Second Order Circuits
First: A Recap of First Order Circuits
We know how to find the response, x(t), of any first order circuit.
(And by response, we mean a current or a voltage,
usually an inductor current or a capacitor voltage.)
The response will always look like this:
where – t/！• The first term (Ae ) is the natural response or called the transient part of the solution –t/！ t/！• The second term (e ， y(t) e dt) is the forced response or called the steady state solution.
• ！ is called the time constant = RC -or- = L/R
• y(t) is called the forcing function = v(t) / ！ -or- = i(t) / ！ ss
• A is an arbitrary constant evaluated from the initial condition of the circuit,
i.e. either the initial capacitor voltage or the initial inductor current.
For the special case of a constant forcing function the response looks like this:
These results came from our previous work.
In case you forgot, this is how we did it:
; First, we obtainted the Thevenin or Norton equivalent circuit from the perspective of the
energy storage element (the inductor or the capacitor).
; Next we found the first order differential equation that describes the circuit:
where x, ！ and y(t) are defined above
; Then we solved this first order differential equation: x(t) = x(t) + x(t). nf
1) To find the transient part of the response, set y(t) = 0 and solve the equation.
Note that, in general, the natural response of a differential equation will have as many
arbitrary constants as the order of the equation.
2) To find the forced part of the response, the original differential equation is solved by
guessing a response that has the same form as the forcing function.
Page 1 of 10
Now: Second Order Circuits
i.e. circuits with two (irreducible) energy storage elements
These circuits are described by a second order differential equation. Hence they are called second order circuits.
The most general form of a 2nd order differential equation is:
The solution to these will be a bit more complex than for first order circuits.
The steps to solving higher order circuits:
1. Find the differential equation (d.e.) that describes the circuit.
2. Solve the d.e. by finding the complete solution, x(t):
i. First find the natural response, x(t). n
(It will have the same number of arbitrary constants as the order of the d.e.
Beware! It is tempting to find the value of the constants at this stage. DON’T!)
ii. Then find the forced response, x(t). f
iii. Finally, put them together to get the complete response: x(t) = x + x nf
3. After you have the complete response, find the arbitrary constants in the solution.
Step 1: Finding the differential equation of a circuit
Use the State Variable Method. (Steps are on the next page.)
This method can be extended to circuits with any number of energy storage elements.
Step 2a: Finding the natural response
The natural response, x(t), is found by setting the forcing function of the differential equation to n
zero. The natural response will have the same number of arbitrary constants as the order of the differential equation. (Don’t try to solve for these constants yet.) Usually this response will die out
as t gets large. So it is also called the transient response.
Step 2b: Finding the forced response
The forced response, x(t), must satisfy the differential equation with no arbitrary constants. Find it f
by guessing a solution that is the most general form as the forcing function and substitute it into the
differential equation to determine the constants.
Step 2c: Finding the complete response
Put them together: x = x + x nf
Step 3: Finding the arbitrary constants in the solution
Once you have the complete response, use the initial conditions of the circuit to solve for the arbitrary constants.
Beware: Don’t try to solve for these constants until you have both the natural response and the
forced response. You need to use the complete response to plug the initial conditions into, not just
the natural response.
The above gives the big picture. Now we will delve into the details.
Page 2 of 10
State Variable Method
thfor finding differential equation of a n order circuit
thFirst find the n order differential equation for the circuit: 1. Identify a set of n state variables.
There are many possible sets.
The best choice is usually all of the capacitor voltages and all of the inductor currents. rdSo for a 3 order circuit with 2 capacitors and 1 inductor the set might look like this: v, v, i 121
2. Obtain n equations in n unknowns.
Note: stThe equations will all be 1 order differential equations.
The unknowns will be the state variables identified in step #1.
(So if you get a variable in one of the equations and it is not a state variable, then you need to
change it so that it is!)