11 Introduction to RF electronics

Q 11.1

The figure below shows a (quite generic) setup of an RF transmitter, consisting of a power amplifier (PA) that is driven by a modulated oscillator and that drives an antenna. Between the PA and the antenna there may be an extra matching network. The complex antenna impedance is $Z_{\mathit{antenna}}=R_a+j{X}_a$. The maximum output current of the PA is $I_{\mathit{max}}$ while the maximum voltage swing is $V_{\mathit{max}}$.

(a)

Explain (briefly & clear) why an antenna that actually transmits has a finite non-zero real part $R_a$ in the antenna impedance $Z_{\mathit{antenna}}=R_a+j{X}_a$.

(b)

The impedance of a(n about) quarter-wave monopole antenna is purely real. Connecting this antenna directly to the output node of a PA – having both the intended RF signal and a non-zero DC-(bias) voltage – without coupling capacitor does not result in any DC power into the antenna. Explain this.

(c)

Derive an expression for the maximum transmit power of the system described above the figure.

(d)

Derive 2 expressions for the ratio between the transmit power (at the same PA output voltage swing) when on one hand a short antenna with $Z_{\mathit{antenna}}=10\mathrm{\Omega }-j1000\mathrm{\Omega }$ and compared to using:
1) a short antenna with “match” for which $Z_{\mathit{antenna},\mathit{matched}}=10\mathrm{\Omega }$ (1st expression)
2) a $\lambda ∕4$ antenna with $Z_{\mathit{antenna},\lambda ∕4}\cong 50\mathrm{\Omega }$ (2nd expression)

Q 11.2

In physical realizations of RF circuits, resistors with some kind of leads are used. The resistor value is $R$, the length of the lead wires is $l$ mm on each side of the resistor. Assume that the inductance per length is $L=2\mathit{nH}∕\mathit{mm}$.

(a)

Derive an equation for the impedance of the resistor with wires, $Z_{\mathit{res}}(\mathit{j\omega })$, at (radian) frequency $\omega$.

(b)

Assuming that $R=100\mathrm{\Omega }$ and lead wires of 2cm at each side of the resistor. Derive the maximum frequency to keep the impedance $|Z_{\mathit{res}}(\mathit{j\omega })|<\sqrt{2}R$.

(c)

Derive the impedance of a resistor with a nominal $R=100\mathrm{\Omega }$ and lead wires of 2cm at each side at the upper limit of the audible audio range, at 20 kHz.

(d)

Assuming that $R=100\mathrm{\Omega }$ and lead wires with length $l$ at each side of the resistor. Derive the lead wire length to get the impedance $|Z_{\mathit{res}}(\mathit{j\omega })|<\sqrt{2}R$ at 20 kHz.

Q 11.3

In physical realizations of RF circuits, capacitors with some kind of leads are used. The capacitor value is $C$, the length of the lead wires is $l$ mm on each side of the capacitor. Assume that the inductance per length is $L=2\mathit{nH}∕\mathit{mm}$.

(a)

Derive an equation for the impedance of the capacitor with wires, $Z_{\mathit{cap}}(\mathit{j\omega })$, at (radian) frequency $\omega$.

(b)

Assuming that $C=50\mathit{pF}$ and lead wires of 1 cm at each side of the capacitor. Derive the impedance of the capacitor with wires at 100 MHz.

(c)

Derive the value of the apparent —or net — capacitance of the 50 pF capacitor with 1 cm wires in the previous question, at 100 MHz.

(d)

Draw the impedance of the base capacitor, and of the total wire length in the impedance plane (2-dimensional, $\mathit{Re}(Z)$) on the x-axis, $\mathit{Im}(Z)$ on the y-axis. From this construct the net capacitance graphically.

(e)

Derive the value of the apparent —or net — capacitance of a 10 pF capacitor with 1 cm wires in the previous question, at 100 MHz.

(f)

Derive the value of the apparent —or net — capacitance of a 1 nF capacitor with 1 cm wires in the previous question, at 100 MHz.

(g)

Assuming that the target net capacitance is 100 pF and that lead wires are 1 cm on each side, derive the capacitance value (without leads) required to get that 100 pF, again at 100 MHz.

Q 11.4

The circuit schematic in the figure below is a more of less regular common emitter amplifier. The circuit contains two inductors; $L_C$ is an explicit inductor while $L_{\mathit{wire}}$ is the inductor that models the wire between emitter and ground. This is the only wired modelled in this question.

(a)

Assuming that $L_{\mathit{wire}}=0$, derive the voltage gain of this circuit, including load $R_L$. For now,

• the capacitors can be assumed to be low ohmic at signal frequencies
• the impedance of $L_C$ can be assumed to be much larger than $R_L$

(b)

Including $L_{\mathit{wire}}$, derive the voltage gain of this circuit, including load $R_L$. For now,

• the capacitors can be assumed to be low ohmic at signal frequencies
• the impedance of $L_C$ can be assumed to be much larger than $R_L$

(c)

Now for $V_{\mathit{CC}}=5V$, $\mathit{\alpha fe}=100$ and $R_B=88k\mathrm{\Omega }$, $T=300K$, derive the (bias) $I_C$ and the $g_m$ of the transistor.

(d)

Using the values in the previous question, calculate the numerical value of the decrease in voltage gain due to 1 cm wire (making up $L_{\mathit{wire}}$ at 100 MHz. You may use the rule-of-thumb wire inductance of 1nH/mm.

(e)

As explained before, any junction can be associated with a (voltage dependent) capacitance. Taking into account only the capacitance associated with the base-emitter junction, $C_{\mathit{be}}$, and assuming for now $L_{\mathit{wire}}=0$, derive an expression for voltage gain.