Sources#
Sources are the electronic components which can supply electronic circuits with power that can be used to perform something useful.
Theoretical Model#
This is a very general definition but it is very much true because any active one-port can, by definition, be operated at a current and voltage which produce negative power, i.e. they feed energy into the network. However, we are usually interested in one-ports which can do this in specific ways.
Ideal Current Sources#
Definition: Ideal Current Source
An ideal current source is a one-port whose I-V characteristic \(\mathcal{F}(t)\) at each time \(t\) is given by
where \(I\) is some function which is completely independent of all external factors as well as the voltage \(v\).
An ideal current source is an independent source whose current \(i\) is given by some function \(I(t)\) at each moment \(t\), regardless of everything else, including the voltage across it. This essentially means that at time \(t\), the ideal current source is injecting current \(I(t)\), regardless of anything else that might be happening.
In general, the graph of the current \(i\) with respect to time might look like the following:
However, at each time \(t\), the I-V characteristic looks like some horizontal line because the current \(i\) of the ideal current source is independent of the voltage across it:
Depending on what the function \(I(t)\) is, we also define two special types of ideal current sources:
Definition: Direct Current Source
A direct current source (DC source) is an ideal current source for which \(I(t) = I_{\text{DC}}\) for some constant \(I_{\text{DC}} \in \mathbb{R}\), i.e. the current \(i\) is always equal to \(I_{\text{DC}}\) at all moments \(t\):
Definition: Alternating Current Source
An alternating current source (AC source) is an ideal current source for which \(I(t)\) satisfies the following conditions:
- It is periodic for some period \(T \in \mathbb{R}\).
- The integral \(\int_{t_0}^{t_0 + T} I(t) \mathop{\mathrm{d}t}\) is zero for all \(t_0 \in \mathbb{R}\).
- The integral \(\int_{t_0}^{t_0 + T} |I(t)|^2 \mathop{\mathrm{d}t}\) is finite for all \(t_0 \in \mathbb{R}\).
We call \(f \overset{\text{def}}{=} \frac{1}{T}\) the frequency of the AC source.
Example: Sinusoidal AC Source
Most commonly, AC sources have a function \(I(t)\) which is given by some sinusoidal wave:
Controlled Current Sources#
The current of an ideal current source is completely independent of anything else. However, we often want this current to change based on some external parameter. The simplest way we model this is by coupling the current either to some other current or to some other voltage in the network.
Definition: Current Controlled Current Source (CCCS)
A current controlled current source (CCCS) is a two-port whose input port is a short circuit and whose output port is an ideal current source for which there exists a real function \(f: \mathbb{R} \to \mathbb{R}\) such that \(i_2 = I(t)\), where \(I(t) = f(i_1(t))\) for each moment \(t\):
Example: CCCS with Linear \(f\)
One very common type of CCCS is one where \(f\) is a linear transformation:
Theorem: Explicit Representations
It has the following hybrid representation:
It has the following forwards transmission representation:
Proof
TODO
Definition: Voltage Controlled Current Source (VCCS)
A voltage controlled current source (VCCS) is a two-port whose input port is an open circuit and whose output port is an ideal current source for which there exists a real function \(f: \mathbb{R} \to \mathbb{R}\) such that \(i_2 = I(t)\), where \(I(t) = f(v_1(t))\) for each moment \(t\):
Example: VCCS with Linear \(f\)
One very common type of VCCS is one where \(f\) is a linear transformation:
Theorem: Explicit Representations
It has the following admittance representation:
It has the following forwards transmission representation:
Proof
TODO
Ideal Voltage Sources#
Definition: Ideal Voltage Source
An ideal voltage source is a one-port whose I-V characteristic \(\mathcal{F}(t)\) at each time \(t\) is given by
where \(V\) is some function which is completely independent of all external factors as well as the current \(i\).
An ideal voltage source is an independent source whose voltage \(v\) is given by some function \(V(t)\) at each moment \(t\), regardless of everything else, including the current flowing through it. This essentially means that at time \(t\), the ideal voltage source creates voltage \(V(t)\), regardless of anything else that might be happening.
In general, the graph of the voltage \(v\) with respect to time might look like the following:
However, at each time \(t\), the I-V characteristic looks like some vertical line because the voltage \(v\) of the ideal voltage source is independent of the current flowing through it:
Controlled Voltage Sources#
The voltage of an ideal voltage source is completely independent of anything else. However, we often want this voltage to change based on some external parameter. The simplest way we model this is by coupling the voltage either to some other current or to some other voltage in the network.
Definition: Current Controlled Voltage Source (CCVS)
A current controlled voltage source (CCVS) is a two-port whose input port is a short circuit and whose output port is an ideal voltage source for which there exists a real function \(f: \mathbb{R} \to \mathbb{R}\) such that \(v_2 = V(t)\), where \(V(t) = f(i_1(t))\) for each moment \(t\):
Example: CCVS with Linear \(f\)
One very common type of CCVS is one where \(f\) is a linear transformation:
Theorem: Explicit Representations
It has the following impedance representation:
It has the following forwards transmission representation:
Proof
TODO
Example: Implementation with \(r \lt 0\)
The simplest implementation of a CCVS with \(r \lt 0\) is just an inverting finite-gain op-amp. In this case, \(r\) is just equal to the voltage gain \(A\):
If this is implemented via an ideal op-amp in the following way, then we have:
Definition: Voltage Controlled Voltage Source (VCVS)
A voltage controlled voltage source (VCVS) is a two-port whose input port is an open circuit and whose output port is an ideal voltage source for which there exists a real function \(f: \mathbb{R} \to \mathbb{R}\) such that \(v_2 = V(t)\), where \(V(t) = f(v_1(t))\) for each moment \(t\):
Example: VCVS with Linear \(f\)
One very common type of VCVS is one where \(f\) is a linear transformation:
Theorem: Explicit Representations
It has the following inverse hybrid representation:
It has the following forwards transmission representation:
Proof
TODO
Example: Amplifier Implementation for \(\mu \lt 0\)
A VCVS can be implemented by connecting an inverting DVA to an op-amp voltage follower. In this case, \(\mu\) is equal to the voltage gain \(A\) of the inverting DVA:
This can be realized using ideal op-amps in the following way:
Example: Amplifier Implementation for \(\mu \gt 0\)
The simplest implementation of a VCVS with \(\mu \gt 0\) is a non-inverting finite-gain op-amp. In this case, \(\mu\) is equal to the voltage gain \(A\).
If we implement this using the following ideal op-amp configuration, then we have \(\mu = \frac{R_0}{R_1}\):
