3 Biasing

Q 3.1

The collector current of a BJT with some finite $\mathit{\alpha fe}$ is set at $I_C$ in the circuit at the right hand side of the figure below.

(a)

Give an approximate expression for the value of the base resistor $R_B$ to get a collector current $I_C$. In other words: express $R_B$ in terms of the target $I_C$ and other (circuit and component) parameters. For the approximate expression, you may assume a fair estimation for the resulting $V_{\mathit{BE}}$.

(b)

For better accuracy, now derive an exact expression for $R_B$, using the element equations of the BJT. For that assume that $I_{C0}$ is known for the BJT. Note that typically $\frac{I_C}{I_{C0}}>>>1$ and hence the ”+1” can be ignored.

(c)

Given is $\mathit{\alpha fe}=200$, $I_C=1\mathit{mA}$, $I_{C0}=10^{-13}A$, $\mathit{kT}∕q=25\mathit{mV}$ and a supply voltage $V_{\mathit{CC}}=10V$. Calculate the base resistor $R_B$ using the relations you found at a) and b).

(d)

$R_B$ is chosen such that $I_C=1\mathit{mA}$. Due to some unforeseen circumstance, the transistor breaks down and is replaced by one that appears to have a collector current of $2\mathit{mA}$. Assume that for this transistor, the $I_{C0}$ is identical to the previous one and that only $\mathit{\alpha fe}$ is different. Give an estimation for $\mathit{\alpha fe}$ of this new transistor.

(e)

In the circuit at the left hand side of Figure 3.1, the BJT is biased at $I_C=1\mathit{mA}$, while its current gain $\mathit{\alpha fe}=200$. We now place an identical transistor parallel to the original one. Will the total collector current be larger than, smaller than or be equal to $2\mathit{mA}$?

Q 3.2

In the circuit schematic below, $Q_1$ and $Q_2$ have very large output resistance, $\mathit{\alpha fe}=100$, $V_{\mathit{CC}}=5V$ and $R_C=1k\mathrm{\Omega }$.

(a)

We choose $R_x$ such that the collector of $Q_2$ is at $3V$. If we then connect the collector of $Q_1$ to its own base terminal instead of to the $5V$, does $I_{C2}$ change considerably? Provide a derivation to illustrate your answer.

Q 3.3

BC546s can be obtained in A, B or C-grade, with as main difference the current gain:

• ${\mathit{\alpha fe}}_A=100$
• ${\mathit{\alpha fe}}_B=250$
• ${\mathit{\alpha fe}}_C=800$

The circuit schematic below — of a crappy FM transmitter [2] — does not indicate whether the BC547 should be type A, B or C, so the current gain of this transistor is very uncertain. This leads to an uncertain collector current $I_C$ and consequently uncertain voltage across the $1k\mathrm{\Omega }$ resistor in the circuit schematic. For proper operation of the BC547, its $V_{\mathit{CE}}$ must be larger than about 100mV and $V_{\mathit{CE}}$ should not be very close to $V_{\mathit{CC}}$. Calculate whether this is the case for all types (A,B,C) of the BC547.

Q 3.4

The circuit schematic below — a clap switch circuit [3] — does not indicated whether $T_1$ and $T_2$ are BC548 A, B or C. Especially the current gain $\mathit{\alpha fe}$ can be anywhere between 110 and 800.

(a)

Derive, from the schematic, the supply voltage $V_{\mathit{CC}}$ for the (left hand side of the) circuit, e.g. for the part left of $R16$.

(b)

Derive an expression for $V_{C,T1}$ as a function of $V_{\mathit{CC}}$, resistor values $R2$, $R3$ and current gain $\mathit{\alpha fe}$.

(c)

Show that $T_1$ and $T_2$ are not properly biased for all of the transistor types (A,B,C).

(d)

Calculate the bias current of $T_4$ and $T_5$

Q 3.5

The figure below shows a single-transistor amplifier stage. For operation in saturation, $i_D=\frac{1}{2}k{(v_{\mathit{GS}}-V_T)}^2$ with $V_T$ the threshold voltage of $M_1$. The supply voltage $V_{\mathit{DD}}>V_T$.

