The transistor

State whether these statements about transistor currents are true or false, justifying each answer: (a) $I_E = I_B + I_C$ always holds true; (b) in the active region $I_C = \beta \cdot I_B$; (c) the base current is usually larger than the collector current; (d) with the base open-circuit, the transistor conducts anyway.

Define the current gain $\beta = \frac{I_C}{I_B}$ and explain what it means for a transistor to have $\beta = 150$.

1. a) State the typical values of $\beta$ in small-signal transistors.

1. b) Explain why manufacturers only guarantee a minimum value of $\beta$ and not an exact value.

A transistor with $\beta = 120$ operates in the active region with a base current of $40\ \mu\text{A}$. a) Calculate its collector current. b) Calculate its emitter current using $I_E = I_B + I_C$. c) Calculate what percentage of the emitter current reaches the collector and comment on the result.

In a transistor in the active region, $I_B = 0{,}5\ \text{mA}$ and $I_C = 85\ \text{mA}$ are measured. a) Calculate its gain $\beta$. b) Calculate the emitter current. c) If the base voltage disappears ($I_B = 0$), reason what $I_C$ and $I_E$ are now worth and which region the transistor enters.

The figure shows the output characteristic curves of a transistor, with its three regions labelled and the operating point $Q$ marked. a) State which region point $Q$ is in. b) Read approximately from the graph the values of $I_C$ and $V_{CE}$ at that point (marked with the question mark). c) Explain which curve in the family corresponds to the highest base current and why.

Determine the operating region of an NPN transistor in each case, justifying your answer: (a) $V_{BE} = 0{,}7\ \text{V}$ and $V_{CE} = 0{,}2\ \text{V}$; (b) $V_{BE} = 0{,}7\ \text{V}$ and $V_{CE} = 6\ \text{V}$ with a supply voltage of $12\ \text{V}$; (c) $V_{BE} = 0\ \text{V}$; (d) $V_{BE} = 0{,}7\ \text{V}$ and $V_{CE} = 11{,}9\ \text{V}$ with a supply voltage of $12\ \text{V}$.

In a circuit with an NPN transistor powered at $V_{CC} = 12\ \text{V}$ with $R_C = 1\ \text{k}\Omega$, $\beta = 200$, and $V_{CE(sat)} = 0{,}2\ \text{V}$:

1. a) Plot and analytically justify the DC load line, calculating accurately the cutoff and actual saturation points.
2. b) A student simulates the circuit and makes the mistake of always applying the active region model ($I_C = \beta \cdot I_B$). For a current $I_{B1} = 30\ \mu\text{A}$, they obtain $V_{CE} = 6\ \text{V}$, while for $I_{B2} = 100\ \mu\text{A}$, they obtain $V_{CE} = -8\ \text{V}$. Critically interpret both results: deduce the actual operating point $Q_1$ in the first case and demonstrate the physical inconsistency of the second result by calculating the actual operating point $Q_2$ ($I_C$, $V_{CE}$).
3. c) Calculate the minimum base current strictly necessary to reach saturation and determine the forced gain or $\beta_{sat}$ with which the transistor operates when $100\ \mu\text{A}$ is injected.

You can do it on paper and submit a photo with your answer.

An NPN transistor switches an LED from a $5\ \text{V}$ digital output. Data: LED rated at $2\ \text{V}$ and $20\ \text{mA}$, supply voltage of $9\ \text{V}$, minimum $\beta$ of $100$, $V_{BE} = 0{,}7\ \text{V}$ and $V_{CE(sat)} = 0{,}2\ \text{V}$. a) Calculate the LED resistor using $R_C = \frac{V_{CC} - V_{LED} - V_{CE(sat)}}{I_C}$. b) Calculate the minimum base current for saturation and, applying a safety margin of $\times 3$, the base resistor $R_B = \frac{V_{control} - V_{BE}}{I_B}$.

The figure shows a transistor driving a relay coil from a control signal. a) Calculate the required collector current. b) Calculate the minimum base current using the indicated minimum $\beta$ and select it with a safety margin of $\times 2$. c) Calculate the base resistance $R_B$, marked with the question mark. d) Explain the function of the diode connected in anti-parallel with the coil.

