This chapter assumes you have completed an introductory study of electronic circuits, which should include operational amplifiers, diodes, and MOSFETs including small-signal analysis and switching circuit theory. Our purpose now is to review the main concepts of those devices and circuits.
An ideal diode is like a valve for electric current. It permits current to flow in the forward direction, and blocks it in the reverse direction.
The exponential model gives a more realistic expression of diode function:
\[I_D = I_S \left[\exp\left(\frac{v_D}{nU_T}\right)-1\right]\]
In this model, \(v_D\) is the the forward voltage across the diode, \(I_S\) is a scale constant (could be anything), \(n\) is the grading coefficient (aka the “ideality” factor), and \(U_T\) is the thermal voltage:
\[U_T=\frac{k_B T}{q}\]
At room temperature, \(U_T\approx 26\)mV, and \(n\) is typically close to 1.
Forward exponential model: when the diode is in forward bias, so that the device’s current is greater than 1nA or so, then \(\exp\left(\frac{v_D}{nU_T}\right)\gg 1\). Therefore:
\[i_D \approx I_S \exp\left(\frac{v_D}{nU_T}\right)\]
The forward exponential model facilitates hand analysis since it can be reversed to solve for the device voltrage:
\[v_D \approx nU_T\ln\left(\frac{i_D}{I_S}\right)\]
Test-point approximation: in forward bias, a standard diode can be characterized by a test point \((I_{D0}, V_{D0})\), representing the current and voltage around the middle of its practical operating range. The forward exponential can be expressed in terms of the test point:
\[i_D \approx I_{D0}\exp\left(\frac{v_D-V_{D0}}{nU_T}\right)\]
\[v_D \approx V_{D0} + nU_T\ln\left(\frac{i_D}{I_{D0}}\right)\]
The test-point approximation allows us to disregard the scale current parameter.
Constant voltage drop model: since the current grows exponentially with \(v_D\), the practical range of device currents occurs within a narrow sliver of voltages. A common example is the 0.7V diode: when the device’s current is 1mA, the corresponding forward voltage is roughly 0.7V.
Reverse leakage current: when a diode is reverse biased, a small “leakage current” flows, which can be in the range from pA to \(\mu\)A depending on the diode. According to the exponential model, the reverse leakage should be equal to \(I_S\), but in practice there are multiple contributors that cause the leakage current to be significantly larger than \(I_S\).
Reverse breakdown: diodes can switch on in reverse bias if the reverse voltage exceeds the device’s Zener breakdown voltage, \(V_Z\). The breakdown voltage can range from less than 1V to over 1000V, depending on the diode. If a resistor (or other component) is used to limit the breakdown current, a diode can be used safely in its Zener breakdown mode for a variety of applications, such as voltage regulators and limiters.
Metal Oxide Semiconductor Field Effect Transistor: can be viewed as a voltage-controlled switch or as a voltage-controlled current source. The Drain and Source terminals comprise an output branch that is controlled by the Gate voltage.
N-type MOSFETs are switched on when the gate voltage is high. More precisely, the device is controlled by the relative Gate-Source voltage \((v_{GS})\), which affects the device current (\(i_{DS}\)) that flows from Drain to Source. There is ideally no gate current in a MOSFET device.
The N-type MOSFET has three major operating modes:
Cutoff mode: When \(v_{GS}\) is less than the MOSFET’s threshold voltage (\(V_{\text{th}})\), then the device is considered off and \(i_{DS}\approx 0\)A.
Triode mode: When the device is truly “on”, like a switch, then the relative Drain-Source voltage \((v_{DS})\) should be very small (this is because the device tries to short the Drain and Source together). In this case the device’s current is \[i_{DS} \approx K_n\left[v_{\text{ov}}v_{DS} - \frac{1}{2}v_{DS}^2\right]\] In this expression, \(v_{\text{ov}}\) is the overdrive voltage, defined as the amount by which \(v_{GS}\) exceeds \(V_{\text{th}}\): \[v_{\text{ov}}\triangleq v_{GS}-v_{\text{th}}\]
and \(K_n\) is a scale constant with units A/V2.
Notice that the current can be brought to zero either when \(v_{\text{ov}}=0\)V or when \(v_{DS}=0\)V.
Saturation mode: When the device is balanced between on/off states, then the \(v_{DS}\) is no longer small. Specifically, when \(v_{DS}>v_{\text{ov}}\), then the device current saturates with respect to \(v_{DS}\), so it only depends on \(v_{\text{ov}}\): \[i_{DS}\approx \frac{1}{2}K_nv_{\text{ov}}^2\] This equation is often called the square law. Since the device has no dependence on \(v_{DS}\) in this mode, a saturation-mode MOSFET can be used as a constant current source, or as a transconductance amplifier (i.e. an amplifier that receives a voltage input and produces a current output).
P-type MOSFETs are switched on when the gate voltage is low, so their behavior is the logical complement of N-type devices. The device can be thought of as an “upside down” version of the N-type MOSFET, controlled by \(v_{SG}\), which affects the device current \((i_{SD})\) that flows from Source to Drain.
