Let me first say that for a rider to be skilled and for a horse to be 'broke' or 'trained', the ability to lope circles in the open away from any fences must be accomplished. You just can't have a 'broke' horse without that horse being willing to lope a round circle, either direction, ANYWHERE, with no fences of any kind.
Circles are not real easy to learn how to do correctly. What I tell people that help me ride horses is "If they were easy to do, everyone could do them." Fact it, very few people can teach a green horse to lope good circles.
First you have to understand what it takes to get a horse to make a circle. You have to know how to teach a horse to 'follow his nose'. You have to know how to 'keep a horse between the reins and between the rider's legs'. Sounds easy, but it is not.
1) You have to have shoulder control. To get a horse to 'follow its nose', you have to be able to control its shoulders. A horse does not HAVE to follow its nose. It DOES HAVE to follow its shoulders.
2) You have to learn how to PUSH a horse into that circle rather than just trying to PULL him into it. It requires a horse moving off of the rider's outside leg to push him into that circle.
3) You have to quit pulling once the horse's head is pointed the right direction. If you continue pulling, the horse will just 'rubber neck' or 'over-bend' and keep on going the wrong way or will just stop. If a horse's nose is farther to the inside than to the point where you can just barely see the corner of his inside eye, THE RIDER IS PULLING TOO MUCH AND PUSHING TOO LITTLE! If the horse's head is pointed the right direction, the inside rein should be loose and the outside rein may need to be tightened to prevent over-bending.
Then, it is the rider's outside leg that has to put enough pressure on the horse to tell him to 'follow his nose.'
4) If you cannot put pressure on the outside of a horse's ribs and have him move into the circle desired, then you need to teach the horse to move off of leg pressure first.
5) The opposite is also true. If the horse drops a shoulder and leans into a circle (usually on the far side away from the gate) forcing the rider to hold the horse out with an outside rein, the rider needs to use inside leg instead of outside rein. The rider needs to PUSH the horse's body out where it belongs -- never pull on the outside rein to get the horse to stay on the outside of the circle.
Both sides of the circle require that a horse not only follow its nose but also stays between the rider's legs. Again, it means more PUSH than PULL.
It has been my experience that riders that have difficulty loping circles just have not learned to use their legs effectively. Teaching leg yielding exercises and riding out in the open (away from any arena fences) will build these skills. It is real easy for both horses and riders to learn to depend on arena fences and walls. Both soon get 'lost' without them.
Hope you find this relevant.