controllers ranged from purely mechanical integrators, through vacuum-tube and solid-state devices, to emulation of analog controllers by microprocessors. Electronic analog computers The similarity between linear mechanical components, such as springs and dashpots (viscous-fluid dampers), and electrical components, such as capacitors, inductors, and resistors is striking in terms of mathematics. They can be modeled using equations of the same form. However, the difference between these systems is what makes analog computing useful. Complex systems often are not amenable to pen-and-paper analysis, and require some form of testing or simulation. Complex mechanical systems, such as suspensions for racing cars, are expensive to fabricate and hard to modify. And taking precise mechanical measurements during high-speed tests adds further difficulty. By contrast, it is very inexpensive to build an electrical equivalent of a complex mechanical system, to simulate its behavior. Engineers arrange a few operational amplifiers (op amps) and some passive linear components to form a circuit that follows the same equations as the mechanical system being simulated. All measurements can be taken directly with an oscilloscope. In the circuit, the (simulated) stiffness of the spring, for instance, can be changed by adjusting the parameters of an integrator. The electrical system is an analogy to the physical system, hence the name, but it is much less expensive than a mechanical prototype, much easier to modify, and generally safer. The electronic circuit can also be made to run faster or slower than the physical system being simulated. Experienced users of electronic analog computers said that they offered a comparatively intimate control and understanding of the problem, relative to digital simulations. Electronic analog computers are especially well-suited to representing situations described by differential equations. Historically, they were often used when a system of differential equations proved very difficult to solve by traditional means. As a simple example, the dynamics of a spring-mass system can be described by the equation , with as the vertical position of a mass , the damping coefficient, the spring constant and the gravity of Earth. For analog computing, the equation is programmed as . The equivalent analog circuit consists of two integrators for the state variables (speed) and (position), one inverter, and three potentiometers. Electronic analog computers have drawbacks: the value of the circuit's supply voltage limits the range over which the variables may vary (since the value of a variable is represented by a voltage on a particular wire). Therefore,