The P-type MOSFET has three major operating modes:
Cutoff mode: When \(v_{SG}<|V_{\text{th}}|\), then the device is considered off and \(i_{SD}\approx 0\)A. The threshold voltage for a P-type device is often reported as a negative value, hence the absolute-value delimiters around \(V_{\text{th}}\).
Triode mode: When the device is truly “on”, like a switch, then the relative Source-Drain voltage \((v_{SD})\) should be small. In this case the device’s current is \[i_{SD} \approx K_p\left[v_{\text{ov}}v_{SD} - \frac{1}{2}v_{SD}^2\right]\] In this expression, \(v_{\text{ov}}\) is the overdrive voltage, defined as the amount by which \(v_{SG}\) exceeds \(V_{\text{th}}\): \[v_{\text{ov}}\triangleq v_{SG}-|V_{\text{th}}|\]
and \(K_p\) is a scale constant with units A/V2.
Notice that the current can be brought to zero either when \(v_{\text{ov}}=0\)V or when \(v_{DS}=0\)V.
Saturation mode: When the device is balanced between on/off states, then the \(v_{SD}\) is no longer small. Specifically, when \(v_{SD}>v_{\text{ov}}\), then the device current saturates with respect to \(v_{SD}\), so it only depends on \(v_{\text{ov}}\): \[i_{SD} \approx \frac{1}{2}K_pv_{\text{ov}}^2\]
Combined model: Since the N-type and P-type device modes and equations are very similar, for ordinary MOSFETs we can combine them into a single model for hand analysis:
\(v_{\text{ov}}\triangleq |v_{GS}| - |V_{\text{th}}|\)
Cutoff: \(v_{\text{ov}}<0\)
Triode: \(v_{\text{ov}}>0\) and \(|v_{DS}| > v_{\text{ov}}\) \[i_D = K\left[v_{\text{ov}}|v_{DS}| - \frac{1}{2}v_{DS}^2\right]\]
Saturation: \[i_{D}\approx \frac{1}{2}Kv_{\text{ov}}^2\]
Channel Length Modulation (CLM): in the saturation mode, the MOSFET’s device current is ideally insensitive to \(v_{DS}\). In practice, there is a small dependence due to a number of subtle physical effects. To account for these effects, the saturation model is amended:
\[i_D \approx \frac{1}{2}Kv_{\text{ov}}^2(1+\lambda|v_{DS}|)\]
The parameter \(\lambda\) has units of V-1, and is typically in the range from 0.01 to 0.1 V-1. This effect determines the output resistance of MOSFET amplifier circuits.
Bipolar JunctionTransistor: can be viewed as a current amplifier. The Collector and Emitter terminals comprise an output branch that is controlled by the Base current. As with MOSFETs, there are two basic types of BJT devices.
NPN transistors: analogous to the N-type MOSFET, the NPN transistor receives a control current (\(i_B\)) into the Base terminal.
The Collector current (\(i_C\)) is proportional to the Base current:
\[i_C = \beta i_B\],
where \(\beta\) is the unitless gain parameter that characterizes the device. Typically \(\beta\) is in the range from 50 to 300.
The Emitter current (\(i_E\)) is the sum of Base and Collector currents:
\[i_E = i_C + i_B = i_B(1+\beta)\]
The Collector current is also proportional to the Emitter current:
\[i_C = \alpha i_E,~\text{where}~\alpha=\frac{\beta}{\beta+1}.\]
Active mode: the above equations only apply in the device’s active mode, when \(v_{CE}>0.4\)V and \(v_{BE}\approx 0.7\)V. Internally, the Base and Emitter terminals form a diode that must be forward-biased for the device to function, hence the Base-Emitter voltage must be in the vicinity of the diode forward voltage, which is often (but not always) 0.7V.
PNP Transistors: analogous to the P-type MOSFET, the PNP transistor receives a control current (\(i_B\)) out from the Base terminal.
The Collector current (\(i_C\)) is proportional to the Base current:
\[i_C = \beta i_B~\text{and}~i_C=\alpha i_E,\]
where \(\alpha\) and \(\beta\) are defined in the same way as for the NPN device.
Active mode: compared to the NPN device, the PNP transistor is “upside-down”, so the active mode is when \(v_{EC}>0.4\)V and \(v_{EB}\approx 0.7\)V.
Early effect: in the active mode, the BJT collector current should depend only on the base current. In practice, there is also a small dependence on \(v_{CE}\):
\[\textbf{NPN:}~i_C = \beta i_B (1+ \frac{v_{CE}}{V_A})\]
\[\textbf{PNP:}~i_C = \beta i_B (1+ \frac{v_{EC}}{V_A}),\]
where \(V_A\) is called the Early voltage (named for James Early), and typically falls in the range from 10V to 150V. The Early effect bears a very close similarity to Channel Length Modulation in MOSFET devices, with \(V_A\) essentially equivalent to \(1/\lambda\).