Dependent sources, amplifier models and two-port networks

Dependent sources were introduced in Lesson 1 as idealized elements whose value is set by a voltage or current elsewhere in the circuit. They are the language engineers use to describe amplification: every transistor, op amp, sensor interface and power converter controller is modelled, at least for small signals, by a network of resistors and dependent sources. This lesson uses that language to build amplifier models, find the input and output resistance of circuits that contain dependent sources, and describe any linear block by its two-port parameters, so that complex systems can be analysed as chains of simple boxes.

The four dependent sources

A dependent (controlled) source produces a voltage or current proportional to a controlling voltage or current:

• Voltage-controlled voltage source (VCVS): \(v = \mu\,v_{x}\), with dimensionless voltage gain \(\mu\). An ideal voltage amplifier.
• Current-controlled current source (CCCS): \(i = \beta\,i_{x}\), with dimensionless current gain \(\beta\). A simple model of a bipolar transistor.
• Voltage-controlled current source (VCCS): \(i = g_{m}\,v_{x}\), with transconductance \(g_{m}\) in siemens. The standard model of a MOSFET.
• Current-controlled voltage source (CCVS): \(v = r_{m}\,i_{x}\), with transresistance \(r_{m}\) in ohms. A model of the transimpedance amplifier (Lesson 8).

Dependent sources are linear elements, so nodal, mesh and Thévenin analysis all apply. Two cautions from Lesson 2 still hold: never "turn off" a dependent source when applying superposition or finding a Thévenin resistance, and always express the controlling variable in terms of the circuit's unknowns.

Amplifier models

Any linear amplifier, however complex inside, can be described at its terminals by three numbers: input resistance \(R_{in}\), open-circuit voltage gain \(A_{vo}\) and output resistance \(R_{out}\). The model is a resistor at the input, and at the output a VCVS of value \(A_{vo}v_{in}\) in series with \(R_{out}\).

Loading occurs at both ends: the source resistance and \(R_{in}\) form a divider at the input, and \(R_{out}\) and the load form another at the output. The overall gain is

A good voltage amplifier therefore has \(R_{in} \gg R_{s}\) and \(R_{out} \ll R_{L}\). Current amplifiers want the opposite: low input resistance and high output resistance.

Resistance of circuits with dependent sources

When a circuit contains dependent sources, its input or Thévenin resistance cannot be found by combining resistors. Use a test source: switch off independent sources only, apply a test voltage \(v_{t}\) (or test current \(i_{t}\)) to the terminals, solve for the resulting current (or voltage), and take the ratio \(R = v_{t}/i_{t}\). Dependent sources can make the result larger or smaller than any resistor in the circuit, and even negative.

Transistor small-signal models

Transistors are nonlinear, but for small signals around a fixed operating point they behave linearly and are modelled with dependent sources (ELE-240 develops the models in detail):

• A MOSFET is modelled by a VCCS \(g_{m}v_{gs}\) from drain to source, with an output resistance \(r_{o}\) in parallel and an open circuit at the gate.
• A bipolar transistor is modelled by a resistance \(r_{\pi}\) between base and emitter and a CCCS \(\beta i_{b}\) (or equivalently a VCCS \(g_{m}v_{be}\)) from collector to emitter.

Two-port networks

Many circuits have an input pair of terminals and an output pair: amplifiers, filters, transformers, transmission lines, sensor interfaces. A two-port network is described completely, for linear circuits, by four parameters relating the two port voltages \(v_{1}, v_{2}\) and currents \(i_{1}, i_{2}\) (both currents defined as flowing into the network). Different parameter sets choose different pairs as independent variables:

Each parameter is found by setting one port to an open or short circuit. For example, \(z_{11} = v_{1}/i_{1}\) with port 2 open (\(i_{2} = 0\)), and \(z_{21} = v_{2}/i_{1}\) also with port 2 open.

Cascading with transmission parameters

When two-ports are connected in a chain, the output of each feeding the input of the next, their ABCD matrices simply multiply: \(\mathbf{T} = \mathbf{T}_{1}\mathbf{T}_{2}\cdots\mathbf{T}_{n}\). A series resistance \(R\) has \(\mathbf{T} = \begin{bmatrix} 1 & R \\ 0 & 1 \end{bmatrix}\) and a shunt conductance \(G\) has \(\mathbf{T} = \begin{bmatrix} 1 & 0 \\ G & 1 \end{bmatrix}\). The input resistance with a load \(R_{L}\) is \((AR_{L} + B)/(CR_{L} + D)\), and the open-circuit voltage gain is \(1/A\).

import numpy as np

def series(R):
    return np.array([[1.0, R], [0.0, 1.0]])

def shunt(R):
    return np.array([[1.0, 0.0], [1.0 / R, 1.0]])

def input_resistance(T, RL):
    A, B, C, D = T.ravel()
    return (A * RL + B) / (C * RL + D)

section = series(1000) @ shunt(1000)        # one rung of an R-R ladder, 1 kOhm each
for n in [1, 2, 3, 5, 10]:
    T = np.linalg.matrix_power(section, n)
    A = T[0, 0]
    print(f"{n:>2} sections: Rin(open end) = {input_resistance(T, 1e15):7.1f} ohm, "
          f"open-circuit gain = {1 / A:.6f}")
print(f"Infinite ladder: {1000 * (1 + 5**0.5) / 2:.1f} ohm")

 1 sections: Rin(open end) =  2000.0 ohm, open-circuit gain = 0.500000
 2 sections: Rin(open end) =  1666.7 ohm, open-circuit gain = 0.200000
 3 sections: Rin(open end) =  1625.0 ohm, open-circuit gain = 0.076923
 5 sections: Rin(open end) =  1618.2 ohm, open-circuit gain = 0.011236
10 sections: Rin(open end) =  1618.0 ohm, open-circuit gain = 0.000091
Infinite ladder: 1618.0 ohm

With each added section the input resistance approaches 1,618 Ω, the infinite-ladder value from Lesson 3, while the voltage reaching the far end falls rapidly. The gains 1/2, 1/5, 1/13 and so on involve the Fibonacci numbers, a curiosity that follows from the matrix multiplication. Power engineers use exactly this method, with impedances instead of resistances, to model long transmission lines as chains of short sections (ELE-380).