Second-order circuit equations
Derive why a circuit with two independent energy-storage states can require two initial conditions and a second-order differential equation. Build the driven series-RLC capacitor-voltage equation directly from KVL, write the equivalent two-state first-order system, derive its characteristic polynomial, define the undamped natural angular frequency and damping coefficient from circuit parameters, prove the natural energy balance, and explain why component count alone does not determine dynamic order.
- Explain dynamic order in terms of independent storage states rather than simply counting capacitors and inductors.
- Derive the series-RLC equation LC d²vC/dt²+RC dvC/dt+vC=vs(t) from KVL and element laws.
- Identify the two independent initial data vC(0+) and iL(0+) and relate dvC/dt to current.
- Rewrite the same circuit as a two-state first-order system in vC and iL.
- Derive the characteristic polynomial s²+(R/L)s+1/(LC)=0 from both the scalar ODE and the state matrix.
- Define ω0=1/√(LC) and α=R/(2L) for the series RLC natural equation, with correct units and physical interpretation.
- Derive the natural energy balance d/dt[½CvC²+½Li²]=−Ri² for a passive series RLC.
- Explain why topology constraints can reduce the number of independent states even when several storage components are present.
- Use limiting cases R→0 and removal of one storage element to check the second-order model.
- State the scope of the lumped linear RLC abstraction before classifying damping regimes in chapter 02.
A first-order RC or RL circuit needed one state variable and one initial condition. A series RLC circuit contains both a capacitor and an inductor. Does that automatically make it second order, and what equation actually follows from Kirchhoff’s laws?
We derive the order from independent storage states, not from a memorized label.
01.1Dynamic order counts independent storage states
An ideal capacitor carries state through its voltage; an ideal inductor carries state through its current. If these quantities can vary independently, the circuit has two independent storage coordinates and generally needs two initial data.
Figure 1.1 — Why is a series RLC second order?
The one question is: which two quantities can independently store the circuit state?
The capacitor stores its voltage state and the inductor stores its current state. Two independent states generally require two initial conditions and produce two natural modes.
However, the number of C and L components is only an upper bound. Two ideal capacitors directly in parallel share one voltage and can often be replaced by one equivalent capacitance. Two ideal inductors constrained to the same series current can similarly collapse to one equivalent inductance. Topology and algebraic constraints determine the number of independent states.
01.2Derive the series-RLC capacitor-voltage equation
Consider a series R–L–C loop driven by vs(t), with loop current i(t), and take vC as the output.
Reference schematic — driven series RLC
The arrows and polarities below define exactly the variables used in KVL.
With the references shown, KVL gives
Using passive element laws,
Substitute i=C dvC/dt and di/dt=C d²vC/dt²:
The second derivative appears because the loop contains two independent storage mechanisms. No transform was needed.
01.3Two initial conditions are physically supplied by C and L
Under finite non-impulsive excitation,
Because the series current equals capacitor current,
Thus the scalar second-order ODE receives its two initial data from the two physical storage states. Arbitrarily specifying vC(0+) and dvC/dt without checking iL can hide inconsistent initial conditions.
01.4The same circuit is two coupled first-order equations
Choose the state vector
The capacitor law gives
KVL gives the inductor-current derivative:
Therefore
A second-order scalar ODE and a two-state first-order system are two descriptions of the same dynamics.
01.5Derive the characteristic polynomial twice
For the source-free scalar equation, divide by LC:
Try vC=Vest:
Now check the state matrix A. Natural modes satisfy det(sI−A)=0:
The same polynomial emerges independently. Its two roots are the two natural modal exponents. Chapter 02 will classify them.
01.6Canonical parameters come from the circuit
Write the natural equation as
For the series RLC capacitor-voltage equation, identify the undamped natural angular frequency and damping coefficient
Both have units of inverse seconds. It is also useful to define the dimensionless damping ratio
These definitions do not yet tell us whether the response oscillates. That conclusion comes from the roots, not from vocabulary alone.
01.7Energy gives the physical meaning of damping
For the source-free series RLC, stored energy is
Differentiate:
Using C dvC/dt=i and L di/dt=−Ri−vC,
Resistance removes stored electromagnetic energy monotonically. If R=0, dw/dt=0 and the ideal LC network conserves energy. This energy result anticipates why resistance controls damping.
01.8Worked equation build and independent checks
Pedagogical values: R=100 Ω, L=10 mH, C=1 μF. Then
The natural equation is
and
Units and coefficient reconstruction provide two independent checks: 2α=R/L and ω0²=1/(LC).
If vC(0+)=2 V and iL(0+)=3 mA, then
These two initial data are sufficient to determine the later source-free response once the roots are known.
01.9Order can change when topology or the model changes
Do not infer order by counting symbols. Examples:
- parallel ideal capacitors with the same terminal voltage form one equivalent capacitive state;
- series ideal inductors constrained to the same current can form one equivalent inductive state;
- controlled-source constraints can couple storage variables and change independence;
- adding parasitic capacitance or another uncoupled storage path can raise the order;
- physically removing C or L changes the model structure and the differential equation must be rederived rather than obtained by blindly forcing a singular parameter limit.
Order is a property of the model’s independent dynamics, not of the drawing’s component count.
01.10Mastery exercises
- Starting from KVL, rederive the series-RLC equation for vC.
- Why are vC(0+) and iL(0+) more physically useful initial data than two arbitrary constants?
- Write the two state equations for vC and iL.
- Compute the characteristic polynomial from the state matrix.
- For R=40 Ω, L=20 mH, C=2 μF, calculate ω0, α and ζ.
- Show that the source-free stored energy cannot increase when R>0.
- What happens to the energy equation when R=0?
- Give one example where two storage components do not create two independent states.
Solutions
1. vs=Ri+Ldi/dt+vC with i=C dvC/dt gives LCvC''+RCvC'+vC=vs.
2. They are the actual continuous storage states; moreover vC'(0+)=iL(0+)/C.
3. vC'=i/C and i'=(vs−Ri−vC)/L.
4. det(sI−A)=s²+(R/L)s+1/(LC).
5. LC=4×10⁻⁸, so ω0=5000 rad/s; α=40/(0.04)=1000 s⁻¹; ζ=0.2.
6. dw/dt=−Ri²≤0.
7. dw/dt=0: the ideal LC stores constant total energy while exchanging it between electric and magnetic forms.
8. Two ideal capacitors directly in parallel share one voltage and can be replaced by one equivalent capacitance.
Chapter 02 will solve the characteristic quadratic and show exactly how two real roots, one repeated root, or a complex-conjugate pair produce overdamped, critically damped and underdamped natural responses.