Circuit Theory

Lumped Element Modeling and Fundamental Laws

Now that we have covered a majority of the concepts found in circuits from a field perspective (using Maxwell's equations), we can now make various simplifications to the quantities and ideas found in smaller scale circuits. It would be cumbersome to try to compute the various electric fields and other things all the time so instead we make some realistic assumptions which allows us to get more straightforward deductions of the nature of the circuit.

Ohm's Law

As mentioned, the most useful aspect of our assumptions will be simplification of electromagnetic phenomena found in small scale (and typically DC) circuits. With these reductions we can readily compute things like voltage, current, resistance, etc. The first of which is Ohm's Law. Ohm's Law gives us a nice relationship between voltage, resistance, and current. It is given by

\[ V = I R \]

Where \(V\) is voltage (measured in volts V), \(I\) is current (measured in amps A), and \(R\) is resistance (measured in Ohms \(\Omega\)).

Kirchhoff's Laws

Kirchhoff's laws are not really "laws" in the sense that nature follows them. Instead, they are useful and very accurate approximations for DC circuits and for low frequency AC currents, since the wavelengths of electromagnetic waves are much larger than the physical size of the circuits. These can be mathematically deduced from Maxwell's equations and are not just guesses.

Kirchhoff's Current Law (KCL)

This law is also known as the conservation of charge. It states that the sum of currents at a node \(= 0\). This basically means that the amount of current entering a point must be the same as the amount of current leaving it. A node is basically a point where two or more circuit elements connect together. It can be thought of as a junction of the circuit. More specifically, a node is a set of points that have the same voltage. In between nodes is where we put all of our circuit elements. This law at a node can be represented as,

\sum_{j=1}^{n}I_j = 0

The sign convention does not relate to the actual sign of the current flow. The current is always a negative flow, since it's the flow of electrons. Although, the actual convention is that current flows in the direction that positive charge would, but this is not relevant to the Kirchhoff's Current Law. We can consider the currents entering a point as positive, and those leaving as negative in this law. This convention is unique for each node.

Kirchhoff's Voltage Law (KVL)

Kirchhoff's Voltage Law is also known as the conservation of energy (in this context). It states that the sum of voltages around a loop must be \(0\). Mathematically this is expressed as,

\sum_{k=1}^{n}V_k = 0

Where \(V_k\) is the voltage across the \(k\)-th element of the circuit. A loop is simply any closed path that starts at a node and returns to the same node. These can appear as sub-loops inside the circuit where this law still holds. When we say the "voltage across" an element, we mean the change in electric potential from one side of the element to the other side.

Deriving Kirchhoff's Laws from Maxwell's Equations

Kirchhoff's Laws can be derived from Maxwell's equations under certain assumptions. This helps explain why they work so well for DC circuits and low-frequency AC circuits.

Kirchhoff's Current Law (KCL)

KCL is based on the continuity equation, which comes from the law of charge conservation: \[ \nabla \cdot \mathbf{J} = -\frac{\partial \rho}{\partial t} \] Here, \(\mathbf{J}\) is the current density and \(\rho\) is the charge density. In a steady-state DC circuit, the charge density doesn't change over time, so \(\partial \rho / \partial t = 0\). This gives \(\nabla \cdot \mathbf{J} = 0\), meaning the net current into any point (node) is zero, exactly KCL.

Kirchhoff's Voltage Law (KVL)

KVL comes from Faraday's law of induction: \[ \oint \mathbf{E} \cdot d\mathbf{l} = -\frac{d\Phi_B}{dt} \] For DC circuits or circuits with slowly changing currents, the magnetic flux \(\Phi_B\) is approximately constant, so \(\frac{d\Phi_B}{dt} \approx 0\). This simplifies to \(\oint \mathbf{E} \cdot d\mathbf{l} = 0\), meaning the sum of voltage drops around any closed loop is zero, exactly KVL.

Circuit Analysis

Circuits are usually drawn as diagrams with nodes (junctions) and elements (resistors, batteries, capacitors, etc.) connected by wires. By convention, current flows in the direction that positive charge would move. Even though the actual charge carriers (electrons) move opposite to this direction, this convention simplifies calculations and aligns with the definitions used in Kirchhoff's Laws and Ohm's Law.

Resistors

Resistors are essentially current throttles; they can limit the amount of current flowing for a set voltage. Resistors dissipate electrical energy into heat. As mentioned in the previous page, this is via the interaction between the current flow and atomic lattice.

Series Resistors

Resistors in series are connected end-to-end, so the same current flows through all of them. Their equivalent resistance is:

R_{\text{eq}} = R_1 + R_2 + ... + R_n

Parallel Resistors

Resistors in parallel are connected across the same two nodes, so the voltage across each is the same. The equivalent resistance is:

\frac{1}{R_{\text{eq}}} = \frac{1}{R_1} + \frac{1}{R_2} + ... + \frac{1}{R_n}

Capacitors

Capacitors can be thought of as a component that stores charge. Physically, the capacitor is typically comprised of two separated metal sheets. As current flows in on one side, the charge builds up on that plate. This build up of charge creates a small electric field and makes the plate negatively charged. The charge will not flow across the gap, instead opposite charge will build up on the other plate. The capacitor stores energy in this electric field.

The Capacitor Equation

The fundamental equation for a capacitor relates its current to the rate of change of voltage:

I(t) = C \frac{dV_C}{dt}

RC Charging Circuit

For a resistor \(R\) in series with our capacitor, we use KVL to find:

V_s = RC\frac{dV_C}{dt} + V_C

This is a first order linear ODE. The solution for the voltage across the capacitor as it charges from zero is:

V_C(t) = V_s(1-e^{-t/\tau})

Where \(\tau = RC\) is the time constant. This determines the speed of charge and discharge. The discharge function (when the source is removed) is:

V_C(t) = V_s e^{-t/\tau}

Capacitor Combinations

When capacitors are connected in series, the total capacitance decreases:

\frac{1}{C_\text{total}} = \frac{1}{C_1} + \frac{1}{C_2} + \dots

When connected in parallel, the total capacitance is simply the sum:

C_\text{total} = C_1 + C_2 + \dots

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