A classmate has built a circuit to turn on an indicator LED ($V_D = 2\ \text{V}$, $I_D = 25\ \text{mA}$) from the $5\ \text{V}$ digital output of a microcontroller using a $12\ \text{V}$ power supply and an NPN transistor ($\beta_{mín} = 80$, $V_{BE} = 0{,}7\ \text{V}$, $V_{CE(sat)} = 0{,}2\ \text{V}$). They have selected the values $R_C = 470\ \Omega$ and $R_B = 33\ \text{k}\Omega$. During laboratory tests, the LED barely glows and the transistor heats up abnormally.

1. a) Quantitatively justify the circuit failure by determining at what operating point ($I_B$, $I_C$, $V_{CE}$) the transistor is operating and the power it dissipates.
2. b) Redesign the circuit by calculating the standard commercial values from the E24 series for $R_C$ and $R_B$ that ensure the nominal current of the LED and force the transistor into deep saturation (with an overdrive factor $k = 2$).
3. c) Compare the power dissipated by the transistor in your design with that of the faulty setup and justify why switching in saturation minimizes thermal losses.

A solid-state switch is assembled to control a resistive load $R_C = 220\ \Omega$ powered at $V_{CC} = 9\ \text{V}$. The circuit is controlled with a $5\ \text{V}$ digital signal through a resistor $R_B = 47\ \text{k}\Omega$. The transistor has $\beta = 150$, $V_{BE} = 0{,}7\ \text{V}$, and $V_{CE(sat)} = 0{,}2\ \text{V}$.

1. a) Demonstrate by calculating $I_B$, $I_C$, and $V_{CE}$ that the transistor is not acting as an ideal closed switch, but is instead in the active region.
2. b) Analytically deduce the inequality that $R_B$ must satisfy in terms of the circuit parameters to ensure that the transistor enters saturation.
3. c) Choose a standard value (E24 series) for $R_B$ that applies a saturation factor $k = 2$ ($I_B = 2 \cdot I_{B(sat)}$) and calculate the power that the transistor will dissipate in saturation, comparing it with the power it dissipated in the original $47\ \text{k}\Omega$ design.

The Darlington configuration connects two transistors so that the emitter of the first drives the base of the second (the collectors are joined together). Explain in your own words:

1. a) What total current gain results from the pair as a function of the gains $\beta_1$ and $\beta_2$ of each transistor.

1. b) What problem this configuration solves when driving large loads with very small control currents.

1. c) What drawback it has compared with a single transistor regarding the base-emitter voltage needed for it to conduct.

Challenge of the unit. Think about the value of the base-emitter voltage of a silicon transistor when conducting and what happens when you place two in series at the input.

You will be assessed on: stating the expression for the total gain, justifying what the configuration is used for, and explaining the drawback, with its approximate value.

Explain what the operating point $Q$ of an amplifier is and why, to amplify a signal, it is placed near the centre of the load line: what happens to the output signal if point $Q$ is too close to cut-off or too close to saturation (clipping of a half-cycle).

Two transistors with $\beta_1 = 100$ and $\beta_2 = 80$ are connected in a Darlington configuration.

1. a) Estimate the total gain of the pair.

1. b) Calculate the total collector current if the base of the first transistor receives $10\ \mu\text{A}$.

1. c) Give two applications where such a huge gain is useful (for example, touch sensors and power output stages) and explain the drawback of the double $V_{BE}$.

To control a DC servomotor ($12\ \text{V}$, $500\ \text{mA}$) from a microcontroller pin limited to a maximum source current of $0{,}5\ \text{mA}$ at $5\ \text{V}$, the use of an integrated Darlington pair is evaluated ($T_1$ and $T_2$ with $\beta_1 = 50$, $\beta_2 = 40$, $V_{BE1} = V_{BE2} = 0{,}7\ \text{V}$, and $V_{CE1(sat)} = 0{,}2\ \text{V}$).

1. a) Analytically demonstrate from the node equations of the transistor pair why the total gain is equivalent to $\beta_{total} \approx \beta_1 \cdot \beta_2$, and justify why it would be impossible to switch the load with a single bipolar transistor of $\beta = 100$.
2. b) Calculate the base resistor $R_B$ (E24 standard commercial value) with an overdrive factor $k = 2$ and verify that the current drawn from the microcontroller does not exceed its specifications.
3. c) Demonstrate through internal mesh analysis why the overall saturation voltage $V_{CE(sat)}$ of the Darlington pair is necessarily greater than that of a single BJT ($V_{CE(sat)} = V_{BE2} + V_{CE1(sat)} = 0{,}9\ \text{V}$). Calculate the thermal power dissipated during conduction and argue why, in high-current applications, it is preferred to replace this stage with a power MOSFET transistor.