/(2πRC) = {fc:.1f} Hz")
for f in (fc / 10, fc, 3e3, 10 * fc):
H = 1 / (1 + 1j * 2 * np.pi * f * R6 * C6)
print(f" f = {f:8.1f} Hz: |H| = {20 * np.log10(abs(H)):7.2f} dB, phase {np.degrees(np.angle(H)):7.2f}°")
# ---------------- 7. Series resonance ----------------
print("\n=== 7. Series RLC: L = 2.2 μH, resonate at 7.1 MHz, R = 5 Ω ===")
L7, R7, F0 = 2.2e-6, 5.0, 7.1e6
C7 = resonant_c(F0, L7)
w0 = 2 * np.pi * F0
Q7 = w0 * L7 / R7
print(f"C = 1/(ω0^2 L) = {C7 * 1e12:.2f} pF")
print(f"Q = ω0 L / R = {Q7:.2f}, also sqrt(L/C)/R = {np.sqrt(L7 / C7) / R7:.2f}")
print(f"Characteristic impedance sqrt(L/C) = {np.sqrt(L7 / C7):.2f} Ω, bandwidth f0/Q = {F0 / Q7 / 1e3:.1f} kHz")
f = np.linspace(6e6, 8.2e6, 2_000_001)
Z = R7 + 1j * (2 * np.pi * f * L7 - 1 / (2 * np.pi * f * C7))
Imag = 1 / np.abs(Z)
band = f[Imag >= Imag.max() / np.sqrt(2)]
print(f"Numerical check: current maximum at {f[np.argmax(Imag)] / 1e6:.4f} MHz, "
f"half-power points {band[0] / 1e6:.4f} ~ {band[-1] / 1e6:.4f} MHz, bandwidth {(band[-1] - band[0]) / 1e3:.1f} kHz")
print(f"Signal source 1 V (RMS): at resonance current {1 / R7:.2f} A, voltage across inductor and capacitor each {Q7:.2f} V")
# ---------------- 8. Parallel resonance ----------------
print("\n=== 8. Parallel resonance: same L, C, inductor loss r = 5 Ω ===")
r8 = 5.0
zp = lambda f: (r8 + 1j * 2 * np.pi * f * L7) / (1 - (2 * np.pi * f) ** 2 * L7 * C7 + 1j * 2 * np.pi * f * C7 * r8)
Zp = zp(f)
k = np.argmax(np.abs(Zp))
print(f"Formula L/(C r) = {L7 / (C7 * r8):.1f} Ω, Q^2 r = {Q7**2 * r8:.1f} Ω")
print(f"Numerical: |Z| max {abs(Zp[k]):.1f} Ω at {f[k] / 1e6:.4f} MHz; "
f"at 1/(2π sqrt(LC)) |Z| = {abs(zp(F0)):.1f} Ω, phase {np.degrees(np.angle(zp(F0))):.2f}°")
# ---------------- 9. Free oscillation and Q ----------------
print("\n=== 9. Free oscillation: capacitor charged to 1 V, then let the RLC circuit ring ===")
def ring_down(L, C, R, cycles=40, steps_per_cycle=400):
"""RK4 solve L di/dt + R i + q/C = 0, return time and capacitor voltage."""