(a)

Show that $M_1$ works in saturation in this circuit.

(b)

Derive an expression for the value of $R_D$ to set the bias current of $M_1$ to $I_D$.

Q 3.6

Derive expression for the bias current of $Q_1$ and $Q_2$ in the circuit schematic — a door alarm circuit [4] — below.

Q 3.7

In the circuit schematic below, transistor $M_1$ is to be biased at a drain current $I_{\mathit{BIAS}}$.

(a)

Derive an expression for the value of $R_{G1}$, required to set a drain current $I_D=I_{\mathit{BIAS}}$, as a function of properties of the components in the circuit (resistor values, transistor parameters, supply voltage, ...).

(b)

Calculate the numerical value of $R_{G1}$ for $I_{\mathit{BIAS}}=1\mathit{mA}$, $R_S=1k\mathrm{\Omega }$, $K=0.5\mathit{mA}∕V^2$, $V_T=2V$, $R_{G2}=10k\mathrm{\Omega }$, $V_{\mathit{DD}}=10V$. Note: you can also answer this question if you didn’t answer a) (yet). In that case, explain each step in your calculation.

(c)

We use the circuit with the values above as an amplifier, where $V_D$ will be the output voltage. Therefore we want to maximize the voltage swing at the $V_D$ node. Derive and equation AND derive the numerical value of $R_D$ for maximum voltage swing such that the output signal will not clip to the supply or push $M_1$ out of saturation.

Q 3.8

Given is the amplifier circuit schematic below. The capacitors have an impedance that is negligible at the signal frequency, nd the transistor is assumed to operate according to the square-law ($i_D=\frac{1}{2}K\cdot {(v_{\mathit{GS}}-V_T)}^2$).

(a)

We want to set $I_{D1}=I_{\mathit{BIAS}}$. Derive a (parametric) equation for the required value for $R_S$ as a function of various component values and component properties.

(b)

For $I_{D1}=I_{\mathit{BIAS}}=4\mathit{mA}$. $V_{\mathit{DD}}=10V$, $K=2\mathit{mA}∕V^2$, $V_T=1V$, $R_{G1}=220k\mathrm{\Omega }$, $R_{G2}=220k\mathrm{\Omega }$, calculate the numerical value of the required $R_S$. Calculate the required value of $R_S$.

(c)

To have some signal swing at the output and to have the transistor operating in saturation, determine the allowed range for $R_D$ values.

(d)

We want to increase the voltage gain of the amplifier. Does the gain get higher from putting a capacitor parallel to $R_S$? What about putting an inductor in series with $R_S$? What about putting an inductor in series with $R_D$?
Note: this question is related to chapters 4 and 5.

Q 3.9

Given is the amplifier circuit schematic below. Derive an expression for the collector bias current $I_C$ expressed in terms of circuit parameters (component values, component parameters, supply voltage, etc.).

Q 3.10

See the circuit of an antenna amplifier [5], shown below.

(a)

Derive an expression for the bias collector current of $Q_2$ and $Q_3$. The current gain of the BJTs is finite $\mathit{\alpha fe}$.

(b)

Calculate their bias currents numerically; assume that $\mathit{\alpha fe}=100$.

(c)

$R_9$ is in parallel with $C_5$. Why is that? Why is $R_6$ not decoupled? Note: this question is related to chapters 4 and 5.

Q 3.11

Derive an expression for the bias collector current of $Q_1$, $I_C$, in the radio receiver circuit [6] below. For simplicity reasons you may assume $\mathit{\alpha fe}\to \infty$.

Q 3.12

Given is the metal detector circuit [7] in the figure below. Derive an equation for the bias collector current of $T1$, calculate its numerical value and try to find the (then obvious) typo in the schematic.

Q 3.13

Given is the FM transmitter circuit[8] below. Derive an equation for the bias collector current of $T1$, and after that calculate its numerical value. Assume $\mathit{\alpha fe}=600$.