Input Matrix

The input matrix, usually written as B, is the part of a state-space model that shows how external inputs affect each state variable in a linear circuit or system. In Electrical Circuits and Systems II, it connects control signals to the system's state equations.

What is the Input Matrix?

The input matrix is the B matrix in a state-space model, and it tells you how each input enters the system's state equations in Electrical Circuits and Systems II. If the state vector x(t) describes the internal condition of a circuit, B tells you how an input like a voltage source or current source pushes those states over time.

In the standard form, you write the system as x'(t) = Ax(t) + Bu(t). The A matrix describes what the system does on its own, while B describes what the outside input does. That means B does not replace the circuit equations you already know from Kirchhoff's laws, it packages the input effects into matrix form so you can analyze the whole system cleanly.

The size of B depends on the model. If you have n state variables and m inputs, then B has n rows and m columns. Each entry shows how strongly one input affects one state variable. A zero in a spot means that input does not directly enter that state equation, while a nonzero value means the input appears there with some scaling.

A simple way to read B is to ask, “Which input actually feeds which state derivative?” For example, in a circuit model, one input might directly drive the capacitor voltage equation, while another input affects the inductor current equation. The matrix keeps those relationships organized, especially when the system has more than one input.

In linear time-invariant systems, B is constant, so the input-to-state relationship does not change with time. If the system is time-varying, B can change as conditions change. Either way, B works together with A, the state matrix, to describe the full dynamic response of the circuit or control system.

Why the Input Matrix matters in Electrical Circuits and Systems II

The input matrix matters because it is the bridge between an outside signal and the internal behavior of the system. In state-space analysis, you are not just solving for a voltage or current at one point, you are tracking how the whole set of state variables evolves when a source is applied.

That makes B central to modeling real circuits with more than one input. If you are analyzing a filter, a feedback loop, or a multi-input circuit, B shows whether a source acts directly on one state, spreads across several states, or does not enter a particular equation at all. That changes the form of the differential equations and the final response.

B also matters when you compare different state-space models of the same circuit. Two models can look different on paper, but if they represent the same physical system, the way inputs enter the states should still make sense physically. This is where you connect the math back to the circuit elements, like sources, resistors, capacitors, and inductors.

The matrix also becomes useful when you move toward control design and simulation. Once you know how the input enters the system, you can predict how changing a source changes the transient response, steady behavior, or controllability of the system.

How the Input Matrix connects across the course

State Matrix

The state matrix A and the input matrix B are usually read together. A describes the system's internal dynamics without forcing, while B describes how external inputs enter those dynamics. If you only know B, you still do not know how the states evolve on their own. If you only know A, you miss the effect of the source or control signal.

State Vector

The state vector is the collection of variables whose values fully describe the system at a given time. B tells you how the input affects those state variables through the derivative equation. In a circuit, the state vector often includes quantities like capacitor voltage or inductor current, and B shows which of those variables is directly driven by the input.

State-Space Model

The input matrix is one piece of the full state-space model. The model also includes the state matrix A and often an output matrix C, so you can describe both the internal motion and the measured output. When you build the model from a circuit, B is the part that captures the source terms in matrix form.

Controllable Canonical Form

In controllable canonical form, the matrices are arranged to make the input path through the system easy to see. That makes B especially easy to interpret because the input is structured to act in a predictable way on the states. This form is often used to study controllability and to simplify state-space calculations.

Is the Input Matrix on the Electrical Circuits and Systems II exam?

A problem set question usually asks you to build or read B from a state-space equation or from a circuit diagram. You may be given a set of differential equations and asked to identify which terms belong in A and which belong in B, or you may need to translate source connections into matrix entries. On quizzes, the common move is to check dimensions first, then match each input to the state equation it affects. If a source does not appear directly in a particular state derivative, that entry in B should be zero. If the course gives you a circuit with a single input and two states, you should be ready to write B as a 2 by 1 column that shows how that source enters each state equation. In longer problems, B is often used before finding the time response from the state equations.

Key things to remember about the Input Matrix

• The input matrix B shows how external inputs enter a state-space system.

• Its rows match the number of state variables, and its columns match the number of inputs.

• A nonzero entry in B means that input directly affects that state equation.

• B works with the state matrix A to describe the full dynamic response of a circuit or control system.

• In Electrical Circuits and Systems II, you use B when turning a circuit or differential equation into state-space form.

Frequently asked questions about the Input Matrix

What is the input matrix in Electrical Circuits and Systems II?

The input matrix, usually called B, is the matrix in a state-space model that shows how external inputs affect the system's state variables. It appears in the equation x'(t) = Ax(t) + Bu(t). In circuit problems, it tells you how sources like voltages or currents enter the dynamics.

How do you find the input matrix from a state-space equation?

Look at the terms multiplying the input vector u(t) in the state equation. Those coefficients form the columns of B. A common mistake is mixing them with the A matrix, but A contains only the state-to-state terms, while B contains the input-to-state terms.

What does each entry in the input matrix mean?

Each entry shows how strongly one input affects one state variable. A larger value means that input has a bigger direct effect on that state derivative. A zero means there is no direct input path to that state equation.

Is the input matrix the same as the state matrix?

No. The state matrix A describes the system's internal behavior without the input, while the input matrix B describes how the input enters the system. They work together, but they answer different questions in the model.