T0 = 2 * np.pi * np.sqrt(L * C)
dt = T0 / steps_per_cycle
n = int(cycles * steps_per_cycle)
q, i = C * 1.0, 0.0
vc = np.empty(n)
def deriv(q, i):
return i, -(R * i + q / C) / L
for m in range(n):
vc[m] = q / C
k1 = deriv(q, i)
k2 = deriv(q + dt / 2 * k1[0], i + dt / 2 * k1[1])
k3 = deriv(q + dt / 2 * k2[0], i + dt / 2 * k2[1])
k4 = deriv(q + dt * k3[0], i + dt * k3[1])
q += dt / 6 * (k1[0] + 2 * k2[0] + 2 * k3[0] + k4[0])
i += dt / 6 * (k1[1] + 2 * k2[1] + 2 * k3[1] + k4[1])
return np.arange(n) * dt, vc
t9, vc9 = ring_down(L7, C7, R7)
peaks = [0] + [m for m in range(1, len(vc9) - 1)
if vc9[m] > vc9[m - 1] and vc9[m] >= vc9[m + 1] and vc9[m] > 0]
pk = vc9[peaks] # pk[n]: peak at the end of the nth cycle
delta = np.mean(np.log(pk[:-1] / pk[1:]))
print(f"Logarithmic decrement δ (log of ratio of adjacent positive peaks) = {delta:.5f}")
print(f"Q measured from decay = π/δ = {np.pi / delta:.2f}, formula ω0L/R = {Q7:.2f}")
for n in (5, 10, 20):
print(f"After {n:2d} cycles, peak remains {pk[n]:.4f} V (predicted by e^(-nπ/Q) = {np.exp(-n * np.pi / Q7):.4f})")
# ---------------- 10. Medium-wave tuning circuit ----------------
print("\n=== 10. Medium-wave tuning: L = 240 μH, cover 530-1600 kHz ===")
L10 = 240e-6
c_lo, c_hi = resonant_c(1600e3, L10), resonant_c(530e3, L10)
print(f"Required capacitance: {c_lo * 1e12:.1f} pF (1600 kHz) ~ {c_hi * 1e12:.1f} pF (530 kHz), "
f"ratio {c_hi / c_lo:.2f} = (1600/530)^2 = {(1600 / 530) ** 2:.2f}")
d10, pitch = 50e-3, 0.35e-3 # coil former diameter, length per turn (0.3 mm enameled wire close-wound)
r_in, p_in, L_uh = d10 / 2 / INCH, pitch / INCH, L10 * 1e6
N = (10 * L_uh * p_in + np.sqrt((10 * L_uh * p_in) ** 2 + 36 * L_uh * r_in**3)) / (2 * r_in**2)
print(f"Close-wound on 50 mm diameter former: need {N:.1f} turns, winding length {N * pitch * 1e3:.1f} mm, "
f"check L = {wheeler_uh(d10, N * pitch, N):.1f} μH")
for cmin in (15e-12, 25e-12):
print(f"Variable capacitor {cmin * 1e12:.0f} ~ 365 pF with 240 μH: tuning range "
f"{1 / (2 * np.pi * np.sqrt(L10 * 365e-12)) / 1e3:.0f} ~ {1 / (2 * np.pi * np.sqrt(L10 * cmin)) / 1e3:.0f} kHz")
# ---------------- Plot: series resonance curve ----------------
fig, ax = plt.subplots(figsize=(8, 4.5))
fx = np.linspace(6e6, 8.2e6, 2000)
for R_ in (20.0, 5.0, 1.0):
Zx = R_ + 1j * (2 * np.pi * fx * L7 - 1 / (2 * np.pi * fx * C7))
ax.plot(fx / 1e6, 20 * np.log10(R_ / np.abs(Zx)), label=f"R = {R_:g} Ω, Q = {w0 * L7 / R_:.0f}")
ax.set_xlabel("Frequency / MHz")
ax.set_ylabel("Current (relative to resonance) / dB")
ax.set_title("Series RLC resonance curve: higher Q gives sharper peak")
ax.set_ylim(-40, 2)
ax.grid(True, color="#e6e5e1")
ax.legend()
fig.tight_layout()
plt.show()
Full output after running:
=== 1. Maximum power transfer: open-circuit voltage 100 V (RMS), internal resistance 50 Ω ===
R_L = 12.5 Ω: current 1.600 A, load power 32.00 W, efficiency 20.0 %
R_L = 25.0 Ω: current 1.333 A, load power 44.44 W, efficiency 33.3 %
R_L = 50.0 Ω: current 1.000 A, load power 50.00 W, efficiency 50.0 %
R_L = 100.0 Ω: current 0.667 A, load power 44.44 W, efficiency 66.7 %
R_L = 200.0 Ω: current 0.400 A, load power 32.00 W, efficiency 80.0 %
Theoretical maximum V^2/(4R_s) = 50.00 W
=== 2. Capacitors ===
Parallel plate: area 10 cm^2, gap 1 mm, air: C = εA/d = 8.854 pF
RC charging: R = 10 kΩ, C = 100 nF, τ = RC = 1.0 ms
t = 1τ: capacitor voltage reaches 63.2 % of supply
t = 2τ: capacitor voltage reaches 86.5 % of supply
t = 3τ: capacitor voltage reaches 95.0 % of supply
t = 5τ: capacitor voltage reaches 99.3 % of supply
100 pF and 220 pF: parallel 320.0 pF, series 68.75 pF
=== 3. Inductors ===
Wheeler formula: diameter 20 mm, length 20 mm, 15 turns air-core coil L = 3.055 μH
Same coil wound with 30 turns (length also doubled): L = 7.231 μH
RL circuit: L = 100 μH, R = 10 Ω, τ = L/R = 10.0 μs
=== 4. Reactance X_L = ωL, X_C = 1/(ωC) ===
f = 1.0 MHz: X_L(1 μH) = 6.28 Ω, X_C(100 pF) = 1591.55 Ω
f = 7.1 MHz: X_L(1 μH) = 44.61 Ω, X_C(100 pF) = 224.16 Ω
f = 30.0 MHz: X_L(1 μH) = 188.50 Ω, X_C(100 pF) = 53.05 Ω
f = 145.0 MHz: X_L(1 μH) = 911.06 Ω, X_C(100 pF) = 10.98 Ω
Frequency where they are equal f = 1/(2π sqrt(LC)) = 15.915 MHz
=== 5. Complex impedance (f = 7.1 MHz) ===
50 Ω series 1 μH: Z = 50 + j44.61 Ω = 67.01 Ω∠41.74°
Driven by 10 V (RMS): I = 0.1492 A∠-41.74°
Real power P = |I|^2 R = 1.114 W, apparent power S = |V||I| = 1.492 VA, power factor cos φ = 0.746
50 Ω parallel 100 pF: Z = 47.63 - j10.62 Ω = 48.8 Ω∠-12.57°
=== 6. RC low-pass: R = 1 kΩ, C = 47 nF ===
Cutoff frequency f_c = 1/(2πRC) = 3386.3 Hz
f = 338.6 Hz: |H| = -0.04 dB, phase -5.71°
f = 3386.3 Hz: |H| = -3.01 dB, phase -45.00°
f = 3000.0 Hz: |H| = -2.52 dB, phase -41.54°
f = 33862.8 Hz: |H| = -20.04 dB, phase -84.29°
=== 7. Series RLC: L = 2.2 μH, resonate at 7.1 MHz, R = 5 Ω ===
C = 1/(ω0^2 L) = 228.40 pF
Q = ω0 L / R = 19.63, also sqrt(L/C)/R = 19.63
Characteristic impedance sqrt(L/C) = 98.14 Ω, bandwidth f0/Q = 361.7 kHz
Numerical check: current maximum at 7.1000 MHz, half-power points 6.9214 ~ 7.2832 MHz, bandwidth 361.7 kHz
Signal source 1 V (RMS): at resonance current 0.20 A, voltage across inductor and capacitor each 19.63 V
=== 8. Parallel resonance: same L, C, inductor loss r = 5 Ω ===
Formula L/(C r) = 1926.4 Ω, Q^2 r = 1926.4 Ω
Numerical: |Z| max 1928.9 Ω at 7.1000 MHz; at 1/(2π sqrt(LC)) |Z| = 1928.9 Ω, phase -2.92°
=== 9. Free oscillation: capacitor charged to 1 V, then let the RLC circuit ring ===
Logarithmic decrement δ (log of ratio of adjacent positive peaks) = 0.16010
Q measured from decay = π/δ = 19.62, formula ω0L/R = 19.63
After 5 cycles, peak remains 0.4491 V (predicted by e^(-nπ/Q) = 0.4492)
After 10 cycles, peak remains 0.2017 V (predicted by e^(-nπ/Q) = 0.2018)
After 20 cycles, peak remains 0.0407 V (predicted by e^(-nπ/Q) = 0.0407)
=== 10. Medium-wave tuning: L = 240 μH, cover 530-1600 kHz ===
Required capacitance: 41.2 pF (1600 kHz) ~ 375.7 pF (530 kHz), ratio 9.11 = (1600/530)^2 = 9.11
Close-wound on 50 mm diameter former: need 66.9 turns, winding length 23.4 mm, check L = 240.0 μH
Variable capacitor 15 ~ 365 pF with 240 μH: tuning range 538 ~ 2653 kHz
Variable capacitor 25 ~ 365 pF with 240 μH: tuning range 538 ~ 2055 kHz